A287699 -id:A287699 - 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 *)

A336622 a(n) = Sum_{k=0..n} Sum_{i=0..k} Sum_{j=0..i} (binomial(n,k) * binomial(k,i) * binomial(i,j))^n.

Original entry on oeis.org

1, 4, 28, 1192, 591460, 3441637504, 219272057247376, 185528149944660881488, 2405748000504972140803769860, 349789137657321307953339196885516144, 652520795984468974632890750361094911319873648

Offset: 0

Ilya Gutkovskiy, Jul 29 2020

nonn

G. C. Greubel, Table of n, a(n) for n = 0..42

Cf. A000302, A002895, A167010, A287699, A336270.

B:=Binomial; [(&+[(&+[(&+[(B(n,j)*B(n-j,k-j)*B(k-j,k-i))^n: j in [0..i]]): i in [0..k]]): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 31 2022

Table[Sum[Sum[Sum[(Binomial[n, k] Binomial[k, i] Binomial[i, j])^n, {j, 0, i}], {i, 0, k}], {k, 0, n}], {n, 0, 10}]
Table[(n!)^n SeriesCoefficient[Sum[x^k/(k!)^n, {k, 0, n}]^4, {x, 0, n}], {n, 0, 10}]

b=binomial
def A336622(n): return sum(sum(sum( (b(n,j)*b(n-j,k-j)*b(k-j,k-i))^n for j in (0..i)) for i in (0..k)) for k in (0..n))
[A336622(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022

a(n) = (n!)^n * [x^n] (Sum_{k>=0} x^k / (k!)^n)^4.

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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A336622 a(n) = Sum_{k=0..n} Sum_{i=0..k} Sum_{j=0..i} (binomial(n,k) * binomial(k,i) * binomial(i,j))^n.

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