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 1-4 of 4 results.

A386502 E.g.f. A(x) satisfies A(x) = exp(x + x^4*A'''(x)).

Original entry on oeis.org

1, 1, 1, 1, 25, 3121, 1141921, 967142401, 1632504592321, 4951351715986369, 25004252825639317441, 198308457113999900437441, 2358282522829655305887600961, 40498770303734530275747011026561, 973509226030256545543333641850364737, 31906760631850274511535853878168004240641
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, (1+j)*sum(k=1, 3, stirling(3, k, 1)*j^k)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * (2*k - 3*k^2 + k^3) * binomial(n-1,k) * a(k) * a(n-1-k).
a(n) == 1 (mod 24). - Hugo Pfoertner, Jul 24 2025

A386503 E.g.f. A(x) satisfies A(x) = exp(x + x^5*A''''(x)).

Original entry on oeis.org

1, 1, 1, 1, 1, 121, 87841, 221971681, 1493423016961, 22593988839985921, 683468095232158346881, 37898988106295372711276161, 3602374572375663444650415755521, 556397556871212729711470761587498241, 133676738300734051631377763872501373230081, 48173754506706929414138973409107160269088573441
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, (1+j)*sum(k=1, 4, stirling(4, k, 1)*j^k)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * (-6*k + 11*k^2 - 6*k^3 + k^4) * binomial(n-1,k) * a(k) * a(n-1-k).
a(n) == 1 (mod 120). - Hugo Pfoertner, Jul 24 2025

A386504 E.g.f. A(x) satisfies A(x) = exp(x + x^6*A'''''(x)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 721, 3638881, 73388931841, 4439222967191041, 671254901566891891201, 223293614016982999277652481, 148555455012284806644741491166721, 183545166980276574090600617506568885761, 396856587856894056179855967245699021196188161, 1430118352830649099320069857966516939680956145171201
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, (1+j)*sum(k=1, 5, stirling(5, k, 1)*j^k)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * (24*k - 50*k^2 + 35*k^3 - 10*k^4 + k^5) * binomial(n-1,k) * a(k) * a(n-1-k).
a(n) == 1 (mod 720). - Hugo Pfoertner, Jul 24 2025

A386531 E.g.f. A(x) satisfies A(x) = exp(x + x^3/6 * A''(x)).

Original entry on oeis.org

1, 1, 1, 2, 13, 181, 4551, 188021, 11924753, 1103029649, 142906232381, 25095114042461, 5813156139567261, 1736262706526700925, 655797361805578202939, 308047913827328021014851, 177358895717746915172030241, 123578165227603044619210348321
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, (1+j)*sum(k=1, 2, stirling(2, k, 1)*j^k)*binomial(i-1, j)*v[j+1]*v[i-j])/6); v;

Formula

a(0) = 1; a(n) = a(n-1) + (1/6) * Sum_{k=0..n-1} (-k + k^3) * binomial(n-1,k) * a(k) * a(n-1-k).
Showing 1-4 of 4 results.