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.

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A253255 G.f. satisfies: A(x) = (1 - x^3*A(x)^3)^2 / (1 - x*A(x))^4.

Original entry on oeis.org

1, 4, 26, 202, 1731, 15780, 150117, 1473292, 14807363, 151638550, 1576616125, 16598802248, 176599380271, 1895767748376, 20508188211018, 223348309510194, 2446792909432683, 26944972018189698, 298111489130625351, 3312016395569631402, 36935315970911333184, 413308467174788509668
Offset: 0

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Author

Paul D. Hanna, May 31 2015

Keywords

Comments

Self-convolution of A253256.

Examples

			G.f.: A(x) = 1 + 4*x + 26*x^2 + 202*x^3 + 1731*x^4 + 15780*x^5 +...
where A(x) = (1 - x^3*A(x)^3)^2 / (1 - x*A(x))^4.
The logarithm begins:
log(A(x)) = 4*x + 36*x^2/2 + 358*x^3/3 + 3748*x^4/4 + 40404*x^5/5 + 443886*x^6/6 + 4941654*x^7/7 +...+ A168595(n)*x^n/n +...

Crossrefs

Cf. A168595, A106228, A064641, A253256.

Programs

  • PARI
    {a(n) = local(A=1); A = (1/x)*serreverse( x*(1-x)^4/(1-x^3)^2 +x^2*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {A168595(n) = sum(k=0, 2*n, binomial(2*n, k) * polcoeff((1+x+x^2)^n, k) )}
{a(n) = local(A=1); A = exp( sum(k=1,n+1, A168595(k)*x^k/k) +x*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} A168595(n)*x^n/n ), where A168595(n) = Sum_{k=0..2*n} binomial(n,k)*trinomial(n,k).
(2) A(x) = (1/x)*Series_Reversion( x*(1-x)^4/(1-x^3)^2 ).

A253256 G.f. satisfies: A(x) = (1 - x^3*A(x)^6) / (1 - x*A(x)^2)^2.

Original entry on oeis.org

1, 2, 11, 79, 647, 5727, 53367, 515802, 5123303, 51977485, 536320688, 5610909773, 59379328267, 634538481389, 6837466955193, 74210071037031, 810527496757335, 8901979424068377, 98253966680382102, 1089260346498608721, 12123804391067414676, 135427509933882292680, 1517725698030921469890
Offset: 0

Views

Author

Paul D. Hanna, May 31 2015

Keywords

Comments

Self-convolution yields A253255.

Examples

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 79*x^3 + 647*x^4 + 5727*x^5 + 53367*x^6 +...
where A(x) = (1 - x^3*A(x)^6) / (1 - x*A(x)^2)^2.
The logarithm begins:
log(A(x)) = 2*x + 18*x^2 + 179*x^3 + 1874*x^4 + 20202*x^5 + 221943*x^6 + 2470827*x^7/7 +...+ A168595(n)/2*x^n/n +...

Crossrefs

Cf. A168595, A106228, A064641, A253255.

Programs

  • PARI
    {a(n) = local(A=1); A = sqrt( (1/x)*serreverse( x*(1-x)^4/(1-x^3)^2 +x^2*O(x^n))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {A168595(n) = sum(k=0, 2*n, binomial(2*n, k) * polcoeff((1+x+x^2)^n, k) )}
{a(n) = local(A=1); A = exp( sum(k=1,n+1, A168595(k)/2 * x^k/k) +x*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} A168595(n)/2 * x^n/n ), where A168595(n) = Sum_{k=0..2*n} binomial(n,k)*trinomial(n,k).
(2) A(x) = sqrt( (1/x)*Series_Reversion( x*(1-x)^4/(1-x^3)^2 ) ).
(3) A(x) = sqrt( (1-x*A(x) - sqrt(1 - 6*x*A(x) - 3*x^2*A(x)^2)) / (2*x*(1+x*A(x))) ).

A386301 a(n) = 1 + Sum_{k=0..n-1} (-1)^k * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 1, -1, 1, 8, 1, -28, 1, 134, 1, -649, 1, 3320, 1, -17497, 1, 94526, 1, -520507, 1, 2910896, 1, -16487794, 1, 94393106, 1, -545337199, 1, 3175320608, 1, -18615098836, 1, 109783526822, 1, -650884962907, 1, 3877184797784, 1, -23193307022860, 1, 139271612505362
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2025

Keywords

Crossrefs

Cf. A007317, A348857, A348858, A348859.
Cf. A064641, A386300.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=1+sum(j=0, i-1, (-1)^j*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies:
(1) A(x) = 1/( (1-x) * (1-x*A(-x)) ).
(2) A(x)*A(-x) = B(-x^2), where B(x) is the g.f. of A064641.
(3) A(x) = 1/(1-x) + 2*x/(1+x^2 + sqrt(1+6*x^2-3*x^4)).
a(2*n) = 1 and a(2*n+1) = 1 + (-1)^n * A064641(n) for n >= 0.
Previous Showing 11-13 of 13 results.

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