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.

A357227 a(n) = coefficient of x^n, n >= 0, in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(n-1).

Original entry on oeis.org

1, 1, 5, 27, 156, 961, 6145, 40546, 273784, 1883468, 13153544, 93012247, 664640794, 4791939802, 34816034143, 254659426691, 1873698891024, 13858201221637, 102975937795619, 768385165594607, 5755185884844403, 43253819566052165, 326093530416255178, 2465456045342545908
Offset: 0

Views

Author

Paul D. Hanna, Oct 17 2022

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 27*x^3 + 156*x^4 + 961*x^5 + 6145*x^6 + 40546*x^7 + 273784*x^8 + 1883468*x^9 + 13153544*x^10 + 93012247*x^11 + 664640794*x^12 + ...
where
1 = ... + x^(-3)/(2*A(x) - x^(-3))^4 + x^(-2)/(2*A(x) - x^(-2))^3 + x^(-1)/(2*A(x) - x^(-1))^2 + 1/(2*A(x) - 1) + x + x^2*(2*A(x) - x^2) + x^3*(2*A(x) - x^3)^2 + x^4*(2*A(x) - x^4)^3 + ... + x^n*(2*A(x) - x^n)^(n-1) + ...
SPECIFIC VALUES.
A(1/9) = 1.30108724398914093656591796643458817060949...
A(1/10) = 1.22176622612326449515553495048940456186175...

Links

Crossrefs

Cf. A358961, A358962, A358963, A358964, A358965, A358937.
Cf. A355868, A357232, A355865.
Cf. A363312, A363313, A363314, A363315.

Programs

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

Formula

Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(n-1).
(2) 2*A(x) = Sum_{n=-oo..+oo} x^(2*n) * (2*A(x) - x^n)^(n-1).
(3) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - 2*x^n*A(x))^(n+1).
(4) 2*A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - 2*x^n*A(x))^(n+1).

A358961 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(2*n+1))^(n-1).

Original entry on oeis.org

1, 3, 7, 33, 163, 858, 4708, 26662, 154699, 914885, 5494719, 33423598, 205493244, 1274928510, 7972042450, 50188844583, 317861388939, 2023777490895, 12945901676736, 83163975425669, 536279878717858, 3470134399230086, 22525040920670333, 146633283078321531
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + 3*x + 7*x^2 + 33*x^3 + 163*x^4 + 858*x^5 + 4708*x^6 + 26662*x^7 + 154699*x^8 + 914885*x^9 + 5494719*x^10 + ...
where A = A(x) satisfies the doubly infinite sum
1 = ... + x^(-2)*(A - x^(-3))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x)^(-1) + x*(A - x^3)^0 + x^2*(A - x^5) + x^3*(A - x^7)^2 + x^4*(A - x^9)^3 + ... + x^n * (A - x^(2*n+1))^(n-1) + ...
also,
A(x) = ... + x^24/(1 - x^(-7)*A)^(-2) - x^12/(1 - x^(-5)*A)^(-1) + x^4 - 1/(1 - x^(-1)*A) + 1/(1 - x*A)^2 - x^4/(1 - x^3*A)^3 + x^12/(1 - x^5*A)^4 - x^24/(1 - x^7*A)^5 + ... + (-1)^(n+1)*x^(2*n*(n-1))/(1 - x^(2*n-1)*A(x))^(n+1) + ...

Links

Crossrefs

Cf. A357227, A358962, A358963, A358964, A358965, A358937.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(n=-#A,#A, x^n * (Ser(A) - x^(2*n+1))^(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:
(1) 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(2*n+1))^(n-1).
(2) x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n^2) / (1 - x^(2*n-1)*A(x))^(n+1).
(3) A(x) = Sum_{n=-oo..+oo} x^(3*n+1)* (A(x) - x^(2*n+1))^(n-1).
(4) A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n*(n-1)) / (1 - x^(2*n-1)*A(x))^(n+1).
(5) 0 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n*(n-1)) / (1 - x^(2*n-1)*A(x))^n.

A366229 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (x^(3*n+1) - A(x))^n.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 55, 138, 349, 904, 2377, 6323, 16993, 46036, 125625, 344973, 952565, 2643257, 7366942, 20613366, 57884187, 163071852, 460769168, 1305466309, 3707928596, 10555941648, 30115379589, 86087330322, 246541672062, 707274898726, 2032285666846
Offset: 0

Views

Author

Paul D. Hanna, Oct 29 2023

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 23*x^5 + 55*x^6 + 138*x^7 + 349*x^8 + 904*x^9 + 2377*x^10 + 6323*x^11 + 16993*x^12 + ...

Links

Crossrefs

Cf. A358937, A355866.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, x^n * (x^(3*n+1) - Ser(A))^n ), #A) ); 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 formulas.
(1) 1 = Sum_{n=-oo..+oo} x^n * (x^(3*n+1) - A(x))^n.
(2) 1 = Sum_{n=-oo..+oo} x^(n*(3*n-2)) / (1 - x^(3*n-1)*A(x))^n.
(3) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (x^(3*n+2) - A(x))^n (trivial).
(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n*(n-1)) / (1 - x^(3*n-1)*A(x))^n (trivial).
Showing 1-3 of 3 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.