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 1-3 of 3 results.

A357206 Coefficients in the power series A(x) such that: x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

Original entry on oeis.org

1, 1, 6, 39, 267, 1949, 14927, 118517, 966840, 8055107, 68247637, 586231174, 5093508706, 44685394843, 395287384067, 3521909281230, 31576985230764, 284687856687607, 2579319718212675, 23472206080648463, 214448766193151410, 1966300700448875377, 18088031500652556354
Offset: 0

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Author

Paul D. Hanna, Sep 18 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 39*x^3 + 267*x^4 + 1949*x^5 + 14927*x^6 + 118517*x^7 + 966840*x^8 + 8055107*x^9 + 68247637*x^10 + ...
where
x*A(x)^2 = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...

Crossrefs

Cf. A355361, A357207, A357208, A357209.

Programs

  • PARI
    {a(n) = my(A=[1, 1], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( x*Ser(A)^2 - sum(m=-t, t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following relations.
(1) x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) -x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(3) x*A(x)^2 = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n), due to the Jacobi triple product identity.
(4) -x*A(x)^3 = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n), due to the Jacobi triple product identity.

A357207 Coefficients in the power series A(x) such that: x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

Original entry on oeis.org

1, 1, 7, 55, 469, 4307, 41678, 418872, 4330275, 45754091, 491916135, 5364166402, 59186372395, 659556170091, 7412556531714, 83921355689635, 956228695216241, 10957322339242547, 126189988012692329, 1459793848341094130, 16955390069787782159, 197653935181097885580
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 55*x^3 + 469*x^4 + 4307*x^5 + 41678*x^6 + 418872*x^7 + 4330275*x^8 + 45754091*x^9 + 491916135*x^10 + ...
where
x*A(x)^3 = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...

Crossrefs

Cf. A355361, A357206, A357208, A357209.

Programs

  • PARI
    {a(n) = my(A=[1, 1], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( x*Ser(A)^3 - sum(m=-t, t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following relations.
(1) x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) -x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(3) x*A(x)^3 = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n), due to the Jacobi triple product identity.
(4) -x*A(x)^4 = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n), due to the Jacobi triple product identity.

A357208 Coefficients in the power series A(x) such that: x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

Original entry on oeis.org

1, 1, 8, 74, 758, 8412, 98605, 1201739, 15075377, 193374064, 2524704727, 33440460233, 448246477551, 6069174992443, 82884604316537, 1140361539606239, 15791577929661603, 219930850717175458, 3078540089119391233, 43287917046150591163, 611156850554916771425
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 74*x^3 + 758*x^4 + 8412*x^5 + 98605*x^6 + 1201739*x^7 + 15075377*x^8 + 193374064*x^9 + 2524704727*x^10 + ...
where
x*A(x)^4 = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...

Crossrefs

Cf. A355361, A357206, A357207, A357209.

Programs

  • PARI
    {a(n) = my(A=[1, 1], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( x*Ser(A)^4 - sum(m=-t, t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following relations.
(1) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) -x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(3) x*A(x)^4 = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n), due to the Jacobi triple product identity.
(4) -x*A(x)^5 = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n), due to the Jacobi triple product identity.
Showing 1-3 of 3 results.

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