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.

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

A137970 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^3)^6.

Original entry on oeis.org

1, 1, 6, 33, 236, 1776, 14148, 117070, 995568, 8653068, 76508562, 686035674, 6223653276, 57018806567, 526802616954, 4902775644477, 45919926029588, 432511043009679, 4094087001128088, 38927025591433926, 371607779425490280
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137969, A137971; A137968, A137972, A137974.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^3)^6);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(6*(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)^6 where B(x) is the g.f. of A137969.
a(n) = Sum_{k=0..n-1} C(6*(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(6*s*(1-s)*(3-4*s) / ((102*s - 72)*Pi)) / (n^(3/2) * r^n), where r = 0.0971328555591006631243189792661187629516513365080... and s = 1.377827066365760014851094517875193622070040930150... are real roots of the system of equations s = 1 + r*(1 + r*s^3)^6, 18 * r^2 * s^2 * (1 + r*s^3)^5 = 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

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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