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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. 21
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

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-Fran├žois 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 * A064085 A256692 A228637

Adjacent sequences:  A294743 A294744 A294745 * A294747 A294748 A294749

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Nov 07 2017

STATUS

approved

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Last modified May 24 07:50 EDT 2018. Contains 304500 sequences. (Running on oeis4.)