A292842 -id:A292842 - cp's OEIS frontend

A292804 Number A(n,k) of sets 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.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 5, 2, 0, 1, 4, 12, 16, 2, 0, 1, 5, 22, 55, 42, 3, 0, 1, 6, 35, 132, 225, 116, 4, 0, 1, 7, 51, 260, 729, 927, 310, 5, 0, 1, 8, 70, 452, 1805, 4000, 3729, 816, 6, 0, 1, 9, 92, 721, 3777, 12376, 21488, 14787, 2121, 8, 0

Offset: 0

Alois P. Heinz, Sep 23 2017

			A(2,2) = 5: {aa}, {ab}, {ba}, {bb}, {a,b}.
Square array A(n,k) begins:
  1, 1,   1,     1,      1,      1,       1,       1, ...
  0, 1,   2,     3,      4,      5,       6,       7, ...
  0, 1,   5,    12,     22,     35,      51,      70, ...
  0, 2,  16,    55,    132,    260,     452,     721, ...
  0, 2,  42,   225,    729,   1805,    3777,    7042, ...
  0, 3, 116,   927,   4000,  12376,   31074,   67592, ...
  0, 4, 310,  3729,  21488,  83175,  250735,  636517, ...
  0, 5, 816, 14787, 113760, 550775, 1993176, 5904746, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000007, A000009, A102866, A256142, A292838, A292839, A292840, A292841, A292842, A292843, A292844.
Rows n=0-2 give: A000012, A001477, A000326.
Main diagonal gives A292805.
Cf. A144074, A292795, A319501.

h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
    end:
A:= (n, k)-> h(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

h[n_, i_, k_] := h[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[h[n-i*j, i-1, k]* Binomial[k^i, j], {j, 0, n/i}]]];
A[n_, k_] := h[n, n, k];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)

G.f. of column k: Product_{j>=1} (1+x^j)^(k^j).

A(n,k) = Sum_{i=0..k} C(k,i) * A319501(n,i).

A343365 Expansion of Product_{k>=1} (1 + x^k)^(8^(k-1)).

Original entry on oeis.org

1, 1, 8, 72, 604, 5148, 43544, 368408, 3112262, 26273542, 221605240, 1867736120, 15730022540, 132385106956, 1113413229000, 9358220560136, 78606905495809, 659886123312449, 5536404584185376, 46424396382193376, 389074608184431328, 3259085506224931424, 27286163457927575200

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A098407, A292842, A343353, A343360, A343361, A343362, A343363, A343364, A343366.

h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(8^(i-1), j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 12 2021

nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(8^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 8^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(8^(k-1))))} \\ Andrew Howroyd, Apr 12 2021

a(n) ~ exp(sqrt(n/2) - 1/16 - c/8) * 2^(3*n - 7/4) / (sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (8^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

cp's OEIS Frontend

A292804 Number A(n,k) of sets 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.

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