A172434 -id:A172434 - cp's OEIS frontend

A336270 a(n) = Sum_{k=0..n} Sum_{j=0..k} (binomial(n,k) * binomial(k,j))^n.

1, 3, 15, 381, 67635, 83118753, 813824623689, 58040410068847251, 32150480245981639533315, 154935057570894645075940703673, 5474671509704049919709361235659936825, 1600436120524545216094358662984789029130593831

Offset: 0

Ilya Gutkovskiy, Jul 15 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..47

Cf. A000244, A002893, A141057, A167010, A172434, A180350.

Table[Sum[Sum[(Binomial[n, k] Binomial[k, j])^n, {j, 0, k}], {k, 0, n}], {n, 0, 11}]
Table[(n!)^n SeriesCoefficient[Sum[x^k/(k!)^n, {k, 0, n}]^3, {x, 0, n}], {n, 0, 11}]

a(n) = sum(k=0, n, sum(j=0, k, (binomial(n,k) * binomial(k,j))^n)); \\ Michel Marcus, Jul 16 2020

a(n) = (n!)^n * [x^n] (Sum_{k>=0} x^k / (k!)^n)^3.

A180350 G.f.: Sum_{n>=0} a(n)*x^n/n!^5 = [ Sum_{n>=0} x^n/n!^5 ]^3.

Original entry on oeis.org

1, 3, 99, 9237, 775971, 83118753, 10657602909, 1463886204147, 215566192274211, 33677584957306713, 5492032622227428849, 928229455634614797447, 161727023896151286167901, 28905146810167510775300463

Offset: 0

Paul D. Hanna, Jan 20 2011

nonn

			G.f.: A(x) = 1 + 3*x + 99*x^2/2!^5 + 9237*x^3/3!^5 + 775971*x^4/4!^5 +...
A(x)^(1/3) = 1 + x + x^2/2!^5 + x^3/3!^5 + x^4/4!^5 +...

Seiichi Manyama, Table of n, a(n) for n = 0..424

Cf. A002893, A005261, A141057, A172434, A336270.

{a(n)=if(n<0, 0, n!^5*polcoeff(sum(m=0, n, x^m/m!^5+x*O(x^n))^3, n))}

{a(n)=sum(k=0, n, binomial(n, k)^5*sum(j=0, k, binomial(k, j)^5))}

a(n) = Sum_{k=0..n} C(n,k)^5 * Sum_{j=0..k} C(k,j)^5 = Sum_{k=0..n} C(n,k)^5 * A005261(k).

cp's OEIS Frontend

A336270 a(n) = Sum_{k=0..n} Sum_{j=0..k} (binomial(n,k) * binomial(k,j))^n.

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A180350 G.f.: Sum_{n>=0} a(n)*x^n/n!^5 = [ Sum_{n>=0} x^n/n!^5 ]^3.

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