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.

A137957 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^4)^3.

Original entry on oeis.org

1, 1, 3, 15, 79, 468, 2895, 18670, 123765, 838860, 5785503, 40473729, 286504086, 2048388112, 14770313397, 107290913232, 784380664232, 5766985753620, 42614014459911, 316304429143995, 2357275139670183, 17631888703154172
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Crossrefs

Cf. A137958, A137956; A137953, A137962, A137969.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[3*(n-k),k]/(n-k)*Binomial[4*k,n-k-1],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Nov 22 2017 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^4)^3);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(3*(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)^3 where B(x) is the g.f. of A137958.
a(n) = Sum_{k=0..n-1} C(3*(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(3*s*(1-s)*(4-5*s) / ((88*s - 72)*Pi)) / (n^(3/2) * r^n), where r = 0.1243879037293364492255197677726812528516871521834... and s = 1.373172215091866448521512759142574301075022413158... are real roots of the system of equations s = 1 + r*(1 + r*s^4)^3, 12 * r^2 * s^3 * (1 + r*s^4)^2 = 1. - Vaclav Kotesovec, Nov 22 2017

A137955 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^4)^2.

Original entry on oeis.org

1, 1, 2, 9, 36, 172, 842, 4310, 22676, 121896, 666884, 3699973, 20771096, 117765084, 673367034, 3878538930, 22483446152, 131070712924, 767929882240, 4519387797894, 26704456819984, 158367557278412, 942285096541344, 5623496055739052, 33653373190735484
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137956, A137954; A137952, A137960, A137967.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[2*(n-k),k]/(n-k)*Binomial[4*k,n-k-1],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Sep 18 2013 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^4)^2);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(2*(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)^2 where B(x) is the g.f. of A137956.
a(n) = Sum_{k=0..n-1} C(2*(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(2*s*(1-s)*(4-5*s) / ((56*s - 48)*Pi)) / (n^(3/2) * r^n), where r = 0.1569043698639381952962655091205241634381480571697... and s = 1.444765371242615455251538467189577278901629278244... are real roots of the system of equations s = 1 + r*(1 + r*s^4)^2, 8 * r^2 * s^3 * (1 + r*s^4) = 1. - Vaclav Kotesovec, Nov 22 2017

A137964 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^4.

Original entry on oeis.org

1, 1, 4, 26, 184, 1451, 12020, 103734, 921132, 8364877, 77317704, 725029730, 6880482816, 65955731874, 637703938860, 6211709281162, 60900108419200, 600486291654444, 5950951929703520, 59242473406384472, 592166933647780576
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137965, A137963; A137956, A137958, A137971.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^5)^4);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(4*(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)^4 where B(x) is the g.f. of A137965.
a(n) = Sum_{k=0..n-1} C(4*(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(4*s*(1-s)*(5-6*s) / ((190*s - 160)*Pi)) / (n^(3/2) * r^n), where r = 0.0927175295193852172913829423030505161354091369581... and s = 1.270497495855793662015513509713357933752729700697... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^4, 20 * r^2 * s^4 * (1 + r*s^5)^3 = 1. - Vaclav Kotesovec, Nov 22 2017

A137971 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^6)^4.

Original entry on oeis.org

1, 1, 4, 30, 232, 2037, 18720, 179454, 1770380, 17864490, 183510672, 1912621814, 20175123732, 214980182783, 2310645275932, 25021270486830, 272717638241172, 2989549949264304, 32938634975109864, 364566094737276708
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137972, A137970; A137956, A137958, A137964.

Programs

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

Formula

G.f.: A(x) = 1 + x*B(x)^4 where B(x) is the g.f. of A137972.
a(n) = Sum_{k=0..n-1} C(4*(n-k),k)/(n-k) * C(6*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(4*s*(1-s)*(6-7*s) / ((276*s - 240)*Pi)) / (n^(3/2) * r^n), where r = 0.0833821738312503523008482260558417829257343369560... and s = 1.229254439060935443156800948762443928645579909446... are real roots of the system of equations s = 1 + r*(1 + r*s^6)^4, 24 * r^2 * s^5 * (1 + r*s^6)^3 = 1. - Vaclav Kotesovec, Nov 22 2017

A137958 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^3)^4.

Original entry on oeis.org

1, 1, 4, 18, 100, 587, 3660, 23640, 157076, 1066281, 7363620, 51568732, 365369868, 2614235293, 18862816112, 137096744232, 1002785827620, 7376023180645, 54525165453672, 404858512190316, 3018190533410664, 22581907465905018
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Crossrefs

Cf. A137957, A137959; A137956, A137964, A137971.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[4*(n-k),k]/(n-k)*Binomial[3*k,n-k-1],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Nov 22 2017 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^3)^4);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(4*(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)^4 where B(x) is the g.f. of A137957.
a(n) = Sum_{k=0..n-1} C(4*(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(4*s*(1-s)*(3-4*s) / ((66*s - 48)*Pi)) / (n^(3/2) * r^n), where r = 0.1243879037293364492255197677726812528516871521834... and s = 1.442260525872978775674461288363175530136608288804... are real roots of the system of equations s = 1 + r*(1 + r*s^3)^4, 12 * r^2 * s^2 * (1 + r*s^3)^3 = 1. - Vaclav Kotesovec, Nov 22 2017
Showing 1-5 of 5 results.

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