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.

A137961 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^2)^5.

Original entry on oeis.org

1, 1, 5, 20, 105, 575, 3306, 19760, 121035, 757230, 4815530, 31039402, 202335855, 1331569725, 8834918160, 59035907480, 396937508816, 2683518356850, 18230570856710, 124390574548960, 852074529347120, 5857453791938135
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137960, A137962; A137963, A137965, A137973.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^2)^5);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(5*(n-k),k)/(n-k)*binomial(2*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^5 where B(x) is the g.f. of A137960.
a(n) = Sum_{k=0..n-1} C(5*(n-k),k)/(n-k) * C(2*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(5*s*(1-s)*(2-3*s) / ((36*s - 20)*Pi)) / (n^(3/2) * r^n), where r = 0.1354712934479194768810666044866029126617104117352... and s = 1.618016818966151244027202981456410137451426090894... are real roots of the system of equations s = 1 + r*(1 + r*s^2)^5, 10 * r^2 * s * (1 + r*s^2)^4 = 1. - Vaclav Kotesovec, Nov 22 2017

A137963 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^3)^5.

Original entry on oeis.org

1, 1, 5, 25, 160, 1075, 7671, 56760, 431865, 3357790, 26558520, 213032988, 1728808700, 14168337265, 117096909495, 974842628790, 8167462511193, 68813778610350, 582675107162175, 4955767502292960, 42318868510894860
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137962, A137964; A137961, A137965, A137973.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^3)^5);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(5*(n-k),k)/(n-k)*binomial(3*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^5 where B(x) is the g.f. of A137962.
a(n) = Sum_{k=0..n-1} C(5*(n-k),k)/(n-k) * C(3*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(5*s*(1-s)*(3-4*s) / ((84*s - 60)*Pi)) / (n^(3/2) * r^n), where r = 0.1085884782751570249717333800652227343328635496829... and s = 1.404764002126311415321709718173984955120001713401... are real roots of the system of equations s = 1 + r*(1 + r*s^3)^5, 15 * r^2 * s^2 * (1 + r*s^3)^4 = 1. - Vaclav Kotesovec, Nov 22 2017

A137965 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^4)^5.

Original entry on oeis.org

1, 1, 5, 30, 220, 1725, 14356, 124020, 1102770, 10023680, 92722620, 870039474, 8261024380, 79225392830, 766302511445, 7466883915800, 73227699088806, 722228333119200, 7159117292177840, 71284856957207030, 712673042497177450
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137964, A137966; A137961, A137963, A137973.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^4)^5);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(5*(n-k),k)/(n-k)*binomial(4*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^5 where B(x) is the g.f. of A137964.
a(n) = Sum_{k=0..n-1} C(5*(n-k),k)/(n-k) * C(4*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(5*s*(1-s)*(4-5*s) / ((152*s - 120)*Pi)) / (n^(3/2) * r^n), where r = 0.0927175295193852172913829423030505161354091369581... and s = 1.306924048536121339538817141295744998528778296640... are real roots of the system of equations s = 1 + r*(1 + r*s^4)^5, 20 * r^2 * s^3 * (1 + r*s^4)^4 = 1. - Vaclav Kotesovec, Nov 22 2017

A137972 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^4)^6.

Original entry on oeis.org

1, 1, 6, 39, 320, 2787, 25788, 247731, 2449188, 24753960, 254610962, 2656496133, 28046838948, 299085697722, 3216723340218, 34852657892685, 380063012970680, 4168108473073596, 45941874232280862, 508664757809869052
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137971, A137973; A137968, A137970, A137974.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^4)^6);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(6*(n-k),k)/(n-k)*binomial(4*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^6 where B(x) is the g.f. of A137971.
a(n) = Sum_{k=0..n-1} C(6*(n-k),k)/(n-k) * C(4*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(6*s*(1-s)*(4-5*s) / ((184*s - 144)*Pi)) / (n^(3/2) * r^n), where r = 0.0833821738312503523008482260558417829257343369560... and s = 1.287689442730957770948767878255357456556632139740... are real roots of the system of equations s = 1 + r*(1 + r*s^4)^6, 24 * r^2 * s^3 * (1 + r*s^4)^5 = 1. - Vaclav Kotesovec, Nov 22 2017

A137974 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^6.

Original entry on oeis.org

1, 1, 6, 45, 410, 4020, 41826, 452207, 5033910, 57300285, 663912420, 7804131660, 92838682242, 1115595461915, 13521340799310, 165104951405235, 2029162664033790, 25081468301798301, 311593507408597920
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Comments

In general, if g.f. satisfies A(x) = 1 + x*(1 + x*A(x)^p)^q, p >= 1, q >= 1, p + q > 2, then a(n) ~ sqrt(q*s*(1-s)*(p*(1-s)-s) / (2*Pi*p*(q-s-p*q*(1-s)))) / (n^(3/2) * r^n), where r and s are real roots of the system of equations s = 1 + r*(1 + r*s^p)^q, p*q * r^2 * s^(p-1) * (1 + r*s^p)^(q-1) = 1. - Vaclav Kotesovec, Nov 22 2017

Links

Crossrefs

Cf. A137973; A137968, A137970, A137972.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^5)^6);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(6*(n-k),k)/(n-k)*binomial(5*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^6 where B(x) is the g.f. of A137973.
a(n) = Sum_{k=0..n-1} C(6*(n-k),k)/(n-k) * C(5*k,n-k-1) for n>0 with a(0)=1. [Paul D. Hanna, Jun 16 2009]
a(n) ~ sqrt(6*s*(1-s)*(5-6*s) / ((290*s - 240)*Pi)) / (n^(3/2) * r^n), where r = 0.0739607593319208338998816978154858830062403258604... and s = 1.234938729532398384561936758596402363403570701060... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^6, 30 * r^2 * s^4 * (1 + r*s^5)^5 = 1. - Vaclav Kotesovec, Nov 22 2017
Showing 1-5 of 5 results.

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