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A294746 Number A(n,k) of compositions (ordered partitions) of 1 into exactly k*n+1 powers of 1/(k+1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A294746 #27 Sep 20 2019 05:13:09
%S A294746 1,1,1,1,1,1,1,1,3,1,1,1,10,13,1,1,1,35,217,75,1,1,1,126,4245,8317,
%T A294746 525,1,1,1,462,90376,1239823,487630,4347,1,1,1,1716,2019836,216456376,
%U A294746 709097481,40647178,41245,1,1,1,6435,46570140,41175714454,1303699790001,701954099115,4561368175,441675,1
%N A294746 Number A(n,k) of compositions (ordered partitions) of 1 into exactly k*n+1 powers of 1/(k+1); square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A294746 Row r >= 2, is asymptotic to r^(r*n + 3/2) / (2*Pi*n)^((r-1)/2). - _Vaclav Kotesovec_, Sep 20 2019
%H A294746 Alois P. Heinz, <a href="/A294746/b294746.txt">Antidiagonals n = 0..45, flattened</a>
%F A294746 A(n,k) = [x^((k+1)^n)] (Sum_{j=0..k*n+1} x^((k+1)^j))^(k*n+1) for k>0, A(n,0) = 1.
%e A294746 A(3,1) = 13: [1/4,1/4,1/4,1/4], [1/2,1/4,1/8,1/8], [1/2,1/8,1/4,1/8], [1/2,1/8,1/8,1/4], [1/4,1/2,1/8,1/8], [1/4,1/8,1/2,1/8], [1/4,1/8,1/8,1/2], [1/8,1/2,1/4,1/8], [1/8,1/2,1/8,1/4], [1/8,1/4,1/2,1/8], [1/8,1/4,1/8,1/2], [1/8,1/8,1/2,1/4], [1/8,1/8,1/4,1/2].
%e A294746 Square array A(n,k) begins:
%e A294746   1,   1,      1,         1,             1,                1, ...
%e A294746   1,   1,      1,         1,             1,                1, ...
%e A294746   1,   3,     10,        35,           126,              462, ...
%e A294746   1,  13,    217,      4245,         90376,          2019836, ...
%e A294746   1,  75,   8317,   1239823,     216456376,      41175714454, ...
%e A294746   1, 525, 487630, 709097481, 1303699790001, 2713420774885145, ...
%p A294746 b:= proc(n, r, p, k) option remember;
%p A294746       `if`(n<r, 0, `if`(r=0, `if`(n=0, p!, 0), add(
%p A294746        b(n-j, k*(r-j), p+j, k)/j!, j=0..min(n, r))))
%p A294746     end:
%p A294746 A:= (n, k)-> `if`(k=0, 1, b(k*n+1, 1, 0, k+1)):
%p A294746 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A294746 b[n_, r_, p_, k_] := b[n, r, p, k] = If[n < r, 0, If[r == 0, If[n == 0, p!, 0], Sum[b[n - j, k*(r - j), p + j, k]/j!, {j, 0, Min[n, r]}]]];
%t A294746 A[n_, k_] := If[k == 0, 1, b[k*n + 1, 1, 0, k + 1]];
%t A294746 Table[A[n, d - n], {d, 0, 10}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A294746 Columns k=0-10 give: A000012, A007178(n+1), A294850, A294851, A294852, A294853, A294854, A294855, A294856, A294857, A294858.
%Y A294746 Rows n=0+1, 2-10 give: A000012, A001700, A294982, A294983, A294984, A294985, A294986, A294987, A294988, A294989.
%Y A294746 Main diagonal gives: A294747.
%Y A294746 Cf. A294775.
%K A294746 nonn,tabl
%O A294746 0,9
%A A294746 _Alois P. Heinz_, Nov 07 2017