This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A287698 #24 Jan 06 2018 18:40:32
%S A287698 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,
%T A287698 85,1192,6219,2252,1,0,1,7,126,2705,36628,111753,15184,1,0,1,8,175,
%U A287698 5136,124405,1297504,2151549,104960,1,0
%N A287698 Square array A(n,k) = (n!)^3 [x^n] hypergeom([], [1, 1], z)^k read by antidiagonals.
%C A287698 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.
%C 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).
%C A287698 A287696 provide polynomials and A287697 rational functions generating the columns of the array.
%e A287698 Array starts:
%e A287698 k\n| 0 1 2 3 4 5 6 7
%e A287698 ---|-------------------------------------------------------------------
%e A287698 k=0| 1, 0, 0, 0, 0, 0, 0, 0, ... A000007
%e A287698 k=1| 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
%e A287698 k=2| 1, 2, 10, 56, 346, 2252, 15184, 104960, ... A000172
%e A287698 k=3| 1, 3, 27, 381, 6219, 111753, 2151549, 43497891, ... A141057
%e A287698 k=4| 1, 4, 52, 1192, 36628, 1297504, 50419096, 2099649808, ... A287699
%e A287698 k=5| 1, 5, 85, 2705, 124405, 7120505, 464011825, 33031599725, ...
%e A287698 k=6| 1, 6, 126, 5136, 316206, 25461756, 2443835736, 263581282656, ...
%e A287698 A001107,A287702,A287700, A287701, A055733
%p A287698 A287698_row := (k, len) -> seq(A287696_poly(j)(k), j=0..len):
%p A287698 A287698_row := proc(k, len) hypergeom([], [1, 1], x):
%p A287698 series(%^k, x, len); seq((i!)^3*coeff(%, x, i), i=0..len-1) end:
%p A287698 for k from 0 to 6 do A287698_row(k, 9) od;
%p A287698 A287698_col := proc(n, len) local k, x; hypergeom([], [1, 1], z);
%p A287698 series(%^x, z=0, n+1): unapply(n!^3*coeff(%, z, n), x); seq(%(j), j=0..len) end:
%p A287698 for n from 0 to 7 do A287698_col(n, 9) od;
%t A287698 Table[Table[SeriesCoefficient[HypergeometricPFQ[{},{1,1},x]^k, {x, 0, n}] (n!)^3, {n, 0, 6}], {k, 0, 9}] (* as a table of rows *)
%Y A287698 Rows: A000007 (k=0), A000012 (k=1), A000172 (k=2), A141057 (k=3), A287699 (k=4).
%Y A287698 Columns: A000172 (n=1), A001477(n=1), A001107 (n=2), A287702 (n=3), A287700 (n=4), A287701 (n=5).
%Y A287698 Cf. A055733 (diagonal), A287696, A287697, A003992, A287316, A245397.
%K A287698 nonn,tabl
%O A287698 0,8
%A A287698 _Peter Luschny_, May 30 2017