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 1-5 of 5 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

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

Views

Author

Paul D. Hanna, Jan 25 2011

Keywords

Examples

			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 +...

Links

Crossrefs

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

Programs

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

Formula

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

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).

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

Original entry on oeis.org

1, 112, 90720, 105100800, 142542960000, 211337613527040, 331831362513530880, 542307255307827609600, 912855634598629193472000, 1571864775032876891607040000, 2755743023914838714304931102720
Offset: 0

Views

Author

Paul D. Hanna, Jan 25 2011

Keywords

Examples

			G.f.: A(x) = 1 + 112*x + 90720*x^2 + 105100800*x^3 +...
A(x)^(1/2) = 1 + 56*x + 43792*x^2 + 50098048*x^3 +...+ A184897(n)*x^n +...

Crossrefs

Cf. A184897; variants: A184423, A008977, A184892, A001421, A184896.

Programs

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

Formula

Self-convolution of A184897, where A184897(n) = (8^n/n!^2) * Product_{k=0..n-1} (16k+1)*(16k+7).
a(n) ~ sqrt(2-sqrt(2)) * 2^(11*n - 1) / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Oct 05 2020

A184887 a(n) = (8^n/n!^2) * Product_{k=0..n-1} (16k+3)*(16k+5).

Original entry on oeis.org

1, 120, 95760, 110230400, 148976385600, 220389705801216, 345522083206128640, 564061275098462085120, 948680557056225919411200, 1632480132897839426558156800, 2860496988068910156792264671232
Offset: 0

Views

Author

Paul D. Hanna, Jan 25 2011

Keywords

Examples

			G.f.: A(x) = 1 + 120*x + 95760*x^2 + 110230400*x^3 +...
A(x)^2 = 1 + 240*x + 205920*x^2 + 243443200*x^3 +...+ A184888(n)*x^n +...

Links

Crossrefs

Cf. A184888, A184897.

Programs

  • Mathematica
    FullSimplify[Table[2^(11*n) * Gamma[n+3/16] * Gamma[n+5/16] / (Gamma[n+1]^2 * Gamma[3/16] * Gamma[5/16]), {n, 0, 15}]] (* Vaclav Kotesovec, Jul 03 2014 *)
  • PARI
    {a(n)=(8^n/n!^2)*prod(k=0,n-1,(16*k+3)*(16*k+5))}

Formula

Self-convolution yields Sum_{k=0..n} a(n-k)*a(k) = A184888(n) where: A184888(n) = C(2n,n) * (8^n/n!^2)*Product_{k=0..n-1} (8k+3)*(8k+5).
Showing 1-5 of 5 results.

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