A287697 -id:A287697 - cp's OEIS frontend

A287698 Square array A(n,k) = (n!)^3 [x^n] hypergeom([], [1, 1], z)^k read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 10, 1, 0, 1, 4, 27, 56, 1, 0, 1, 5, 52, 381, 346, 1, 0, 1, 6, 85, 1192, 6219, 2252, 1, 0, 1, 7, 126, 2705, 36628, 111753, 15184, 1, 0, 1, 8, 175, 5136, 124405, 1297504, 2151549, 104960, 1, 0

Offset: 0

Peter Luschny, May 30 2017

Let A_m(n,k) = (n!)^m [x^n] hypergeom([], [1,…,1], z)^k where [1,…,1] lists (m-1) times 1. These arrays can be seen as generalizations of the power functions n^k. For m = 1 -> A003992, m = 2 -> A287316, m = 3 -> A287698.
A_m(n,n) is the sum of m-th powers of coefficients in the full expansion of (z_1+z_2+...+z_n)^n (compare A245397).
A287696 provide polynomials and A287697 rational functions generating the columns of the array.

			Array starts:
k\n| 0  1    2       3       4         5           6             7
---|-------------------------------------------------------------------
k=0| 1, 0,   0,      0,      0,        0,          0,            0, ... A000007
k=1| 1, 1,   1,      1,      1,        1,          1,            1, ... A000012
k=2| 1, 2,  10,     56,    346,     2252,      15184,       104960, ... A000172
k=3| 1, 3,  27,    381,   6219,   111753,    2151549,     43497891, ... A141057
k=4| 1, 4,  52,   1192,  36628,  1297504,   50419096,   2099649808, ... A287699
k=5| 1, 5,  85,   2705, 124405,  7120505,  464011825,  33031599725, ...
k=6| 1, 6, 126,   5136, 316206, 25461756, 2443835736, 263581282656, ...
       A001107,A287702,A287700,  A287701,                               A055733

Rows: A000007 (k=0), A000012 (k=1), A000172 (k=2), A141057 (k=3), A287699 (k=4).
Columns: A000172 (n=1), A001477(n=1), A001107 (n=2), A287702 (n=3), A287700 (n=4), A287701 (n=5).
Cf. A055733 (diagonal), A287696, A287697, A003992, A287316, A245397.

A287698_row := (k, len) -> seq(A287696_poly(j)(k), j=0..len):
A287698_row := proc(k, len) hypergeom([], [1, 1], x):
series(%^k, x, len); seq((i!)^3*coeff(%, x, i), i=0..len-1) end:
for k from 0 to 6 do A287698_row(k, 9) od;
A287698_col := proc(n, len) local k, x; hypergeom([], [1, 1], z);
series(%^x, z=0, n+1): unapply(n!^3*coeff(%, z, n), x); seq(%(j), j=0..len) end:
for n from 0 to 7 do A287698_col(n, 9) od;

Table[Table[SeriesCoefficient[HypergeometricPFQ[{},{1,1},x]^k, {x, 0, n}] (n!)^3, {n, 0, 6}], {k, 0, 9}] (* as a table of rows *)

cp's OEIS Frontend

A287698 Square array A(n,k) = (n!)^3 [x^n] hypergeom([], [1, 1], z)^k read by antidiagonals.

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