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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A218299 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..2*n} A084606(n,k)^2 * x^k * A(x)^k ), where A084606(n,k) = [x^k] (1 + 2*x + 2*x^2)^n.

Original entry on oeis.org

1, 1, 5, 21, 109, 573, 3209, 18425, 108649, 652425, 3979805, 24583853, 153488501, 966993893, 6139832385, 39249227569, 252400089361, 1631676380497, 10597809743477, 69123464993925, 452567027633853, 2973269053045197, 19595030047168569, 129509530910221737
Offset: 0

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Author

Paul D. Hanna, Oct 28 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 109*x^4 + 573*x^5 + 3209*x^6 +...
Let A = g.f. A(x), then the logarithm of the g.f. equals the series:
log(A(x)) = (1 + 2^2*x*A + 2^2*x^2*A^2)*x +
(1 + 4^2*x*A + 8^2*x^2*A^2 + 8^2*x^3*A^3 + 4^2*x^4*A^4)*x^2/2 +
(1 + 6^2*x*A + 18^2*x^2*A^2 + 32^2*x^3*A^3 + 36^2*x^4*A^4 + 24^2*x^5*A^5 + 8^2*x^6*A^6)*x^3/3 +
(1 + 8^2*x*A + 32^2*x^2*A^2 + 80^2*x^3*A^3 + 136^2*x^4*A^4 + 160^2*x^5*A^5 + 128^2*x^6*A^6 + 64^2*x^7*A^7 + 16^2*x^8*A^8)*x^4/4 +...
which involves the squares of the trinomial coefficients A084606(n,k):
1;
1, 2, 2;
1, 4, 8, 8, 4;
1, 6, 18, 32, 36, 24, 8;
1, 8, 32, 80, 136, 160, 128, 64, 16;
1, 10, 50, 160, 360, 592, 720, 640, 400, 160, 32; ...

Crossrefs

Cf. A218298, A200475, A084606.

Programs

  • PARI
    /* G.f. A(x) using the squares of the trinomial coefficients A084606: */
{A084606(n, k)=polcoeff((1 + 2*x + 2*x^2)^n, k)}
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, A084606(m, k)^2*x^k*(A+x*O(x^n))^k)*x^m/m))); polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=local(A=serreverse( x*(1 - 2*x^2)^2 / ((1 + x)*(1 + 4*x^3)*(1 + 4*x^4 +x*O(x^n))))/x);polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

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

A218619 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..2*n} A200536(n,k)^2 * x^k * A(x)^(2*k) ), where A200536(n,k) = [x^k] (1 + 3*x + 2*x^2)^n.

Original entry on oeis.org

1, 1, 11, 72, 734, 6994, 74641, 803196, 8989482, 102192197, 1184211027, 13897707080, 165052834584, 1978844990494, 23924151189858, 291313067897212, 3569576082827250, 43981925261314302, 544590342185545146, 6772925262506494672, 84567358373934285042
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2012

Keywords

Comments

More generally, given that A(x) satisfies:
A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..2*n} TC(n,k)^2*x^k*A(x)^(2*k) ),
where TC(n,k) = [x^k] (1 + b*x + c*x^2)^n, then A(x) satisfies:
(1) A(x) = (1+x*A(x)^2)*(1+c^2*x^3*A(x)^6)*(1+(b^2-2*c)*x^2*A(x)^4+c^2*x^4*A(x)^8) / (1-2*x^2*A(x)^4)^2.
(2) A(x) = sqrt( (1/x)*Series_Reversion( x*(1-c*x^2)^4 / ((1+x)*(1+c^2*x^3)*(1+(b^2-2*c)*x^2+c^2*x^4))^2 ) ).

Examples

			G.f.: A(x) = 1 + x + 11*x^2 + 72*x^3 + 734*x^4 + 6994*x^5 + 74641*x^6 +...
Let A = g.f. A(x), then the logarithm of the g.f. equals the series:
log(A(x)) = (1 + 3^2*x*A^2 + 2^2*x^2*A^4)*x*A +
(1 + 6^2*x*A^2 + 13^2*x^2*A^4 + 12^2*x^3*A^6 + 4^2*x^4*A^8)*x^2*A^2/2 +
(1 + 9^2*x*A^2 + 33^2*x^2*A^4 + 63^2*x^3*A^6 + 66^2*x^4*A^8 + 36^2*x^5*A^10 + 8^2*x^6*A^12)*x^3*A^3/3 +
(1 + 12^2*x*A^2 + 62^2*x^2*A^4 + 180^2*x^3*A^6 + 321^2*x^4*A^8 + 360^2*x^5*A^10 + 248^2*x^6*A^12 + 96^2*x^7*A^14 + 16^2*x^8*A^16)*x^4*A^4/4 +...
which involves the squares of the trinomial coefficients A200536(n,k):
1;
1, 3, 2;
1, 6, 13, 12, 4;
1, 9, 33, 63, 66, 36, 8;
1, 12, 62, 180, 321, 360, 248, 96, 16;
1, 15, 100, 390, 985, 1683, 1970, 1560, 800, 240, 32; ...

Crossrefs

Cf. A200536, A200537, A218298.

Programs

  • PARI
    /* G.f. A(x) using the squares of the trinomial coefficients */
{A200536(n, k)=polcoeff((1 + 3*x + 2*x^2)^n, k)}
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, A200536(m, k)^2*x^k*(A+x*O(x^n))^(2*k))*x^m*A^m/m))); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=local(A=sqrt(serreverse( x*(1-2*x^2)^4/((1+x)*(1+x^2)*(1+4*x^2)*(1+4*x^3+x*O(x^n)))^2 )/x));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. satisfies:
(1) A(x) = (1+x*A(x)^2)*(1+x^2*A(x)^4)*(1+4*x^2*A(x)^4)*(1+4*x^3*A(x)^6) / (1-2*x^2*A(x)^4)^2.
(2) A(x) = sqrt( (1/x)*Series_Reversion( x*(1-2*x^2)^4 / ((1+x)*(1+x^2)*(1+4*x^2)*(1+4*x^3))^2 ) ).
Showing 1-2 of 2 results.

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