A185402 -id:A185402 - cp's OEIS frontend

A185401 a(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+2)*(14k+5).

1, 70, 37240, 28674800, 25943525300, 25700693903192, 26985728598351600, 29506966670254735200, 33241442139458850123750, 38316302306082901242642500, 44974142994787866162564060800

Offset: 0

Paul D. Hanna, Jan 25 2011

nonn

			G.f.: A(x) = 1 + 70*x + 37240*x^2 + 28674800*x^3 +...
A(x)^2 = 1 + 140*x + 79380*x^2 + 62563200*x^3 +...+ A185402(n)*x^n +...

G. C. Greubel, Table of n, a(n) for n = 0..320

Cf. A184895, A185402, A185403.

Table[(7^n/(n!)^2)*Product[(14*k + 2)*(14*k + 5), {k, 0, n - 1}], {n, 0, 50}] (* G. C. Greubel, Jun 29 2017 *)

{a(n)=(7^n/n!^2)*prod(k=0,n-1,(14*k+2)*(14*k+5))}

Self-convolution yields Sum_{k=0..n} a(n-k)*a(k) = A185402(n) where A185402(n) = C(2n,n) * (7^n/n!^2) * Product_{k=0..n-1} (7k+2)*(7k+5).

a(n) ~ 2^(2*n) * 7^(3*n) / (Gamma(1/7) * Gamma(5/14) * n^(3/2)). - Vaclav Kotesovec, Nov 19 2023

A185404 a(n) = C(2n,n) * (7^n/n!^2) * Product_{k=0..n-1} (7k+3)*(7k+4).

Original entry on oeis.org

1, 168, 97020, 76969200, 70715452500, 70710926711040, 74713950839848320, 82063363963278297600, 92763657280052631873000, 107208829261440251585240000, 126104599836427618807641720480

Offset: 0

Paul D. Hanna, Jan 26 2011

nonn

			G.f.: A(x) = 1 + 168*x + 97020*x^2 + 76969200*x^3 +...
A(x)^(1/2) = 1 + 84*x + 44982*x^2 + 34706112*x^3 +...+ A185403(n)*x^n +...

G. C. Greubel, Table of n, a(n) for n = 0..320

Cf. A184896, A185402, A185403.

Table[Binomial[2*n, n]*(7^n/(n!)^2)*Product[(7*k + 3)*(7*k + 4), {k, 0, n - 1}], {n, 0, 50}] (* G. C. Greubel, Jun 29 2017 *)

{a(n)=(2*n)!/n!^2*(7^n/n!^2)*prod(k=0,n-1,(7*k+3)*(7*k+4))}

Self-convolution of A185403:
A185403(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+3)*(14k+4).
a(n) ~ cos(Pi/14) * 2^(2*n) * 7^(3*n) / (Pi*n)^(3/2). - Vaclav Kotesovec, Oct 23 2020

cp's OEIS Frontend

A185401 a(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+2)*(14k+5).

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