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.

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

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

A137967 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^6)^2.

Original entry on oeis.org

1, 1, 2, 13, 66, 406, 2602, 17271, 118444, 829514, 5914980, 42791085, 313277294, 2316793170, 17281455882, 129867946828, 982293317064, 7472406051744, 57132051350160, 438797394096378, 3383870656327576, 26191385476141936
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137968, A137966; A137952, A137955, A137960.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^6)^2);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(2*(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)^2 where B(x) is the g.f. of A137968.
a(n) = Sum_{k=0..n-1} C(2*(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(2*s*(1-s)*(6-7*s) / ((132*s - 120)*Pi)) / (n^(3/2) * r^n), where r = 0.1201742080825038015263858974579392344239858277873... and s = 1.297009871974239150024579315539982910111693413337... are real roots of the system of equations s = 1 + r*(1 + r*s^6)^2, 12 * r^2 * s^5 * (1 + r*s^6) = 1. - Vaclav Kotesovec, Nov 22 2017

A137952 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^3)^2.

Original entry on oeis.org

1, 1, 2, 7, 24, 95, 386, 1641, 7150, 31844, 144216, 662228, 3076044, 14427582, 68235078, 325049475, 1558212804, 7511319253, 36387218312, 177050945886, 864912345340, 4240388439744, 20857232340528, 102896737106415
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Crossrefs

Cf. A137953, A019497; A137955, A137960, A137967.

Programs

  • Mathematica
    Flatten[{1, Table[Sum[Binomial[2*(n-k),k]/(n-k) * Binomial[3*k,n-k-1], {k,0,n-1}], {n,1,30}]}] (* Vaclav Kotesovec, Nov 18 2017 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^3)^2);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(2*(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)^2 where B(x) is the g.f. of A137953.
a(n) = Sum_{k=0..n-1} C(2*(n-k),k)/(n-k) * C(3*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
Recurrence: 5*n*(5*n - 4)*(5*n - 3)*(5*n - 1)*(5*n + 3)*(8845200*n^11 - 252428400*n^10 + 3221192232*n^9 - 24137808840*n^8 + 117463352781*n^7 - 387964460127*n^6 + 882822962553*n^5 - 1374856808005*n^4 + 1422227015434*n^3 - 915895407668*n^2 + 320324023880*n - 42693386400)*a(n) = - 360*(5*n - 2)*(5670000*n^13 - 63714600*n^12 - 1032645960*n^11 + 24848001198*n^10 - 218480624507*n^9 + 1101741928166*n^8 - 3582401014336*n^7 + 7865579681092*n^6 - 11836392808433*n^5 + 12130520012664*n^4 - 8236278842764*n^3 + 3497924862840*n^2 - 827741189520*n + 81691545600)*a(n-1) + 180*(2653560000*n^16 - 86342760000*n^15 + 1284348733200*n^14 - 11544882534000*n^13 + 69915022739748*n^12 - 301277354913324*n^11 + 951521048997123*n^10 - 2235356609743737*n^9 + 3921814538564296*n^8 - 5108337175422974*n^7 + 4854490688899951*n^6 - 3250616687965913*n^5 + 1431302003002666*n^4 - 349408874612852*n^3 + 16089460853736*n^2 + 12240998632800*n - 2031289747200)*a(n-2) + 72*(17672709600*n^16 - 601551846000*n^15 + 9383367519936*n^14 - 88661500185240*n^13 + 565349613141438*n^12 - 2565633937621131*n^11 + 8513410651166583*n^10 - 20875837005697545*n^9 + 37705724089968084*n^8 - 49181218885648923*n^7 + 44098626888119141*n^6 - 23771481353637565*n^5 + 3467317211974378*n^4 + 4824415011450004*n^3 - 3654086377331160*n^2 + 1070168332564800*n - 116760296016000)*a(n-3) + 144*(8597534400*n^16 - 305543145600*n^15 + 4975684360704*n^14 - 49077873815616*n^13 + 326509076764188*n^12 - 1543742190898488*n^11 + 5321067950386782*n^10 - 13479709842928188*n^9 + 24903384308348709*n^8 - 32579354322085314*n^7 + 27941366702438094*n^6 - 11913061039189846*n^5 - 3157851308946897*n^4 + 7647346836930652*n^3 - 4534021704525180*n^2 + 1245319349576400*n - 132684717816000)*a(n-4) + 72*(2*n - 7)*(3*n - 14)*(3*n - 10)*(6*n - 25)*(6*n - 23)*(8845200*n^11 - 155131200*n^10 + 1183394232*n^9 - 5046898752*n^8 + 12951310413*n^7 - 19922972292*n^6 + 16394061984*n^5 - 2858995378*n^4 - 7011543813*n^3 + 6369403462*n^2 - 2180183136*n + 267092640)*a(n-5). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ sqrt((1 + 4*r*s^3 + 3*r^2*s^6) / (3*Pi*s*(2 + 5*r*s^3))) / (2*n^(3/2) * r^(n + 1/2)), where r = 0.1898739884773465982357897900946346962414966313829... and s = 1.607584028097173055359903977736399386285943742600... are roots of the system of equations 1 + r*(1 + r*s^3)^2 = s, 6*r^2*s^2*(1 + r*s^3) = 1. - Vaclav Kotesovec, Nov 18 2017

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
Showing 1-5 of 5 results.

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