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.

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, 525, 1, 1, 1, 462, 90376, 1239823, 487630, 4347, 1, 1, 1, 1716, 2019836, 216456376, 709097481, 40647178, 41245, 1, 1, 1, 6435, 46570140, 41175714454, 1303699790001, 701954099115, 4561368175, 441675, 1

Offset

0,9

Comments

Row r >= 2, is asymptotic to r^(r*n + 3/2) / (2*Pi*n)^((r-1)/2). - Vaclav Kotesovec, Sep 20 2019

Links

Alois P. Heinz, Antidiagonals n = 0..45, flattened

Formula

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.

Example

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].
Square array A(n,k) begins:
  1,   1,      1,         1,             1,                1, ...
  1,   1,      1,         1,             1,                1, ...
  1,   3,     10,        35,           126,              462, ...
  1,  13,    217,      4245,         90376,          2019836, ...
  1,  75,   8317,   1239823,     216456376,      41175714454, ...
  1, 525, 487630, 709097481, 1303699790001, 2713420774885145, ...

Maple

b:= proc(n, r, p, k) option remember;
      `if`(n<r, 0, `if`(r=0, `if`(n=0, p!, 0), add(
       b(n-j, k*(r-j), p+j, k)/j!, j=0..min(n, r))))
    end:
A:= (n, k)-> `if`(k=0, 1, b(k*n+1, 1, 0, k+1)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

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]}]]];
A[n_, k_] := If[k == 0, 1, b[k*n + 1, 1, 0, k + 1]];
Table[A[n, d - n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A000012, A007178(n+1), A294850, A294851, A294852, A294853, A294854, A294855, A294856, A294857, A294858.

Rows n=0+1, 2-10 give: A000012, A001700, A294982, A294983, A294984, A294985, A294986, A294987, A294988, A294989.

Main diagonal gives: A294747.

Cf. A294775.

Sequence in context: A263383 A185620 A096066 * A326180 A064085 A256692

Adjacent sequences: A294743 A294744 A294745 * A294747 A294748 A294749

Keyword

nonn,tabl

Author

Alois P. Heinz, Nov 07 2017

Status

approved