A185404 -id:A185404 - cp's OEIS frontend

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

1, 140, 79380, 62563200, 57288340200, 57169180452384, 60324072262534080, 66193973824733314560, 74770747698820830356700, 86365239335124673905181200, 101541339191092781603799640464

Offset: 0

Paul D. Hanna, Jan 26 2011

nonn

			G.f.: A(x) = 1 + 140*x + 79380*x^2 + 62563200*x^3 +...
A(x)^(1/2) = 1 + 70*x + 37240*x^2 + 28674800*x^3 +...+ A185401(n)*x^n +...

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

Cf. A184896, A185401, A185404.

Table[Binomial[2n,n] 7^n/(n!)^2 Product[(7k+2)(7k+5),{k,0,n-1}],{n,0,10}] (* Harvey P. Dale, May 10 2012 *)

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

Self-convolution of A185401:
A185401(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+2)*(14k+5).
a(n) ~ cos(3*Pi/14) * 2^(2*n) * 7^(3*n) / (Pi*n)^(3/2). - Vaclav Kotesovec, Oct 23 2020

A185403 a(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+3)*(14k+4).

Original entry on oeis.org

1, 84, 44982, 34706112, 31430722680, 31154132320416, 32723954432339184, 35790656447712684672, 40328240610474258475572, 46491988990198595758628560, 54576945875594131561054066584

Offset: 0

Paul D. Hanna, Jan 26 2011

nonn

			G.f.: A(x) = 1 + 84*x + 44982*x^2 + 34706112*x^3 +...
A(x)^2 = 1 + 168*x + 97020*x^2 + 76969200*x^3 +...+ A185404(n)*x^n +...

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

Cf. A184895, A185401, A185404.

Table[(7^n/(n!)^2)*Product[(14*k + 3)*(14*k + 4), {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+3)*(14*k+4))}

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

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

cp's OEIS Frontend

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

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