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A144074
Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.
15
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 3, 0, 1, 4, 15, 20, 5, 0, 1, 5, 26, 64, 59, 7, 0, 1, 6, 40, 148, 276, 162, 11, 0, 1, 7, 57, 285, 843, 1137, 449, 15, 0, 1, 8, 77, 488, 2020, 4632, 4648, 1200, 22, 0, 1, 9, 100, 770, 4140, 13876, 25124, 18585, 3194, 30, 0, 1, 10, 126
OFFSET
0,8
COMMENTS
Column k > 1 is asymptotic to k^n * exp(2*sqrt(n) - 1/2 + c(k)) / (2 * sqrt(Pi) * n^(3/4)), where c(k) = Sum_{m>=2} 1/(m*(k^(m-1)-1)). - Vaclav Kotesovec, Mar 14 2015
LINKS
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
N. J. A. Sloane, Transforms
FORMULA
G.f. of column k: Product_{j>=1} 1/(1-x^j)^(k^j).
Column k is Euler transform of the powers of k.
T(n,k) = Sum_{i=0..k} C(k,i) * A257740(n,k-i). - Alois P. Heinz, May 08 2015
EXAMPLE
A(4,1) = 5: {aaaa}, {aaa,a}, {aa,aa}, {aa,a,a}, {a,a,a,a}.
A(2,2) = 7: {aa}, {a,a}, {bb}, {b,b}, {ab}, {ba}, {a,b}.
A(2,3) = 15: {aa}, {a,a}, {bb}, {b,b}, {cc}, {c,c}, {ab}, {ba}, {a,b}, {ac}, {ca}, {a,c}, {bc}, {cb}, {b,c}.
A(3,2) = 20: {aaa}, {a,aa}, {a,a,a}, {bbb}, {b,bb}, {b,b,b}, {aab}, {aba}, {baa}, {a,ab}, {a,ba}, {aa,b}, {a,a,b}, {bba}, {bab}, {abb}, {b,ba}, {b,ab}, {bb,a}, {b,b,a}.
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 2, 7, 15, 26, 40, ...
0, 3, 20, 64, 148, 285, ...
0, 5, 59, 276, 843, 2020, ...
0, 7, 162, 1137, 4632, 13876, ...
MAPLE
with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n, k)-> etr(j->k^j)(n); seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
a[n_, k_] := SeriesCoefficient[ Product[1/(1-x^j)^(k^j), {j, 1, n}], {x, 0, n}]; a[0, _] = 1; a[_?Positive, 0] = 0;
Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 15 2014 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d p[d], {d, Divisors[j]}] b[n-j], {j, 1, n}]/n]; b];
A[n_, k_] := etr[k^#&][n];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 30 2020, after Alois P. Heinz *)
CROSSREFS
Rows n=0-2 give: A000012, A001477, A005449.
Main diagonal gives A252654.
Sequence in context: A265609 A362125 A261718 * A261780 A124540 A386558
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 09 2008
EXTENSIONS
Name changed by Alois P. Heinz, Sep 21 2018
STATUS
approved