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 1-8 of 8 results.

A214763 G.f. satisfies: A(x) = 1/A(-x*A(x)^3).

Original entry on oeis.org

1, 2, 8, 40, 224, 1280, 7168, 40000, 231296, 1436928, 9773056, 72242176, 563679232, 4491707904, 35735001088, 280941652992, 2178641254400, 16710771339264, 127402021142528, 970887186407424, 7436390169329664, 57531833133899776, 451525691751628800, 3608174274928951296
Offset: 0

Views

Author

Paul D. Hanna, Jul 27 2012

Keywords

Comments

Compare to: G(x) = 1/G(-x*G(x)^3) when G(x) = 1 + x*G(x)^2 (A000108).
Compare to: B(x) = 1/B(-x*B(x)^3) when B(x) = 1/(1-9*x)^(1/3) = g.f. of A004987.
An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^3); for example, (*) is satisfied by G(x) = C(m*x) = (1-sqrt(1-4*m*x))/(2*m*x) for all m, where C(x) is the Catalan function.

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 40*x^3 + 224*x^4 + 1280*x^5 + 7168*x^6 +...
A(x)^2 = 1 + 4*x + 20*x^2 + 112*x^3 + 672*x^4 + 4096*x^5 + 24640*x^6 +...
A(x)^3 = 1 + 6*x + 36*x^2 + 224*x^3 + 1440*x^4 + 9312*x^5 + 59456*x^6 +...
1/A(x) = A(-x*A(x)^3) = 1 - 2*x - 4*x^2 - 16*x^3 - 80*x^4 - 384*x^5 - 1664*x^6 - 7360*x^7 - 40832*x^8 - 304128*x^9 - 2667008*x^10 -...

Links

Crossrefs

Cf. A214761, A214762, A214764, A214765, A214766, A214767, A214768, A214769.

Programs

  • PARI
    {a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^3+x*O(x^n)))/2);polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

The g.f. of this sequence is the limit of the recurrence:
(*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)^3))/2 starting at G_0(x) = 1+2*x.

A214762 G.f. satisfies: A(x) = 1/A(-x*A(x)^2).

Original entry on oeis.org

1, 2, 6, 24, 110, 496, 2156, 9216, 38742, 160032, 664532, 2898848, 13923468, 75361600, 450629592, 2844358656, 18224898790, 116051632704, 728724233988, 4509502911328, 27569637798116, 167072272244352, 1006431412676456, 6037728817690112, 36101656922629500
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2012

Keywords

Comments

Compare to: W(x) = 1/W(-x*W(x)^2) when W(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
Compare to: B(x) = 1/B(-x*B(x)^2) when B(x) = Sum_{n>=0} (2*n)!*x^n/n!^2.
An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^2); for example, (*) is satisfied by G(x) = W(m*x) = LambertW(-m*x)/(-m*x) for all m, where W(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 110*x^4 + 496*x^5 + 2156*x^6 +...
Related expansions:
A(x)^2 = 1 + 4*x + 16*x^2 + 72*x^3 + 352*x^4 + 1720*x^5 + 8192*x^6 +...
1/A(x) = A(-x*A(x)^2) = 1 - 2*x - 2*x^2 - 8*x^3 - 34*x^4 - 112*x^5 - 324*x^6 - 896*x^7 - 1866*x^8 - 800*x^9 + 5540*x^10 +...

Links

Crossrefs

Cf. A214761, A214763, A214764, A214765, A214766, A214767, A214768, A214769.

Programs

  • PARI
    {a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^2+x*O(x^n)))/2);polcoeff(A,n)}
for(n=0,31,print1(a(n),", "))

Formula

The g.f. of this sequence is the limit of the recurrence:
(*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)^2))/2 starting at G_0(x) = 1+2*x.

A214764 G.f. satisfies: A(x) = 1/A(-x*A(x)^4).

Original entry on oeis.org

1, 2, 10, 60, 390, 2660, 18772, 138984, 1107686, 9576100, 87944188, 830857464, 7876505340, 73967614584, 685644645896, 6289047266480, 57465415636166, 528315307772004, 4947263762389484, 47785581838822232, 480797992896880788, 5058812497153271912
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2012

Keywords

Comments

Compare to: W(x) = 1/W(-x*W(x)^4) when W(x) = Sum_{n>=0} (2*n+1)^(n-1)*x^n/n!.
Compare to: B(x) = 1/B(-x*B(x)^4) when B(x) = 1/(1-8*x)^(1/4) = g.f. of A004981.
An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^4); for example, (*) is satisfied by G(x) = W(m*x), where W(x) = Sum_{n>=0} (2*n+1)^(n-1)*x^n/n!.

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 60*x^3 + 390*x^4 + 2660*x^5 + 18772*x^6 +...
A(x)^4 = 1 + 8*x + 64*x^2 + 512*x^3 + 4096*x^4 + 32800*x^5 + 263168*x^6 +...

Crossrefs

Cf. A214761, A214762, A214763, A214765, A214766, A214767, A214768, A214769.

Programs

  • PARI
    {a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^4+x*O(x^n)))/2);polcoeff(A,n)}
for(n=0,31,print1(a(n),", "))

Formula

The g.f. of this sequence is the limit of the recurrence:
(*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)^4))/2 starting at G_0(x) = 1+2*x.

A214765 G.f. satisfies: A(x) = 1/A(-x*A(x)^5).

Original entry on oeis.org

1, 2, 12, 84, 616, 4832, 42112, 410368, 4316800, 46899648, 512004480, 5554843904, 59657443584, 633013100288, 6639969848320, 69332566233088, 733169635126272, 8068863012833280, 95049764691595264, 1213724245095528448, 16619899465108049920, 238054738089559379968
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2012

Keywords

Comments

Compare g.f. to: G(x) = 1/G(-x*G(x)^5) when G(x) = 1 + x*G(x)^3 (A001764).
An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^5); for example, (*) is satisfied by G(x) = F(m*x) = 1 + m*x*F(m*x)^3 for all m, where F(x) is the g.f. of A001764.

Examples

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 84*x^3 + 616*x^4 + 4832*x^5 + 42112*x^6 +...
A(x)^3 = 1 + 6*x + 48*x^2 + 404*x^3 + 3432*x^4 + 29808*x^5 + 271056*x^6 +...
A(x)^5 = 1 + 10*x + 100*x^2 + 980*x^3 + 9400*x^4 + 89632*x^5 + 866080*x^6 +...

Crossrefs

Cf. A214761, A214762, A214763, A214764, A214766, A214767, A214768, A214769.

Programs

  • PARI
    {a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^5+x*O(x^n)))/2);polcoeff(A,n)}
for(n=0,31,print1(a(n),", "))

Formula

The g.f. of this sequence is the limit of the recurrence:
(*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)^5))/2 starting at G_0(x) = 1+2*x.

A214766 G.f. satisfies: A(x) = 1/A(-x*A(x)^6).

Original entry on oeis.org

1, 2, 14, 112, 910, 8008, 84588, 1059296, 13998070, 179505848, 2193386772, 26007310560, 306461781228, 3616653947520, 42388643986040, 493154764709376, 5905712543971814, 78382075059128216, 1209853310234969668, 20945651586098921696, 378625571347575985092
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2012

Keywords

Comments

Compare to: W(x) = 1/W(-x*W(x)^6) when W(x) = Sum_{n>=0} (3*n+1)^(n-1)*x^n/n!.
An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^6); for example, (*) is satisfied by G(x) = W(m*x), where W(x) = Sum_{n>=0} (3*n+1)^(n-1)*x^n/n!.

Examples

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 112*x^3 + 910*x^4 + 8008*x^5 + 84588*x^6 +...
A(x)^6 = 1 + 12*x + 144*x^2 + 1672*x^3 + 18720*x^4 + 207000*x^5 + 2339072*x^6 +...

Crossrefs

Cf. A214761, A214762, A214763, A214764, A214765, A214767, A214768, A214769.

Programs

  • PARI
    {a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^6+x*O(x^n)))/2);polcoeff(A,n)}
for(n=0,31,print1(a(n),", "))

Formula

The g.f. of this sequence is the limit of the recurrence:
(*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)^6))/2 starting at G_0(x) = 1+2*x.

A214767 G.f. satisfies: A(x) = 1/A(-x*A(x)^7).

Original entry on oeis.org

1, 2, 16, 144, 1280, 12416, 156288, 2445952, 39005696, 569584128, 7551139840, 94905663488, 1200235880448, 15657039026176, 204235121909760, 2589347043356672, 34080849916796928, 554466780012625920, 11679936697324273664, 269604415927633805312, 6025264829519275556864
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2012

Keywords

Comments

Compare g.f. to: G(x) = 1/G(-x*G(x)^7) when G(x) = 1 + x*G(x)^4 (A002293).
An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^7); for example, (*) is satisfied by G(x) = F(m*x) = 1 + m*x*F(m*x)^4 for all m, where F(x) is the g.f. of A002293.

Examples

			G.f.: A(x) = 1 + 2*x + 16*x^2 + 144*x^3 + 1280*x^4 + 12416*x^5 + 156288*x^6 +...
A(x)^4 = 1 + 8*x + 88*x^2 + 992*x^3 + 10896*x^4 + 121600*x^5 + 1492480*x^6 +...
A(x)^7 = 1 + 14*x + 196*x^2 + 2632*x^3 + 33712*x^4 + 424032*x^5 + 5484864*x^6 +...

Crossrefs

Cf. A214761, A214762, A214763, A214764, A214765, A214766, A214768, A214769.

Programs

  • PARI
    {a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^7+x*O(x^n)))/2);polcoeff(A,n)}
for(n=0,31,print1(a(n),", "))

Formula

The g.f. of this sequence is the limit of the recurrence:
(*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)^7))/2 starting at G_0(x) = 1+2*x.

A214768 G.f. satisfies: A(x) = 1/A(-x*A(x)^8).

Original entry on oeis.org

1, 2, 18, 180, 1734, 18300, 270420, 5151720, 96203910, 1565102844, 22108977596, 287976684088, 3835267955036, 55283720348664, 804522994149032, 10849701753955856, 150403977728200774, 3086256025416536700, 91156710989444409004, 2687925748932854737432
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2012

Keywords

Comments

Compare to: W(x) = 1/W(-x*W(x)^8) when W(x) = Sum_{n>=0} (4*n+1)^(n-1)*x^n/n!.
An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^8); for example, (*) is satisfied by G(x) = W(m*x), where W(x) = Sum_{n>=0} (4*n+1)^(n-1)*x^n/n!.

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 180*x^3 + 1734*x^4 + 18300*x^5 + 270420*x^6 +...
A(x)^8 = 1 + 16*x + 256*x^2 + 3904*x^3 + 56320*x^4 + 793984*x^5 + 11567104*x^6 +...

Crossrefs

Cf. A214761, A214762, A214763, A214764, A214765, A214766, A214767, A214769.

Programs

  • PARI
    {a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^8+x*O(x^n)))/2);polcoeff(A,n)}
for(n=0,31,print1(a(n),", "))

Formula

The g.f. of this sequence is the limit of the recurrence:
(*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)^8))/2 starting at G_0(x) = 1+2*x.

A214769 G.f. satisfies: A(x) = 1/A(-x*A(x)^9).

Original entry on oeis.org

1, 2, 20, 220, 2280, 25920, 443744, 10057408, 215047552, 3841564160, 57161584256, 757459114112, 10427052678656, 166827795710208, 2728593278189568, 38108069305433088, 521570277192555520, 14195894062729323520, 594582326909611536384, 21399757674339677249536
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2012

Keywords

Comments

Compare g.f. to: G(x) = 1/G(-x*G(x)^7) when G(x) = 1 + x*G(x)^5 (A002294).
An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^9); for example, (*) is satisfied by G(x) = F(m*x) = 1 + m*x*F(m*x)^5 for all m, where F(x) is the g.f. of A002294.

Examples

			G.f.: A(x) = 1 + 2*x + 20*x^2 + 220*x^3 + 2280*x^4 + 25920*x^5 + 443744*x^6 +...
A(x)^5 = 1 + 10*x + 140*x^2 + 1980*x^3 + 26680*x^4 + 362432*x^5 + 5617920*x^6 +...
A(x)^9 = 1 + 18*x + 324*x^2 + 5532*x^3 + 88776*x^4 + 1386432*x^5 + 22460832*x^6 +...

Crossrefs

Cf. A214761, A214762, A214763, A214764, A214765, A214766, A214767, A214768.

Programs

  • PARI
    {a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^9+x*O(x^n)))/2);polcoeff(A,n)}
for(n=0,31,print1(a(n),", "))

Formula

The g.f. of this sequence is the limit of the recurrence:
(*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)^9))/2 starting at G_0(x) = 1+2*x.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.