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A245397 A(n,k) is the sum of k-th powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n; square array A(n,k), n>=0, k>=0, read by antidiagonals. 12
1, 1, 1, 1, 1, 3, 1, 1, 4, 10, 1, 1, 6, 27, 35, 1, 1, 10, 93, 256, 126, 1, 1, 18, 381, 2716, 3125, 462, 1, 1, 34, 1785, 36628, 127905, 46656, 1716, 1, 1, 66, 9237, 591460, 7120505, 8848236, 823543, 6435, 1, 1, 130, 51033, 11007556, 495872505, 2443835736, 844691407, 16777216, 24310 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Alois P. Heinz, Antidiagonals n = 0..50, flattened

FORMULA

A(n,k) = [x^n] (n!)^k * (Sum_{j=0..n} x^j/(j!)^k)^n.

EXAMPLE

A(3,2) = 93: (z1+z2+z3)^3 = z1^3 +3*z1^2*z2 +3*z1^2*z3 +3*z1*z2^2 +6*z1*z2*z3 +3*z1*z3^2 +z2^3 +3*z2^2*z3 +3*z2*z3^2 +z3^3 => 1^2+3^2+3^2+3^2+6^2+3^2+1^2+3^2+3^2+1^2 = 93.

Square array A(n,k) begins:

0 :    1,    1,      1,       1,         1,           1, ...

1 :    1,    1,      1,       1,         1,           1, ...

2 :    3,    4,      6,      10,        18,          34, ...

3 :   10,   27,     93,     381,      1785,        9237, ...

4 :   35,  256,   2716,   36628,    591460,    11007556, ...

5 :  126, 3125, 127905, 7120505, 495872505, 41262262505, ...

MAPLE

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(b(n-j, i-1, k)*binomial(n, j)^(k-1)/j!, j=0..n)))

    end:

A:= (n, k)-> n!*b(n$2, k):

seq(seq(A(n, d-n), n=0..d), d=0..10);

MATHEMATICA

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1, k] * Binomial[n, j]^(k-1)/j!, {j, 0, n}]]]; A[n_, k_] := n!*b[n, n, k]; Table[ Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-Fran├žois Alcover, Jan 30 2015, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 give: A001700(n-1) for n>0, A000312, A033935, A055733, A055740, A246240, A246241, A246242, A246243, A246244, A246245.

Rows n=0+1, 2 give: A000012, A052548.

Main diagonal gives A245398.

Sequence in context: A160870 A025255 A296006 * A294316 A294761 A145085

Adjacent sequences:  A245394 A245395 A245396 * A245398 A245399 A245400

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 21 2014

STATUS

approved

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Last modified May 26 13:05 EDT 2019. Contains 323586 sequences. (Running on oeis4.)