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A111062
Triangle T(n, k) = binomial(n, k) * A000085(n-k), 0 <= k <= n.
6
1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 10, 16, 12, 4, 1, 26, 50, 40, 20, 5, 1, 76, 156, 150, 80, 30, 6, 1, 232, 532, 546, 350, 140, 42, 7, 1, 764, 1856, 2128, 1456, 700, 224, 56, 8, 1, 2620, 6876, 8352, 6384, 3276, 1260, 336, 72, 9, 1, 9496, 26200, 34380, 27840, 15960, 6552, 2100, 480, 90, 10, 1
OFFSET
0,4
COMMENTS
Triangle related to A000085.
Riordan array [exp(x(2+x)/2),x]. - Paul Barry, Nov 05 2008
Array is exp(S+S^2/2) where S is A132440 the infinitesimal generator for Pascal's triangle. T(n,k) gives the number of ways to choose a subset of {1,2,...,n} of size k and then partitioning the remaining n-k elements into sets each of size 1 or 2. Cf. A122832. - Peter Bala, May 14 2012
T(n,k) is equal to the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the partial Brauer monoid of degree n. - James East, Aug 17 2015
LINKS
Igor Dolinka, James East, Athanasios Evangelou, Des FitzGerald, Nicholas Ham, James Hyde, and Nicholas Loughlin, Enumeration of idempotents in diagram semigroups and algebras, arXiv:1408.2021 [math.GR], 2014.
Igor Dolinka, James East, Athanasios Evangelou, Des FitzGerald, Nicholas Ham, James Hyde, and Nicholas Loughlin, Enumeration of idempotents in diagram semigroups and algebras, J. Combin. Theory Ser. A 131 (2015), 119-152.
Tom Halverson and Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018.
FORMULA
Sum_{k>=0} T(m, k)*T(n, k)*k! = T(m+n, 0) = A000085(m+n).
Sum_{k=0..n} T(n, k) = A005425(n).
Apparently satisfies T(n,m) = T(n-1,m-1) + T(n-1,m) + (m+1) * T(n-1,m+1). - Franklin T. Adams-Watters, Dec 22 2005 [corrected by Werner Schulte, Feb 12 2025]
T(n,k) = (n!/k!)*Sum_{j=0..n-k} C(j,n-k-j)/(j!*2^(n-k-j)). - Paul Barry, Nov 05 2008
G.f.: 1/(1-xy-x-x^2/(1-xy-x-2x^2/(1-xy-x-3x^2/(1-xy-x-4x^2/(1-... (continued fraction). - Paul Barry, Apr 23 2009
T(n,k) = C(n,k)*Sum_{j=0..n-k} C(n-k,j)*(n-k-j-1)!! where m!!=0 if m is even. - James East, Aug 17 2015
From Tom Copeland, Jun 26 2018: (Start)
E.g.f.: exp[t*p.(x)] = exp[t + t^2/2] e^(x*t).
These polynomials (p.(x))^n = p_n(x) are an Appell sequence with the lowering and raising operators L = D and R = x + 1 + D, with D = d/dx, such that L p_n(x) = n * p_(n-1)(x) and R p_n(x) = p_(n+1)(x), so the formalism of A133314 applies here, giving recursion relations.
The transpose of the production matrix gives a matrix representation of the raising operator R.
exp(D + D^2/2) x^n= e^(D^2/2) (1+x)^n = h_n(1+x) = p_n(x) = (a. + x)^n, with (a.)^n = a_n = A000085(n) and h_n(x) the modified Hermite polynomials of A099174.
A159834 with the e.g.f. exp[-(t + t^2/2)] e^(x*t) gives the matrix inverse for this entry with the umbral inverse polynomials q_n(x), an Appell sequence with the raising operator x - 1 - D, such that umbrally composed q_n(p.(x)) = x^n = p_n(q.(x)). (End)
EXAMPLE
Rows begin:
1;
1, 1;
2, 2, 1;
4, 6, 3, 1;
10, 16, 12, 4, 1;
26, 50, 40, 20, 5, 1;
76, 156, 150, 80, 30, 6, 1;
232, 532, 546, 350, 140, 42, 7, 1;
764, 1856, 2128, 1456, 700, 224, 56, 8, 1;
2620, 6876, 8352, 6384, 3276, 1260, 336, 72, 9, 1;
From Paul Barry, Apr 23 2009: (Start)
Production matrix is:
1, 1,
1, 1, 1,
0, 2, 1, 1,
0, 0, 3, 1, 1,
0, 0, 0, 4, 1, 1,
0, 0, 0, 0, 5, 1, 1,
0, 0, 0, 0, 0, 6, 1, 1,
0, 0, 0, 0, 0, 0, 7, 1, 1,
0, 0, 0, 0, 0, 0, 0, 8, 1, 1 (End)
From Peter Bala, Feb 12 2017: (Start)
The infinitesimal generator has integer entries and begins
0
1 0
1 2 0
0 3 3 0
0 0 6 4 0
0 0 0 10 5 0
0 0 0 0 15 6 0
...
and is the generalized exponential Riordan array [x + x^2/2!,x].(End)
MATHEMATICA
a[n_] := Sum[(2 k - 1)!! Binomial[n, 2 k], {k, 0, n/2}]; Table[Binomial[n, k] a[n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 20 2015, after Michael Somos at A000085 *)
PROG
(SageMath)
def A111062_triangle(dim):
M = matrix(ZZ, dim, dim)
for n in (0..dim-1): M[n, n] = 1
for n in (1..dim-1):
for k in (0..n-1):
M[n, k] = M[n-1, k-1]+M[n-1, k]+(k+1)*M[n-1, k+1]
return M
A111062_triangle(9) # Peter Luschny, Sep 19 2012
(GAP) Flat(List([0..10], n->List([0..n], k->(Factorial(n)/Factorial(k))*Sum([0..n-k], j->Binomial(j, n-k-j)/(Factorial(j)*2^(n-k-j)))))); # Muniru A Asiru, Jun 29 2018
CROSSREFS
Cf. A099174, A133314, A159834 (inverse matrix).
Sequence in context: A091869 A112307 A228336 * A369632 A363493 A193597
KEYWORD
easy,nonn,tabl
AUTHOR
Philippe Deléham, Oct 07 2005
EXTENSIONS
Corrected by Franklin T. Adams-Watters, Dec 22 2005
10th row added by James East, Aug 17 2015
STATUS
approved