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.

A218251 G.f. satisfies A(x) = (1 + x*A(x))^2 * (1 + x^3*A(x)).

Original entry on oeis.org

1, 2, 5, 15, 48, 160, 550, 1937, 6954, 25355, 93633, 349490, 1316397, 4997306, 19100278, 73440718, 283876092, 1102466529, 4299673200, 16832894330, 66127276201, 260595497227, 1029913570587, 4081124171097, 16211144100379, 64539011439944, 257474646313530
Offset: 0

Views

Author

Paul D. Hanna, Oct 24 2012

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 15*x^3 + 48*x^4 + 160*x^5 + 550*x^6 +...
where
A(x) = 1 + (2+x^2)*x*A(x) + (1+2*x^2)*x^2*A(x)^2 + x^5*A(x)^3.

Links

Crossrefs

Cf. A218250, A182053.

Programs

  • Mathematica
    nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=2; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1 + x*AGF)^2 * (1 + x^3*AGF) - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 10 2013 *)
  • PARI
    {a(n)=local(A=1); for(i=1, n, A=(1+x*A)^2*(1+x^3*A)+x*O(x^n)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

Formula

Recurrence: (n+2)*(n+3)*(1241*n^5 - 8896*n^4 + 14395*n^3 + 17632*n^2 - 50640*n + 20520)*a(n) = - 6*(n+2)*(1201*n^4 - 9868*n^3 + 26581*n^2 - 24270*n + 2340)*a(n-1) + 2*(12410*n^7 - 64140*n^6 - 41011*n^5 + 524724*n^4 - 340939*n^3 - 550044*n^2 + 232560*n + 81000)*a(n-2) - 6*(2482*n^7 - 16551*n^6 + 12327*n^5 + 105521*n^4 - 209527*n^3 + 39268*n^2 + 134496*n - 70920)*a(n-3) + 2*(4964*n^7 - 40548*n^6 + 79541*n^5 + 175950*n^4 - 881383*n^3 + 1128540*n^2 - 373392*n - 118152)*a(n-4) + 6*(2482*n^7 - 23997*n^6 + 57469*n^5 + 92361*n^4 - 533975*n^3 + 581508*n^2 - 19896*n - 133272)*a(n-5) + 60*(n-5)*(2*n - 7)*(n^3 - 34*n^2 + 132*n - 144)*a(n-6) - 2*(n-6)*(2*n - 9)*(1241*n^5 - 2691*n^4 - 8779*n^3 + 19851*n^2 - 1570*n - 5748)*a(n-7). - Vaclav Kotesovec, Sep 10 2013
a(n) ~ c*d^n/n^(3/2), where d = 4.2142983943967634... is the root of the equation 4 - 12*d^2 - 8*d^3 + 12*d^4 - 20*d^5 + d^7 = 0 and c = 2.164253883870... - Vaclav Kotesovec, Sep 10 2013
a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k,k) * binomial(2*n-4*k+3,n-3*k+1)/(2*n-4*k+3). - Seiichi Manyama, Aug 28 2023

A211854 G.f. satisfies: A(x) = (1+x*A(x)^2)*(1+x^2*A(x)^2)*(1+x^3*A(x)^2).

Original entry on oeis.org

1, 1, 3, 11, 42, 173, 746, 3321, 15155, 70516, 333282, 1595620, 7722036, 37715028, 185661034, 920244770, 4588778327, 23003827327, 115867080623, 586089365947, 2975978506450, 15163583668774, 77507719810688, 397320926569995, 2042152353063874
Offset: 0

Views

Author

Paul D. Hanna, Apr 22 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 42*x^4 + 173*x^5 + 746*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 28*x^3 + 115*x^4 + 496*x^5 + 2211*x^6 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 84*x^3 + 391*x^4 + 1844*x^5 + 8800*x^6 +...
A(x)^6 = 1 + 6*x + 33*x^2 + 176*x^3 + 912*x^4 + 4674*x^5 + 23842*x^6 +...
where A(x) = 1 + x*(1+x+x^2)*A(x)^2 + x^3*(1+x+x^2)*A(x)^4 + x^6*A(x)^6.

Links

Crossrefs

Cf. A182053, A211855.

Programs

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

Formula

a(n) ~ s * sqrt((1 + 2*r + 4*r^3*s^2 + 5*r^4*s^2 + 6*r^5*s^4 + 3*r^2*(1 + s^2)) / (Pi*(1 + r + 6*r^3*s^2 + 6*r^4*s^2 + 15*r^5*s^4 + r^2*(1 + 6*s^2)))) / (2*n^(3/2)*r^n), where r = 0.1829152018931276962733907918487144062831105492965... and s = 1.828118673659452305128580127483211657533668751760... are real roots of the system of equations (1 + r*s^2)*(1 + r^2*s^2)*(1 + r^3*s^2) = s, 2*r*s*(1 + r + 2*r^3*s^2 + 2*r^4*s^2 + 3*r^5*s^4 + r^2*(1 + 2*s^2)) = 1. - Vaclav Kotesovec, Nov 22 2017

A211855 G.f. satisfies: A(x) = (1+x*A(x)^3)*(1+x^2*A(x)^2)*(1+x^3*A(x)).

Original entry on oeis.org

1, 1, 4, 19, 98, 553, 3288, 20287, 128681, 833889, 5496837, 36742204, 248454438, 1696588460, 11682677436, 81031854579, 565614332353, 3970182041035, 28006229772030, 198438070511163, 1411652452459443, 10078529348799106, 72192155099054325, 518659038159324250
Offset: 0

Views

Author

Paul D. Hanna, Apr 22 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 19*x^3 + 98*x^4 + 553*x^5 + 3288*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 9*x^2 + 46*x^3 + 250*x^4 + 1454*x^5 + 8827*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 82*x^3 + 468*x^4 + 2808*x^5 + 17431*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 128*x^3 + 765*x^4 + 4736*x^5 + 30086*x^6 +...
A(x)^5 = 1 + 5*x + 30*x^2 + 185*x^3 + 1155*x^4 + 7376*x^5 + 47970*x^6 +...
A(x)^6 = 1 + 6*x + 39*x^2 + 254*x^3 + 1653*x^4 + 10884*x^5 + 72474*x^6 +...
where A(x) = 1 + x*A(x)^3 + x^2*A(x)^2 + x^3*(A(x)+A(x)^5) + x^4*A(x)^4 + x^5*A(x)^3 + x^6*A(x)^6.

Links

Crossrefs

Cf. A182053, A211854.

Programs

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

Formula

a(n) ~ sqrt(s*(2*r*s + s^2 + 5*r^4*s^2 + 4*r^3*s^3 + 6*r^5*s^5 + 3*r^2*(1 + s^4)) / (Pi*(r + 3*s + 3*r^4*s + 6*r^3*s^2 + 10*r^2*s^3 + 15*r^5*s^4))) / (2*n^(3/2)*r^n), where r = 0.1303652752058746790368151406944165350206179676971... and s = 1.504659035764367744283558911063644754705733371817... are real roots of the system of equations (1 + r^3*s)*(1 + r^2*s^2)*(1 + r*s^3) = s, r*(2*r*s + 3*s^2 + 3*r^4*s^2 + 4*r^3*s^3 + 6*r^5*s^5 + r^2*(1 + 5*s^4)) = 1. - Vaclav Kotesovec, Nov 22 2017

A182267 G.f. satisfies: A(x) = (1+x*A(x))*(1+x^2*A(x)^2)*(1+x^3*A(x)).

Original entry on oeis.org

1, 1, 2, 6, 16, 46, 140, 435, 1382, 4474, 14687, 48787, 163703, 554009, 1888794, 6481220, 22366415, 77575617, 270277602, 945480612, 3319582632, 11693824752, 41318554495, 146399071577, 520042511448, 1851657641932, 6607352892709, 23624965371264
Offset: 0

Views

Author

Paul D. Hanna, Apr 22 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 46*x^5 + 140*x^6 + 435*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 48*x^4 + 148*x^5 + 472*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 102*x^4 + 336*x^5 + 1124*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 52*x^3 + 185*x^4 + 648*x^5 + 2272*x^6 +...
where A(x) = 1 + x*A(x) + x^2*A(x)^2 + x^3*(A(x) + A(x)^3) + x^4*A(x)^2 + x^5*A(x)^3 + x^6*A(x)^4.

Links

Crossrefs

Cf. A182053, A211854, A211855.

Programs

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

Formula

a(n) ~ sqrt(s*(1 + 2*r*s + 4*r^3*s + 5*r^4*s^2 + 6*r^5*s^3 + 3*r^2*(1 + s^2)) / (Pi*(1 + r^2 + 3*r*s + 3*r^3*s + 6*r^4*s^2))) / (2 * n^(3/2) * r^(n + 1/2)), where r = 0.2649675733882333627400730579639429790476557486165... and s = 2.383929237709193665917448862090331200952809331679... are roots of the system of equations (1 + r*s)*(1 + r^3*s)*(1 + r^2*s^2) = s, r*(1 + r^2 + 2*r*s + 2*r^3*s + 3*r^2*s^2 + 3*r^4*s^2 + 4*r^5*s^3) = 1. - Vaclav Kotesovec, Nov 18 2017

A212070 G.f. satisfies: A(x) = (1+x*A(x))*(1+x*A(x)^2)*(1+x*A(x)^3).

Original entry on oeis.org

1, 3, 21, 199, 2166, 25551, 317736, 4101292, 54429850, 738053745, 10180705447, 142408547576, 2015296793331, 28800644332829, 415060115307920, 6025247760182629, 88023011490624217, 1293147320502884759, 19092299095314415811, 283137984006724444796
Offset: 0

Views

Author

Paul D. Hanna, Apr 29 2012

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 21*x^2 + 199*x^3 + 2166*x^4 + 25551*x^5 +..
Related expansions:
A(x)^2 = 1 + 6*x + 51*x^2 + 524*x^3 + 5967*x^4 + 72456*x^5 +...
A(x)^3 = 1 + 9*x + 90*x^2 + 1002*x^3 + 11970*x^4 + 150057*x^5 +...
A(x)^4 = 1 + 12*x + 138*x^2 + 1660*x^3 + 20823*x^4 + 269964*x^5 +...
A(x)^5 = 1 + 15*x + 195*x^2 + 2525*x^3 + 33255*x^4 + 446298*x^5 +...
A(x)^6 = 1 + 18*x + 261*x^2 + 3624*x^3 + 50076*x^4 + 695934*x^5 +...
where A(x) = 1 + x*A(x) + x*A(x)^2 + x*(1+x)*A(x)^3 + x^2*A(x)^4 + x^2*A(x)^5 + x^3*A(x)^6.

Links

Crossrefs

Cf. A182053, A211854.

Programs

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

Formula

G.f.: A(x) = 1 + x*A(x)*(1+A(x)+A(x)^2) + x^2*A(x)^3*(1+A(x)+A(x)^2) + x^3*A(x)^6.
a(n) ~ sqrt((s*(1 + s + (1 + 2*r)*s^2 + 2*r*s^3 + 2*r*s^4 + 3*r^2*s^5)) / (1 + 3*(1 + r)*s + 6*r*s^2 + 10*r*s^3 + 15*r^2*s^4)) / (2*sqrt(Pi) * n^(3/2) * r^n), where r = 0.06228198686712455165459532624572875420874352588006064829276... and s = 1.61944833450852965640457413211207525783408084239130679443147... are roots of the system of equations (1 + r*s) * (1 + r*s^2) * (1 + r*s^3) = s, r*(1 + 2*s + 3*(1+r)*s^2 + 4*r*s^3 + 5*r*s^4 + 6*r^2*s^5) = 1. - Vaclav Kotesovec, Aug 24 2017

A218250 G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x^2*A(x))^2.

Original entry on oeis.org

1, 1, 3, 7, 18, 49, 135, 383, 1104, 3228, 9554, 28557, 86095, 261487, 799323, 2457327, 7592620, 23565444, 73437284, 229691620, 720800824, 2268820824, 7161255962, 22661307317, 71878917199, 228487568175, 727779875401, 2322485254421, 7424488376794, 23773398866825
Offset: 0

Views

Author

Paul D. Hanna, Oct 24 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 18*x^4 + 49*x^5 + 135*x^6 + 383*x^7 +...
where
A(x) = 1 + (1+2*x)*x*A(x) + (2+x)*x^3*A(x)^2 + x^5*A(x)^3.

Crossrefs

Cf. A218251, A182053.

Programs

  • Mathematica
    nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1 + x*AGF) * (1 + x^2*AGF)^2 - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 10 2013 *)
  • PARI
    {a(n)=local(A=1); for(i=1, n, A=(1+x*A)*(1+x^2*A)^2+x*O(x^n)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

Formula

Recurrence: 2*(n+2)*(2*n+5)*(43*n^3 - 48*n^2 - 43*n + 12)*a(n) = 2*(2*n+1)*(2*n+3)*(43*n^3 - 5*n^2 - 94*n + 8)*a(n-1) + 2*(344*n^5 + 132*n^4 - 1303*n^3 - 399*n^2 + 554*n + 168)*a(n-2) + (473*n^5 - 528*n^4 - 1711*n^3 + 1866*n^2 + 1256*n - 960)*a(n-3) - 6*(86*n^5 - 225*n^4 - 321*n^3 + 794*n^2 + 160*n - 416)*a(n-4) + 4*(n-4)*(n-2)*(43*n^3 + 81*n^2 - 10*n - 36)*a(n-5). - Vaclav Kotesovec, Sep 10 2013
a(n) ~ c*d^n/n^(3/2), where d = 3.361963061296269297... is the root of the equation -4 + 12*d - 11*d^2 - 16*d^3 - 8*d^4 + 4*d^5 = 0 and c = 2.227460242885392531198808525530878354... - Vaclav Kotesovec, Sep 10 2013
Showing 1-6 of 6 results.

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