A180350 -id:A180350 - 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.

A172434 G.f.: Sum_{n>=0} a(n)*x^n/n!^4 = [ Sum_{n>=0} x^n/n!^4 ]^3.

Original entry on oeis.org

1, 3, 51, 1785, 67635, 2973753, 146591529, 7735733883, 430208938035, 24954576411225, 1496639801457801, 92241539987122683, 5816057121183700521, 373854785336483200155, 24431647104881328618315, 1619654401178752389082785

Offset: 0

Paul D. Hanna, Jan 20 2011

nonn

			G.f.: A(x) = 1 + 3*x + 51*x^2/2!^4 + 1785*x^3/3!^4 + 67635*x^4/4!^4 +...
A(x)^(1/3) = 1 + x + x^2/2!^4 + x^3/3!^4 + x^4/4!^4 +...

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A002893, A005260, A141057, A180350, A336270.

{a(n)=if(n<0, 0, n!^4*polcoeff(sum(m=0, n, x^m/m!^4+x*O(x^n))^3, n))}

{a(n)=sum(k=0, n, binomial(n, k)^4*sum(j=0, k, binomial(k, j)^4))}

a(n) = Sum_{k=0..n} C(n,k)^4 * Sum_{j=0..k} C(k,j)^4 = Sum_{k=0..n} C(n,k)^4 * A005260(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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A172434 G.f.: Sum_{n>=0} a(n)*x^n/n!^4 = [ Sum_{n>=0} x^n/n!^4 ]^3.

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