cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

1, 60, 39780, 38454000, 43751038500, 54538294552560, 72081445966966800, 99225259048241726400, 140744828381240373790500, 204278086466816584003782000, 301931182921413583820949947280
Offset: 0

Views

Author

Michael Somos, Mar 08 2004

Keywords

Comments

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

Links

Crossrefs

Cf. A001421; variants: A184424, A178529, A184891, A184895, A184897. - Paul D. Hanna, Jan 25 2011
Cf. A289307.

Programs

  • Mathematica
    CoefficientList[ Series[ Hypergeometric2F1[ 1/12, 5/12, 1, 1728 x], {x, 0, 10}], x]
  • PARI
    {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])}
  • PARI
    {a(n)=(12^n/n!^2)*prod(k=0, n-1, (12*k+1)*(12*k+5))} \\ Paul D. Hanna, Jan 25 2011

Formula

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

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

Views

Author

Paul D. Hanna, Jan 25 2011

Keywords

Examples

			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 +...

Links

Crossrefs

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

Programs

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

Formula

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).

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

Original entry on oeis.org

1, 84, 45864, 35672000, 32445913500, 32247604076688, 33935228690034672, 37165308416775931392, 41919854708375196052500, 48365506771435816732770000, 56812832722107710740048677120, 67715433011522917282547695380480
Offset: 0

Views

Author

Paul D. Hanna, Jan 25 2011

Keywords

Examples

			G.f.: A(x) = 1 + 84*x + 45864*x^2 + 35672000*x^3 +...
A(x)^(1/2) = 1 + 42*x + 22050*x^2 + 16909900*x^3 +...+ A184895(n)*x^n +...

Crossrefs

Cf. A184895; variants: A184423, A008977, A184892, A001421, A184898.

Programs

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

Formula

Self-convolution of A184895, where A184895(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+1)*(14k+6).
a(n) ~ sin(Pi/7) * 2^(2*n) * 7^(3*n) / (Pi*n)^(3/2). - Vaclav Kotesovec, Oct 23 2020

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

Views

Author

Paul D. Hanna, Jan 25 2011

Keywords

Examples

			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 +...

Links

Crossrefs

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

Programs

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

Formula

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).

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

Original entry on oeis.org

1, 70, 37240, 28674800, 25943525300, 25700693903192, 26985728598351600, 29506966670254735200, 33241442139458850123750, 38316302306082901242642500, 44974142994787866162564060800
Offset: 0

Views

Author

Paul D. Hanna, Jan 25 2011

Keywords

Examples

			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 +...

Links

Crossrefs

Cf. A184895, A185402, A185403.

Programs

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

Formula

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

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

Views

Author

Paul D. Hanna, Jan 26 2011

Keywords

Examples

			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 +...

Links

Crossrefs

Cf. A184895, A185401, A185404.

Programs

  • Mathematica
    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 *)
  • PARI
    {a(n)=(7^n/n!^2)*prod(k=0,n-1,(14*k+3)*(14*k+4))}

Formula

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
Showing 1-6 of 6 results.

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