A184424 -id:A184424 - cp's OEIS frontend

A092870 Expansion of Hypergeometric function F(1/12, 5/12; 1; 1728*x) in powers of x.

1, 60, 39780, 38454000, 43751038500, 54538294552560, 72081445966966800, 99225259048241726400, 140744828381240373790500, 204278086466816584003782000, 301931182921413583820949947280

Offset: 0

Michael Somos, Mar 08 2004

nonn

Self-convolution yields Sum_{k=0..n} a(n-k)*a(k) = A001421(n). - Paul D. Hanna, Jan 25 2011

Seiichi Manyama, Table of n, a(n) for n = 0..310 (terms 0..100 from Vincenzo Librandi)
R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.29a).

Cf. A001421; variants: A184424, A178529, A184891, A184895, A184897. - Paul D. Hanna, Jan 25 2011

Cf. A289307.

CoefficientList[ Series[ Hypergeometric2F1[ 1/12, 5/12, 1, 1728 x], {x, 0, 10}], x]

{a(n) = local(an); if( n<1, n==0, an = vector(n+1); an[1] = 1; for(k=1, n, an[k+1] = an[k] * 12 * (12*k - 7) * (12*k - 11) / k^2); an[n+1])}

{a(n)=(12^n/n!^2)*prod(k=0, n-1, (12*k+1)*(12*k+5))} \\ Paul D. Hanna, Jan 25 2011

G.f.: F(1/12, 5/12; 1; 1728*x). a(n) * n^2 = a(n-1) * 12 * (12*n - 7) * (12*n - 11).
G.f. A(x) = y satisfies 0 = (1728*x^2 - x) * y" + (2592*x - 1) * y' + 60 * y.
a(n) = (12^n/n!^2) * Product_{k=0..n-1} (12k+1)*(12k+5). - Paul D. Hanna, Jan 25 2011
G.f.: A(x) = 1 + 60*x + 39780*x^2 + 38454000*x^3 +... with  A(x)^2 = 1 + 120*x + 83160*x^2 + 81681600*x^3 +...+ A184894(n)*x^n +... - Paul D. Hanna, Jan 25 2011
a(n) ~ 1728^n * GAMMA(11/12) * GAMMA(7/12) / (4*Pi^2*n^(3/2)). - Vaclav Kotesovec, Apr 20 2014

A184423 a(n) = (2n)!(3*n)!/n!^5.

Original entry on oeis.org

1, 12, 540, 33600, 2425500, 190702512, 15849497664, 1369618398720, 121821136479900, 11079206239530000, 1025579963180813040, 96310511463483233280, 9152842704012278107200, 878622906816654279840000

Offset: 0

Paul D. Hanna, Jan 13 2011

Denoted by h_3[n] by T. Piezas III. He also gives formulas for 1/Pi such as 1/Pi = 2 * Sum_{n>=0} a(n) * (-1)^n * (51*n + 7) / (12^3)^(n + 1/2). - Michael Somos, May 31 2012
Diagonal of the rational function R(x,y,z,w) = 1/(1-(w*y+w*z+x+y+z)). - Gheorghe Coserea, Jul 15 2016
From Peter Bala, Jun 28 2023: (Start)
The supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for primes p >= 5 and positive integers n and r. This follows from Meštrović equation 39, since a(n) = binomial(3*n,n) * binomial(2*n,n)^2.
Inductively define a family of sequences {a(i,n) : n >= 0}, i >= 1, by setting a(1,n) = a(n) and, for i >= 2, a(i,n) = [x^n] ( exp(Sum_{k >= 1} a(i-1,k)*x^k/k) )^n. We conjecture that the sequences {a(i,n) : n >= 0}, i >= 2, also satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for primes p >= 5 and positive integers n and r. Cf. A362730 and A362732. (End)

			G.f.: A(x) = 1 + 12*x + 540*x^2 + 33600*x^3 + 2425500*x^4 +...
G.f. of A184424 equals A(x)^(1/2):
A(x)^(1/2) = 1 + 6*x + 252*x^2 + 15288*x^3 + 1089270*x^4 + 84963060*x^5 +...+ [(3^n/n!^2)*Product_{k=1..n} (6*k-4)*(6*k-5)]*x^n +...

Gheorghe Coserea, Table of n, a(n) for n = 0..200
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012), arXiv:1111.3057 [math.NT], 2011.
Titus Piezas III, 0013: Article 3 (Pi Formulas and the Monster Group).

Cf. A184424, A275047.

Related to diagonal of rational functions: A268545-A268555.

Table[((2n)!(3n)!)/(n!)^5,{n,0,20}] (* Harvey P. Dale, Dec 18 2018 *)

PARI
```
{a(n)=(3*n)!*(2*n)!/n!^5}
```

{a(n)=polcoeff(sum(m=0,n,3^m*prod(k=1,m,(6*k-4)*(6*k-5))/m!^2*x^m+x*O(x^n))^2,n)}

Self-convolution of A184424:
a(n) = Sum_{k=0..n} A184424(k)*A184424(n-k) where A184424(n) = (3^n/n!^2)*Product_{k=1..n} (6*k-4)*(6*k-5).
a(n) = 6 * (2*n - 1) * (3*n - 1) * (3*n - 2) / n^3 * a(n-1) if n>0. - Michael Somos, May 31 2012
0 = (x^2-108*x^3)*y''' + (3*x-486*x^2)*y''+ (1-348*x)*y' - 12*y, where y is g.f. - Gheorghe Coserea, Jul 15 2016
a(n) ~ 3^(1/2)/(2*Pi^(3/2)) * n^(-3/2) * 108^n. - Ilya Gutkovskiy, Jul 15 2016
a(n) = C(2*n,n)^2 * C(3*n,n) = ( [x^n](1 + x)^(2*n) )^2 * ( [x^n](1 + x)^(3*n) ) = [x^n]( F(x)^(12*n) ), where [x^n] is the coefficient extraction operator and where F(x) = 1 + x + 11*x^2 + 350*x^3  + 15293*x^4  + 794433*x^5  + 45958617*x^6 + ... appears to have integral coefficients. Cf. A000897 and A001451. - Peter Bala, Dec 30 2019

A184895 a(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+1)*(14k+6).

Original entry on oeis.org

1, 42, 22050, 16909900, 15269639700, 15109613875944, 15853342647837688, 17325438750851187600, 19510609713302293636050, 22482485054570487449402900, 26382746561837375612125315092, 31419888802098260334367621118904

Offset: 0

Paul D. Hanna, Jan 25 2011

nonn

			G.f.: A(x) = 1 + 42*x + 22050*x^2 + 16909900*x^3 +...
A(x)^2 = 1 + 84*x + 45864*x^2 + 35672000*x^3 +...+ A184896(n)*x^n +...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A184896; variants: A184424, A178529, A184891, A092870, A184897.

FullSimplify[Table[2^(2*n) * 7^(3*n) * Gamma[n+1/14] * Gamma[n+3/7] / (Gamma[3/7] * Gamma[1/14] * Gamma[n+1]^2), {n, 0, 15}]] (* Vaclav Kotesovec, Jul 03 2014 *)

{a(n)=(7^n/n!^2)*prod(k=0,n-1,(14*k+1)*(14*k+6))}

Self-convolution yields Sum_{k=0..n} a(n-k)*a(k) = A184896(n) where A184896(n) = C(2n,n) * (7^n/n!^2)*Product_{k=0..n-1} (7k+1)*(7k+6).

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

A184891 a(n) = (5^n/n!^2) * Product_{k=0..n-1} (10k+1)*(10k+4).

Original entry on oeis.org

1, 20, 3850, 1078000, 355066250, 128107903000, 49001272897500, 19520507080800000, 8012558140822125000, 3365274419145292500000, 1439327869068441602250000, 624739666805574817770000000

Offset: 0

Paul D. Hanna, Jan 25 2011

nonn

			G.f.: A(x) = 1 + 20*x + 3850*x^2 + 1078000*x^3  +...
A(x)^2 = 1 + 40*x + 8100*x^2 + 2310000*x^3  +...+ A184892(n)*x^n +...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A184892; variants: A184424, A178529, A092870, A184895, A184897.

Table[5^n/(n!)^2 Product[(10k+1)(10k+4),{k,0,n-1}],{n,0,20}] (* Harvey P. Dale, Feb 02 2012 *)
FullSimplify[Table[2^(2*n) * 5^(3*n) * Gamma[n+1/10] * Gamma[n+2/5] / (Gamma[2/5] * Gamma[1/10] * Gamma[n+1]^2), {n, 0, 15}]] (* Vaclav Kotesovec, Jul 03 2014 *)

{a(n)=(5^n/n!^2)*prod(k=0,n-1,(10*k+1)*(10*k+4))}

Self-convolution yields Sum_{k=0..n} a(n-k)*a(k) = A184892(n) where

. A184892(n) = C(2n,n) * (5^n/n!^2)*Product_{k=0..n-1} (5k+1)*(5k+4).

A184897 a(n) = (8^n/n!^2) * Product_{k=0..n-1} (16k+1)*(16k+7).

Original entry on oeis.org

1, 56, 43792, 50098048, 67507119680, 99694514343424, 156121609461801984, 254663020429855285248, 428056704465033002591232, 736257531679856764456919040, 1289628692490437108622739390464

Offset: 0

Paul D. Hanna, Jan 25 2011

nonn

			G.f.: A(x) = 1 + 56*x + 43792*x^2 + 50098048*x^3 +...
A(x)^2 = 1 + 112*x + 90720*x^2 + 105100800*x^3 +...+ A184898(n)*x^n +...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A184898; variants: A184424, A178529, A184891, A092870, A184895.

FullSimplify[Table[2^(11*n) * Gamma[n+1/16] * Gamma[n+7/16] / (Gamma[n+1]^2 * Gamma[1/16] * Gamma[7/16]), {n, 0, 15}]] (* Vaclav Kotesovec, Jul 03 2014 *)

{a(n)=(8^n/n!^2)*prod(k=0,n-1,(16*k+1)*(16*k+7))}

Self-convolution yields Sum_{k=0..n} a(n-k)*a(k) = A184898(n) where A184898(n) = C(2n,n) * (8^n/n!^2)*Product_{k=0..n-1} (8k+1)*(8k+7).

cp's OEIS Frontend

A092870 Expansion of Hypergeometric function F(1/12, 5/12; 1; 1728*x) in powers of x.

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A184423 a(n) = (2*n)!*(3*n)!/n!^5.

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A184895 a(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+1)*(14k+6).

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A184891 a(n) = (5^n/n!^2) * Product_{k=0..n-1} (10k+1)*(10k+4).

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A184897 a(n) = (8^n/n!^2) * Product_{k=0..n-1} (16k+1)*(16k+7).

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Mathematica

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Formula

A184423 a(n) = (2n)!(3*n)!/n!^5.