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 1-6 of 6 results.

A137954 G.f. satisfies A(x) = 1 + x + x^2*A(x)^4.

Original entry on oeis.org

1, 1, 1, 4, 10, 32, 107, 360, 1270, 4544, 16537, 61092, 228084, 860056, 3269994, 12521488, 48250690, 186959312, 727989318, 2847167632, 11179394088, 44053232012, 174160578150, 690576010820, 2745713062854, 10944253432600
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137955, A137953; A019497, A137959, A137966.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[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+x^2*A^4);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(n-k,k)/(n-k)*binomial(4*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(4*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
Recurrence: 3*(n-1)*n*(3*n-8)*(3*n-5)*(3*n-2)*(3*n+2)*a(n) = 64*(n-1)^2*(2*n-3)*(2*n-1)*(3*n-8)*(3*n-5)*a(n-2) + 32*(2*n-3)*(3*n-8)*(36*n^4 - 204*n^3 + 364*n^2 - 216*n + 35)*a(n-3) + 16*(3*n-2)*(144*n^5 - 1536*n^4 + 6005*n^3 - 10278*n^2 + 6790*n - 600)*a(n-4) + 8*n*(2*n-7)*(3*n-5)*(3*n-2)*(4*n-19)*(4*n-9)*a(n-5). - Vaclav Kotesovec, Sep 18 2013
a(n) ~ sqrt(s*(1-s)*(4-5*s) / ((24*s - 24)*Pi)) / (n^(3/2) * r^n), where r = 0.2362629484147719796376166796890824064312524895955... and s = 1.648350597886362639516822239585443208575003319460... are real roots of the system of equations s = 1 + r*(1 + r*s^4), 4 * r^2 * s^3 = 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

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

A137960 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^2.

Original entry on oeis.org

1, 1, 2, 11, 50, 275, 1560, 9212, 56082, 348675, 2207120, 14171155, 92075064, 604266000, 3999688050, 26670727220, 178997024610, 1208160130227, 8195828345756, 55849242272130, 382119958804520, 2624041637846210
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137961, A137959; A137952, A137955, A137967.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^5)^2);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(2*(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)^2 where B(x) is the g.f. of A137961.
a(n) = Sum_{k=0..n-1} C(2*(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(2*s*(1-s)*(5-6*s) / ((90*s - 80)*Pi)) / (n^(3/2) * r^n), where r = 0.1354712934479194768810666044866029126617104117352... and s = 1.354660923650925199331121807468321286698258863972... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^2, 10 * r^2 * s^4 * (1 + r*s^5) = 1. - Vaclav Kotesovec, Nov 22 2017

A371609 G.f. satisfies A(x) = ( 1 + x * (1 + x*A(x)^2)^2 )^2.

Original entry on oeis.org

1, 2, 5, 22, 94, 452, 2253, 11640, 61732, 333924, 1836052, 10229434, 57628078, 327711260, 1878658490, 10845298128, 62993496588, 367874945560, 2158717741928, 12722258713956, 75269561054412, 446891212180568, 2661788871400197, 15900644226590952, 95240143776976144
Offset: 0

Views

Author

Seiichi Manyama, Mar 29 2024

Keywords

Crossrefs

Cf. A137955, A371614.

Programs

  • PARI
    a(n, r=2, s=2, t=0, u=4) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

Formula

a(n) = Sum_{k=0..n} binomial(4*(n-k)+2,k) * binomial(2*k,n-k)/(2*(n-k)+1).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A137955.
Showing 1-6 of 6 results.

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