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 1-9 of 9 results.

A213282 G.f. satisfies A(x) = G(x/(1-x)^3) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 6, 36, 236, 1656, 12192, 92960, 727824, 5817696, 47281472, 389533056, 3245867136, 27308274688, 231654031104, 1979205694464, 17016094611712, 147104972637696, 1277988764697600, 11151534242977792, 97692088569096192, 858890594909048832, 7575804347863105536
Offset: 0

Views

Author

Paul D. Hanna, Jun 08 2012

Keywords

Comments

Compare to the g.f. B(x) of A006319 where B(x) = C(x/(1-x)^2) such that C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 36*x^3 + 236*x^4 + 1656*x^5 + 12192*x^6 +...
G.f.: A(x) = G(x/(1-x)^3) where G(x) = 1 + x*G(x)^3 is g.f. of A001764:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...

Links

Crossrefs

Cf. A213281, A001764; variants: A006319 (royal paths in a lattice), A213336.

Programs

  • Maple
    series(RootOf(G = 1 + G^3*x/(1-x)^3, G),x=0,30); # Mark van Hoeij, Apr 18 2013
  • PARI
    /* G.f. A(x) = G(x/(1-x)^3) where G(x) = 1 + x*G(x)^3: */
{a(n)=local(A,G=1+x);for(i=1,n,G=1+x*G^3+x*O(x^n));A=subst(G,x,x/(1-x+x*O(x^n))^3);polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* G.f. A(x) = F(x*A(x)^3) where F(x) = 1 + x/F(-x)^3: */
{a(n)=local(F=1+x+x*O(x^n), A=1); for(i=1, n+1, F=1+x/subst(F^3, x, -x+x*O(x^n))); A=(serreverse(x/F^3)/x)^(1/3); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. satisfies: A(x) = F(x*A(x)^3) where F(x) = 1 + x/F(-x)^3 is the g.f. of A213281.
G.f. A(x) satisfies: A(1 - G(-x)) = G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
a(n) = Sum_{k=0..n} binomial(n+2*k-1,n-k) * binomial(3*k,k)/(2*k+1). - Seiichi Manyama, Oct 03 2023

A213335 G.f. satisfies: A(x) = 1 + x/A(-x)^4.

Original entry on oeis.org

1, 1, 4, -6, -84, 171, 2940, -6576, -124260, 291321, 5810120, -14012244, -289392508, 711239741, 15052561056, -37498302048, -808073773572, 2033589755205, 44436219882252, -112715767473482, -2490257138332712, 6356863001632326, 141706826771491368
Offset: 0

Views

Author

Paul D. Hanna, Jun 09 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 - 6*x^3 - 84*x^4 + 171*x^5 + 2940*x^6 - 6576*x^7 +...
where
1/A(-x) = 1 + x - 3*x^2 - 13*x^3 + 77*x^4 + 402*x^5 - 2849*x^6 - 16040*x^7 +...
1/A(-x)^4 = 1 + 4*x - 6*x^2 - 84*x^3 + 171*x^4 + 2940*x^5 - 6576*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 28*x^3 - 263*x^4 - 476*x^5 + 8740*x^6 +...
The g.f. G(x) of A213336 begins:
G(x) = 1 + x + 8*x^2 + 64*x^3 + 568*x^4 + 5440*x^5 + 54888*x^6 +...
where G(x) = A(x*G(x)^4) and G(x/A(x)^4) = A(x);
also, G(x) = F(x/(1-x)^4) where F(x) = 1 + x*F(x)^4 is g.f. of A002293:
F(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...

Links

Crossrefs

Cf. A213336, A213252, A213281, A143047; A002293.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+x/subst(A^4, x, -x+x*O(x^n))); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f. satisfies: A(x) = G(x/A(x)^4) where G(x) = A(x*G(x)^4) is the g.f. of A213336.
G.f. satisfies: A(x) = ( x/Series_Reversion( x*F(x/(1-x)^4)^4 ) )^(1/4) where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.
G.f. satisfies: A(x) = A(x)*A(-x) + x/A(x)^3.

A366431 G.f. A(x) satisfies A(x) = 1 + x * (A(x) / (1 - x))^(5/2).

Original entry on oeis.org

1, 1, 5, 25, 135, 775, 4651, 28845, 183450, 1190050, 7844230, 52389678, 353770190, 2411324700, 16568343325, 114639216915, 798076174113, 5586035989185, 39287407321075, 277508001643575, 1967816928168265, 14003018984540741, 99965175670335750
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Partial sums give A366400.
Cf. A006319, A071724, A213282, A213336, A366432, A366433, A366434, A366435, A366436, A366437.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+3*k/2-1, n-k)*binomial(5*k/2, k)/(3*k/2+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+3*k/2-1,n-k) * binomial(5*k/2,k) / (3*k/2+1).

A366501 G.f. A(x) satisfies A(x) = 1 + x / ((1+x)^4*A(x)^3).

Original entry on oeis.org

1, 1, -7, 49, -399, 3633, -35511, 363937, -3858079, 41951521, -465296487, 5243459409, -59865074223, 690979478481, -8049598938135, 94522387901505, -1117615459764031, 13294669980012865, -158995530738069703, 1910555096402418545, -23056131790988675279
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2023

Keywords

Crossrefs

Cf. A001006, A366221, A366272, A366273, A366495, A366496, A366497, A366498, A366499, A366500.
Cf. A213336.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(n+3*k-1, n-k)*binomial(4*k-1, k)/(4*k-1));

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A213336.
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(n+3*k-1,n-k) * binomial(4*k-1,k) / (4*k-1).

A214337 Triangle read by rows: T(n,k) = number of rooted maps with n vertices and k faces on a non-orientable surface of type 3/2 (0 <= k <= n).

Original entry on oeis.org

0, 0, 41, 0, 690, 16925, 0, 7150, 237652, 4306778, 0, 58760, 2518957, 56864524, 910734615, 0, 420182, 22417804, 613687758, 11675167470, 174833737848
Offset: 0

Views

Author

N. J. A. Sloane, Jul 27 2012

Keywords

Examples

			Triangle begins:
  0;
  0,     41;
  0,    690,    16925;
  0,   7150,   237652,   4306778;
  0,  58760,  2518957,  56864524,   910734615;
  0, 420182, 22417804, 613687758, 11675167470, 174833737848;
  ...

Links

Crossrefs

Diagonals give A118448, A214335, A213336, A213338.
Cf. A214806.

A364410 G.f. A(x) satisfies A(x) = 1 + x^2 * (A(x) / (1 - x))^4.

Original entry on oeis.org

1, 0, 1, 4, 14, 52, 201, 800, 3260, 13536, 57068, 243664, 1051512, 4579088, 20097526, 88810872, 394811696, 1764477304, 7923087616, 35728412152, 161731039076, 734646128920, 3347600839252, 15298276784648, 70097391229500, 321974115549256, 1482242974320685
Offset: 0

Views

Author

Seiichi Manyama, Oct 15 2023

Keywords

Crossrefs

Partial sums give A186996.
Cf. A213336, A366645, A366646.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(n+2*k-1, n-2*k)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n+2*k-1,n-2*k) * binomial(4*k,k) / (3*k+1).

A366645 G.f. A(x) satisfies A(x) = 1 + x^3 * (A(x) / (1 - x))^4.

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 24, 67, 200, 586, 1704, 5049, 15232, 46284, 141240, 433696, 1340500, 4164830, 12993792, 40697472, 127941300, 403561902, 1276763096, 4050430502, 12882398456, 41068966204, 131211997496, 420056152498, 1347272602056, 4328764460928, 13931034024536
Offset: 0

Views

Author

Seiichi Manyama, Oct 15 2023

Keywords

Crossrefs

Partial sums give A215340.
Cf. A213336, A364410, A366646.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n+k-1, n-3*k)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n+k-1,n-3*k) * binomial(4*k,k) / (3*k+1).

A366646 G.f. A(x) satisfies A(x) = 1 + (x * A(x) / (1 - x))^4.

Original entry on oeis.org

1, 0, 0, 0, 1, 4, 10, 20, 39, 88, 228, 600, 1507, 3652, 8866, 22100, 56365, 144656, 369784, 942480, 2408934, 6196280, 16026652, 41571640, 107959654, 280708560, 731349400, 1910098320, 4999759830, 13109582376, 34421585844, 90500370760, 238272324682
Offset: 0

Views

Author

Seiichi Manyama, Oct 15 2023

Keywords

Crossrefs

Partial sums give A127902.
Cf. A213336, A364410, A366645.
Cf. A002026, A361932.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(n-1, n-4*k)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(n-1,n-4*k) * binomial(4*k,k) / (3*k+1).

A366647 G.f. A(x) satisfies A(x) = 1 + x * (A(x) / (1 - x))^5.

Original entry on oeis.org

1, 1, 10, 100, 1120, 13600, 174352, 2322880, 31846720, 446387200, 6367988480, 92154502912, 1349572428800, 19963252142080, 297843703347200, 4476750466785280, 67724540010278912, 1030392038941573120, 15756269876770734080, 242027462112980172800
Offset: 0

Views

Author

Seiichi Manyama, Oct 15 2023

Keywords

Crossrefs

Partial sums give A349311.
Cf. A006319, A213282, A213336.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+4*k-1, n-k)*binomial(5*k, k)/(4*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+4*k-1,n-k) * binomial(5*k,k) / (4*k+1).
Showing 1-9 of 9 results.

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

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