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A122848
Exponential Riordan array (1, x(1+x/2)).
23
1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 3, 6, 1, 0, 0, 0, 15, 10, 1, 0, 0, 0, 15, 45, 15, 1, 0, 0, 0, 0, 105, 105, 21, 1, 0, 0, 0, 0, 105, 420, 210, 28, 1, 0, 0, 0, 0, 0, 945, 1260, 378, 36, 1, 0, 0, 0, 0, 0, 945, 4725, 3150, 630, 45, 1, 0, 0, 0, 0, 0, 0, 10395, 17325, 6930, 990, 55, 1, 0, 0
OFFSET
0,9
COMMENTS
Entries are Bessel polynomial coefficients. Row sums are A000085. Diagonal sums are A122849. Inverse is A122850. Product of A007318 and A122848 gives A100862.
T(n,k) is the number of self-inverse permutations of {1,2,...,n} having exactly k cycles. - Geoffrey Critzer, May 08 2012
Bessel numbers of the second kind. For relations to the Hermite polynomials and the Catalan (A033184 and A009766) and Fibonacci (A011973, A098925, and A092865) matrices, see Yang and Qiao. - Tom Copeland, Dec 18 2013
Also the inverse Bell transform of the double factorial of odd numbers Product_{k= 0..n-1} (2*k+1) (A001147). For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015
LINKS
Abdelhay Benmoussa, Normal Ordering and Bessel Numbers with Integral Operators, arXiv:2511.09817v2 [math.CO], 2025-2026. See references.
Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Hyuk Han and Seunghyun Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, Eur. J. Combinat. (2008) Vol. 29, No. 7, 1544-1554. [From R. J. Mathar, Mar 20 2009]
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 18.
Sheng Liang Yang and Zhan Ke Qiao, The Bessel Numbers and Bessel Matrices, Journal of Mathematical Research & Exposition, July, 2011, Vol. 31, No. 4, pp. 627-636. [From Tom Copeland, Dec 18 2013]
FORMULA
Number triangle T(n,k) = k!*C(n,k)/((2k-n)!*2^(n-k)).
T(n,k) = A001498(k,n-k). - Michael Somos, Oct 03 2006
E.g.f.: exp(y(x+x^2/2)). - Geoffrey Critzer, May 08 2012
Triangle equals the matrix product A008275*A039755. Equivalently, the n-th row polynomial R(n,x) is given by the Type B Dobinski formula R(n,x) = exp(-x/2)*Sum_{k>=0} P(n,2*k+1)*(x/2)^k/k!, where P(n,x) = x*(x-1)*...*(x-n+1) denotes the falling factorial polynomial. Cf. A113278. - Peter Bala, Jun 23 2014
From Daniel Checa, Aug 28 2022: (Start)
E.g.f. for the m-th column: (x^2/2+x)^m/m!.
T(n,k) = T(n-1,k-1) + (n-1)*T(n-2,k-1) for n>1 and k=1..n, T(0,0) = 1. (End)
Let f be a function that is 2n-times continuously differentiable and vanishes at x_0 along with all of its derivatives up to order 2n-1, then x^k D^k f(x) = Sum_{m=0..k} T(k,m) * (xI)^m D^(2m)f(x), where 0 < k < n, D = d/dx and I(f)(x) = Integral_{x_0..x} f(t) dt. - Abdelhay Benmoussa, Oct 20 2025
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 0 3 1
0 0 3 6 1
0 0 0 15 10 1
0 0 0 15 45 15 1
0 0 0 0 105 105 21 1
0 0 0 0 105 420 210 28 1
0 0 0 0 0 945 1260 378 36 1
From Gus Wiseman, Jan 12 2021: (Start)
As noted above, a(n) is the number of set partitions of {1..n} into k singletons or pairs. This is also the number of set partitions of subsets of {1..n} into n - k pairs. In the first case, row n = 5 counts the following set partitions:
{{1},{2,3},{4,5}} {{1},{2},{3},{4,5}} {{1},{2},{3},{4},{5}}
{{1,2},{3},{4,5}} {{1},{2},{3,4},{5}}
{{1,2},{3,4},{5}} {{1},{2,3},{4},{5}}
{{1,2},{3,5},{4}} {{1,2},{3},{4},{5}}
{{1},{2,4},{3,5}} {{1},{2},{3,5},{4}}
{{1},{2,5},{3,4}} {{1},{2,4},{3},{5}}
{{1,3},{2},{4,5}} {{1},{2,5},{3},{4}}
{{1,3},{2,4},{5}} {{1,3},{2},{4},{5}}
{{1,3},{2,5},{4}} {{1,4},{2},{3},{5}}
{{1,4},{2},{3,5}} {{1,5},{2},{3},{4}}
{{1,4},{2,3},{5}}
{{1,4},{2,5},{3}}
{{1,5},{2},{3,4}}
{{1,5},{2,3},{4}}
{{1,5},{2,4},{3}}
In the second case, we have:
{{1,2},{3,4}} {{1,2}} {}
{{1,2},{3,5}} {{1,3}}
{{1,2},{4,5}} {{1,4}}
{{1,3},{2,4}} {{1,5}}
{{1,3},{2,5}} {{2,3}}
{{1,3},{4,5}} {{2,4}}
{{1,4},{2,3}} {{2,5}}
{{1,4},{2,5}} {{3,4}}
{{1,4},{3,5}} {{3,5}}
{{1,5},{2,3}} {{4,5}}
{{1,5},{2,4}}
{{1,5},{3,4}}
{{2,3},{4,5}}
{{2,4},{3,5}}
{{2,5},{3,4}}
(End)
MAPLE
# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n<2, 1, 0), 9); # Peter Luschny, Jan 27 2016
MATHEMATICA
t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten
(* Alternative: *)
rows = 12;
t = Join[{1, 1}, Table[0, rows]];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
sbs[{}]:={{}}; sbs[set:{i_, ___}]:=Join@@Function[s, (Prepend[#1, s]&)/@sbs[Complement[set, s]]]/@Cases[Subsets[set], {i}|{i, _}];
Table[Length[Select[sbs[Range[n]], Length[#]==k&]], {n, 0, 6}, {k, 0, n}] (* Gus Wiseman, Jan 12 2021 *)
PROG
(PARI) {T(n, k)=if(2*k<n||k>n, 0, n!/(2*k-n)!/(n-k)!*2^(k-n))} /* Michael Somos, Oct 03 2006 */
(SageMath) # uses[inverse_bell_transform from A265605]
multifact_2_1 = lambda n: prod(2*k + 1 for k in (0..n-1))
inverse_bell_matrix(multifact_2_1, 9) # Peter Luschny, Dec 31 2015
CROSSREFS
Row sums are A000085.
Column sums are A001515.
Same as A049403 but with a first column k = 0.
The same set partitions counted by number of pairs are A100861.
Reversing rows gives A111924 (without column k = 0).
A047884 counts standard Young tableaux by size and greatest row length.
A238123 counts standard Young tableaux by size and least row length.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
Sequence in context: A354490 A365970 A144357 * A386635 A272481 A054548
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Sep 14 2006
STATUS
approved