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.

A386209 G.f. A(x) satisfies A(x) = 1/(1-x)^2 + x^2 * (d/dx A(x)^2).

Original entry on oeis.org

1, 2, 7, 40, 329, 3474, 44127, 650144, 10862273, 202632498, 4172689415, 94010964072, 2300682029417, 60787775220578, 1725027567563263, 52338601112555648, 1691028812744005697, 57973475215478590626, 2102150579452302435655, 80389277428829219813864
Offset: 0

Views

Author

Seiichi Manyama, Jul 15 2025

Keywords

Crossrefs

Programs

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

Formula

a(n) = n + 1 + (n-1) * Sum_{k=0..n-1} a(k) * a(n-1-k).
a(n) = n + 1 + 2 * Sum_{k=0..n-1} k * a(k) * a(n-1-k).

A386210 G.f. A(x) satisfies A(x) = 1/(1-x)^3 + x^2 * (d/dx A(x)^2).

Original entry on oeis.org

1, 3, 12, 76, 687, 7917, 108928, 1725768, 30796797, 609902671, 13260956196, 313951142508, 8039710476955, 221467222589241, 6531522562100448, 205381223316464464, 6860540121003113913, 242651957254699927803, 9060531236377877408956, 356208300597443254526892
Offset: 0

Views

Author

Seiichi Manyama, Jul 15 2025

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=binomial(i+2, 2)+(i-1)*sum(j=0, i-1, v[j+1]*v[i-j])); v;

Formula

a(n) = binomial(n+2,2) + (n-1) * Sum_{k=0..n-1} a(k) * a(n-1-k).
a(n) = binomial(n+2,2) + 2 * Sum_{k=0..n-1} k * a(k) * a(n-1-k).

A386265 G.f. A(x) satisfies A(x) = 1/(1+x) + x^2 * (d/dx A(x)^2).

Original entry on oeis.org

1, -1, -1, -3, -11, -61, -469, -4711, -58071, -842361, -13995017, -261468651, -5420285859, -123390215413, -3058937910365, -82014781572431, -2364353822942639, -72922308541928945, -2395804482405465233, -83528123730918481747, -3079941980238262567995
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2025

Keywords

Crossrefs

Programs

  • Mathematica
    terms=21; A[]=1; Do[A[x]=1/(1+x) + 2x^2 * A[x]A'[x] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 17 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-1)^i+(i-1)*sum(j=0, i-1, v[j+1]*v[i-j])); v;

Formula

a(n) = (-1)^n + (n-1) * Sum_{k=0..n-1} a(k) * a(n-1-k).
a(n) = (-1)^n + 2 * Sum_{k=0..n-1} k * a(k) * a(n-1-k).
Showing 1-3 of 3 results.