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

0,8

Comments

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.

Links

Table of n, a(n) for n=0..54.

Example

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

Maple

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;

Mathematica

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

Crossrefs

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.

Sequence in context: A305466 A160114 A213028 * A366834 A109970 A116088

Adjacent sequences: A287695 A287696 A287697 * A287699 A287700 A287701

Keyword

nonn,tabl

Author

Peter Luschny, May 30 2017

Status

approved