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.

A137953 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^2)^3.

Original entry on oeis.org

1, 1, 3, 9, 34, 132, 546, 2327, 10191, 45534, 206788, 951723, 4429182, 20808186, 98550468, 470038119, 2255684699, 10883852112, 52769785320, 256960840946, 1256147650818, 6162349332204, 30328107189312, 149698391878458
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137952, A137954; A137957, A137962, A137969.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[3*(n-k), k]/(n-k)*Binomial[2*k, n-k-1],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec after Paul D. Hanna, Mar 25 2014 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^2)^3);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(3*(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)^3 where B(x) is the g.f. of A137952.
a(n) = Sum_{k=0..n-1} C(3*(n-k),k)/(n-k) * C(2*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
Recurrence: 5*n*(5*n-3)*(5*n-2)*(5*n+1)*(5*n+4)*(2948400*n^11 - 80922240*n^10 + 991552680*n^9 - 7191167904*n^8 + 34388915791*n^7 - 113938412552*n^6 + 266574560812*n^5 - 439214051186*n^4 + 497527715029*n^3 - 367402366838*n^2 + 158427508008*n - 30063700800)*a(n) = -240*(5*n-1)*(3402000*n^13 - 102564900*n^12 + 1682146080*n^11 - 16176231033*n^10 + 95359496344*n^9 - 359981654612*n^8 + 893831335718*n^7 - 1468770570635*n^6 + 1566970769558*n^5 - 1019176919948*n^4 + 331927521052*n^3 + 34505928*n^2 - 32180612832*n + 6541274880)*a(n-1) + 180*(884520000*n^16 - 26930232000*n^15 + 372745486800*n^14 - 3118060887120*n^13 + 17644263763548*n^12 - 71507400823524*n^11 + 214013670957835*n^10 - 480132169105811*n^9 + 810380315383846*n^8 - 1022562903644722*n^7 + 947982058983979*n^6 - 624324084479227*n^5 + 273663045967416*n^4 - 68343334466444*n^3 + 4273926176256*n^2 + 2065304121408*n - 381518968320)*a(n-2) + 72*(5890903200*n^16 - 188191699920*n^15 + 2743292998800*n^14 - 24248455085592*n^13 + 145518104758338*n^12 - 628264374415281*n^11 + 2014705595228766*n^10 - 4876859081303636*n^9 + 8950855221646414*n^8 - 12378944029917433*n^7 + 12665670452628658*n^6 - 9249292270917382*n^5 + 4496305419163048*n^4 - 1229711760456116*n^3 + 68797455703176*n^2 + 53468550934560*n - 10544040864000)*a(n-3) + 72*(5731689600*n^16 - 191702972160*n^15 + 2927459413440*n^14 - 27105381081216*n^13 + 170350803352728*n^12 - 770345146059408*n^11 + 2589617705669352*n^10 - 6581794624393248*n^9 + 12710327685293639*n^8 - 18531898603387194*n^7 + 20012311600272546*n^6 - 15421584075698196*n^5 + 7904537517669183*n^4 - 2290793383663938*n^3 + 159318295564312*n^2 + 94065554487360*n - 19593691084800)*a(n-4) + 72*(2*n-9)*(3*n-11)*(3*n-7)*(6*n-25)*(6*n-23)*(2948400*n^11 - 48489840*n^10 + 344492280*n^9 - 1422208584*n^8 + 3817772239*n^7 - 6909787807*n^6 + 8311308487*n^5 - 6272196721*n^4 + 2621759746*n^3 - 403021048*n^2 - 67705152*n + 22579200)*a(n-5). - Vaclav Kotesovec, Mar 25 2014
a(n) ~ sqrt(3*s*(s-1)*(3*s-2)/(5*s-3)) / (2*sqrt(Pi)*n^(3/2)*r^n), where s = 1.7888356349988794022183... is the root of the equation 216*(s-1)^2 = s*(5*s-6)^4, and r = 1/(s*(5*s-6)) = 0.189873988477346598... - Vaclav Kotesovec, Mar 25 2014

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

A137962 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^3.

Original entry on oeis.org

1, 1, 3, 18, 106, 720, 5085, 37493, 284331, 2204973, 17404720, 139369905, 1129411314, 9244823986, 76326154857, 634847759955, 5314684735045, 44746683774474, 378652035541761, 3218705637379698, 27471657413667780
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137963, A137961; A137953, A137957, A137969.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^5)^3);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(3*(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)^3 where B(x) is the g.f. of A137963.
a(n) = Sum_{k=0..n-1} C(3*(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(3*s*(1-s)*(5-6*s) / ((140*s - 120)*Pi)) / (n^(3/2) * r^n), where r = 0.1085884782751570249717333800652227343328635496829... and s = 1.301018963559115613510052458264916439485131890857... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^3, 15 * r^2 * s^4 * (1 + r*s^5)^2 = 1. - Vaclav Kotesovec, Nov 22 2017

A137968 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^2)^6.

Original entry on oeis.org

1, 1, 6, 27, 158, 981, 6342, 42728, 295008, 2079882, 14908740, 108312873, 795836544, 5903472999, 44151306690, 332552305818, 2520416719368, 19207222744326, 147086508325056, 1131292622149352, 8735383810590486
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137967, A137969; A137970, A137972, A137974.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^2)^6);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(6*(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)^6 where B(x) is the g.f. of A137967.
a(n) = Sum_{k=0..n-1} C(6*(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(6*s*(1-s)*(2-3*s) / ((44*s - 24)*Pi)) / (n^(3/2) * r^n), where r = 0.1201742080825038015263858974579392344239858277873... and s = 1.572098697306844482137442690518486437859864764710... are real roots of the system of equations s = 1 + r*(1 + r*s^2)^6, 12 * r^2 * s * (1 + r*s^2)^5 = 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
Showing 1-5 of 5 results.

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