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 1-2 of 2 results.

A357794 a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^n * (1 - x^(n+1))^n * A(x)^(n+1).

Original entry on oeis.org

1, 3, 15, 114, 1086, 10824, 114382, 1252002, 14083275, 161810358, 1890774909, 22401092826, 268465408738, 3248818848876, 39643793276526, 487251937616006, 6026537732208078, 74954027622814455, 936840765257368687, 11761260253206563461, 148240496011414115676
Offset: 0

Views

Author

Paul D. Hanna, Dec 22 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (y - x^n)^(n-1), which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 114*x^3 + 1086*x^4 + 10824*x^5 + 114382*x^6 + 1252002*x^7 + 14083275*x^8 + 161810358*x^9 + 1890774909*x^10 + ...

Links

Crossrefs

Cf. A357158, A357795, A357796.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, n*(n+1)/2 * x^n * if(n==-1,0, (1 - x^(n+1) +x*O(x^#A) )^n) * Ser(A)^(n+1) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, (-1)^n * n*(n-1)/2 * x^(n*(n-2)) * if(n==1,0, 1/(1 - x^(n-1) +x*O(x^#A) )^n) / Ser(A)^(n-1) ), #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.
(1) 1 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^n * (1 - x^(n+1))^n * A(x)^(n+1).
(2) 1 = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)/2 * x^(n*(n-2)) / ((1 - x^(n-1))^n * A(x)^(n-1)).

A357795 a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^n * (1 - x^(n+2))^n * A(x)^(n+2).

Original entry on oeis.org

1, 4, 26, 300, 4134, 61696, 969660, 15837400, 266125823, 4571229248, 79904206064, 1416736880104, 25418030469904, 460600399886240, 8417980252615072, 154985730303047328, 2871904782258356719, 53519211809275995362, 1002383232008661189884, 18858606600633628740774
Offset: 0

Views

Author

Paul D. Hanna, Dec 22 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^(3*n) * (y - x^n)^(n-1), which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 4*x + 26*x^2 + 300*x^3 + 4134*x^4 + 61696*x^5 + 969660*x^6 + 15837400*x^7 + 266125823*x^8 + 4571229248*x^9 + 79904206064*x^10 + ...

Links

Crossrefs

Cf. A357158, A357794, A357796.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A-1, #A+1, n*(n+1)*(n+2)/3! * x^n * if(n==-2,0, (1 - x^(n+2) +x*O(x^#A) )^n) * Ser(A)^(n+2) ), #A-1) ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A-1, #A+1, (-1)^(n-1) * n*(n-1)*(n-2)/3! * x^(n*(n-3)) * if(n==2,0, 1/(1 - x^(n-2) +x*O(x^#A) )^n) / Ser(A)^(n-2) ), #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.
(1) 1 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^n * (1 - x^(n+2))^n * A(x)^(n+2).
(2) 1 = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)*(n-2)/3! * x^(n*(n-3)) / ((1 - x^(n-2))^n * A(x)^(n-2)).
Showing 1-2 of 2 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.