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

A137959 G.f. satisfies A(x) = 1 + x + x^2*A(x)^5.

Original entry on oeis.org

1, 1, 1, 5, 15, 55, 220, 876, 3645, 15485, 66735, 292155, 1293456, 5782320, 26071435, 118402495, 541150155, 2487204315, 11488482130, 53302256250, 248293549685, 1160794446445, 5444674773325, 25614768620105, 120837493137460
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137960, A137958; A019497, A137954, A137966.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n-k,k]/(n-k)*Binomial[5*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^5));polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(n-k,k)/(n-k)*binomial(5*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

a(n) = Sum_{k=0..n-1} C(n-k,k)/(n-k) * C(5*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
Recurrence: 64*(n-4)*(n-3)*(n-2)*(n-1)*n*(2*n-5)*(2*n-3)*(2*n-1)*(2*n+1)*a(n) = + 5*(n-4)*(n-3)*(n-2)*(2*n-5)*(2*n-3)*(5*n-8)*(5*n-6)*(5*n-4)*(5*n-2)*a(n-2) + 5*(n-4)*(n-3)*(2*n-5)*(5000*n^6 - 45000*n^5 + 157250*n^4 - 267750*n^3 + 227216*n^2 - 87057*n + 11520)*a(n-3) + 15*(n-4)*(5000*n^8 - 80000*n^7 + 532250*n^6 - 1903250*n^5 + 3938648*n^4 - 4710638*n^3 + 3044313*n^2 - 895443*n + 80640)*a(n-4) + 5*(n-2)*(2*n-1)*(5000*n^7 - 95000*n^6 + 734250*n^5 - 2951750*n^4 + 6510194*n^3 - 7505289*n^2 + 3655107*n - 207360)*a(n-5) + 5*(n-3)*(n-2)*n*(2*n-3)*(2*n-1)*(5*n-29)*(5*n-23)*(5*n-17)*(5*n-11)*a(n-6). - Vaclav Kotesovec, Sep 18 2013
a(n) ~ sqrt(s*(1-s)*(5-6*s) / ((40*s - 40)*Pi)) / (n^(3/2) * r^n), where r = 0.1990700277700792324868112833575428736312653553870... and s = 1.498837534712599040608514104196928592039081694233... are real roots of the system of equations s = 1 + r*(1 + r*s^5), 5 * r^2 * s^4 = 1. - Vaclav Kotesovec, Nov 22 2017

A137956 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^2)^4.

Original entry on oeis.org

1, 1, 4, 14, 64, 301, 1500, 7738, 40948, 221278, 1215284, 6765148, 38083556, 216431253, 1240048740, 7155236960, 41542685352, 242513393884, 1422608044604, 8381507029660, 49574494112992, 294260899150492, 1752288415205896
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137955, A137957; A137958, A137964, A137971.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[4*(n-k),k]/(n-k)*Binomial[2*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^2)^4);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(4*(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)^4 where B(x) is the g.f. of A137955.
a(n) = Sum_{k=0..n-1} C(4*(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(4*s*(1-s)*(2-3*s) / ((28*s - 16)*Pi)) / (n^(3/2) * r^n), where r = 0.1569043698639381952962655091205241634381480571697... and s = 1.683635070625292013962854364673077567156937629734... are real roots of the system of equations s = 1 + r*(1 + r*s^2)^4, 8 * r^2 * s * (1 + r*s^2)^3 = 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
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.