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 1-7 of 7 results.

A019497 Number of ternary search trees on n keys.

Original entry on oeis.org

1, 1, 1, 3, 6, 16, 42, 114, 322, 918, 2673, 7875, 23457, 70551, 213846, 652794, 2004864, 6190612, 19207416, 59850384, 187217679, 587689947, 1850692506, 5845013538, 18509607753, 58759391013, 186958014766, 596108115402, 1904387243796, 6095040222192, 19540540075824
Offset: 0

Views

Author

James Fill (jimfill(AT)jhu.edu)

Keywords

Links

Crossrefs

Cf. A137954, A137959, A137966.

Programs

  • Maple
    A:= proc(n) option remember; if n=0 then 1 else convert(series(1+x+x^2*A(n-1)^3, x=0,n+1), polynom) fi end: a:= n-> coeff(A(n), x,n): seq(a(n), n=0..27); # Alois P. Heinz, Aug 22 2008
  • Mathematica
    a[0] = 1; a[n_] := Sum[Binomial[1*(n-k), k]/(n-k)*Binomial[3*k, n-k-1], {k, 0, n-1}]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Apr 07 2015, after Paul D. Hanna *)
  • PARI
    v=vector(50,j,1);for(n=3,50,A=sum(i=1,n,sum(j=1,n,sum(k=1,n,if(i+j+k-n,0,v[i]*v[j]*v[k]))));v[n]=A);a(n)=v[n+1];
  • PARI
    {a(n)= local(A); if(n<0, 0, A= 1+O(x); forstep(k= 1, n, 2, A= 1+x+x*x*A^3); polcoeff(A, n))} /* Michael Somos, Mar 29 2007 */
  • PARI
    {a(n)= if(n<0, 0, (-1)^n* polcoeff( serreverse((1-sqrt(1-4*x+4*x^3+x^2*O(x^n)))/2), n+1))} /* Michael Somos, Mar 29 2007 */
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(1*(n-k),k)/(n-k)*binomial(3*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

a(0)=a(1)=1 and for n>=2 a(n)= sum( i+j+k=n-2, a(i)*a(j)*a(k) ) (i, j, k>=0). - Benoit Cloitre, Jun 14 2004
G.f. A(x) satisfies A(x)= 1+ x+ x^2*A(x)^3. - Michael Somos, Mar 29 2007
Given g.f. A(x), then x*A(-x) is series reversion of A025262(n-1). - Michael Somos, Mar 29 2007
a(n) = Sum_{k=0..n-1} C(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) ~ (8 + 3*sqrt(3))^(1/4) * 3^(n/2 - 3/8) * (3 + sqrt(9 + 8*sqrt(3)))^(n + 1/2) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 2)). - Vaclav Kotesovec, Jul 31 2021

Extensions

More terms from Olivier Gérard, Jul 1997

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

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

A137965 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^4)^5.

Original entry on oeis.org

1, 1, 5, 30, 220, 1725, 14356, 124020, 1102770, 10023680, 92722620, 870039474, 8261024380, 79225392830, 766302511445, 7466883915800, 73227699088806, 722228333119200, 7159117292177840, 71284856957207030, 712673042497177450
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137964, A137966; A137961, A137963, A137973.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^4)^5);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(5*(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)^5 where B(x) is the g.f. of A137964.
a(n) = Sum_{k=0..n-1} C(5*(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(5*s*(1-s)*(4-5*s) / ((152*s - 120)*Pi)) / (n^(3/2) * r^n), where r = 0.0927175295193852172913829423030505161354091369581... and s = 1.306924048536121339538817141295744998528778296640... are real roots of the system of equations s = 1 + r*(1 + r*s^4)^5, 20 * r^2 * s^3 * (1 + r*s^4)^4 = 1. - Vaclav Kotesovec, Nov 22 2017

A295498 G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 - x^2/A(x)^3.

Original entry on oeis.org

1, 1, 1, 6, 11, 51, 106, 492, 1115, 5317, 12912, 62283, 159146, 767083, 2036260, 9765849, 26735811, 127447531, 358219288, 1696410364, 4879284508, 22946311567, 67362378507, 314520916727, 940422623222, 4359165612216, 13252603911289, 60989336178364, 188258217816004, 860270701616648, 2692815154387672, 12220594038311373, 38750249291035303, 174684318231133053, 560585633201038635
Offset: 0

Views

Author

Paul D. Hanna, Nov 22 2017

Keywords

Comments

Note that G(x) such that G(x) = 1 + x*G(x)^2 - x^2/G(x)^4 has negative coefficients.

Examples

			G.f.: A(x) = 1 + x + x^2 + 6*x^3 + 11*x^4 + 51*x^5 + 106*x^6 + 492*x^7 + 1115*x^8 + 5317*x^9 + 12912*x^10 + 62283*x^11 + 159146*x^12 + 767083*x^13 + 2036260*x^14 + 9765849*x^15 + 26735811*x^16 + 127447531*x^17 + 358219288*x^18 + 1696410364*x^19 + 4879284508*x^20 +...
such that A(x) = 1 + x*A(x)^2 - x^2/A(x)^3.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 3*x^2 + 14*x^3 + 35*x^4 + 136*x^5 + 372*x^6 + 1430*x^7 + 4159*x^8 + 16242*x^9 + 49525*x^10 + 196040*x^11 + 618436*x^12 +...
A(x)^3 = 1 + 3*x + 6*x^2 + 25*x^3 + 75*x^4 + 276*x^5 + 868*x^6 + 3159*x^7 + 10293*x^8 + 37851*x^9 + 127023*x^10 + 472767*x^11 + 1622387*x^12 +...
A(x)^4 = 1 + 4*x + 10*x^2 + 40*x^3 + 135*x^4 + 496*x^5 + 1694*x^6 + 6144*x^7 + 21303*x^8 + 77636*x^9 + 273548*x^10 + 1005368*x^11 + 3591432*x^12 +...
A(x)^5 = 1 + 5*x + 15*x^2 + 60*x^3 + 220*x^4 + 826*x^5 + 2985*x^6 + 11010*x^7 + 39785*x^8 + 146525*x^9 + 532601*x^10 + 1969045*x^11 + 7208040*x^12 +...
where x^2 = A(x)^3 - A(x)^4 + x*A(x)^5.
Let F(x) be the series given by
F(x) = (1/x)*Series_Reversion(x*A(x)) = 1 - x + x^2 - 6*x^3 + 21*x^4 - 86*x^5 + 396*x^6 - 1812*x^7 + 8607*x^8 - 41958*x^9 + 207333*x^10 +...+ (-1)^n*A137966(n)*x^n +...
then F(x) = 1 - x + x^2*F(x)^6.

Links

Crossrefs

Cf. A137966, A295404, A295504.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A = 1 + x*A^2 - x^2/A^3  +x*O(x^n)); polcoeff(G=A, n)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f. A(x) satisfies: x^2 = A(x)^3 - A(x)^4 + x*A(x)^5.
a(n) ~ s * sqrt((2*r - s^5) / (Pi*(6*r - s^5))) / (2*n^(3/2)*r^n), where r = 0.2520932825034369933395854430839207204336449133515... and s = 1.957646455287329963796650229699010012363761835805... are real roots of the system of equations 1 + r*s^2 = r^2/s^3 + s, 3*r^2/s^4 + 2*r*s = 1. - Vaclav Kotesovec, Nov 23 2017

A371608 G.f. satisfies A(x) = ( 1 + x * (1 + x*A(x)^3) )^2.

Original entry on oeis.org

1, 2, 3, 14, 55, 226, 1042, 4840, 23103, 113118, 561568, 2826550, 14392534, 73967650, 383271596, 2000096144, 10502029735, 55446004880, 294155761676, 1567371462762, 8384300275607, 45009106969022, 242400290365756, 1309314066314354, 7091306989205453
Offset: 0

Views

Author

Seiichi Manyama, Mar 29 2024

Keywords

Crossrefs

Cf. A137966, A371613.

Programs

  • PARI
    a(n, r=2, s=1, t=0, u=6) = 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(6*(n-k)+2,k) * binomial(k,n-k)/(3*(n-k)+1).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A137966.
Showing 1-7 of 7 results.

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