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.

A381890 Expansion of Product_{k>=1} (1 + k*x)^((1/12) * (3/4)^k).

Original entry on oeis.org

1, 1, -3, 21, -225, 3207, -56821, 1202099, -29558466, 828401462, -26068940938, 910286433318, -34930741605414, 1461245816594058, -66187658069563710, 3227353484661602866, -168557942284281821933, 9388117645333487820387, -555463036269652132509113
Offset: 0

Views

Author

Seiichi Manyama, May 26 2025

Keywords

Crossrefs

Cf. A384343, A384344, A384345.
Cf. A050352, A090353.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, (-1)^(k-1)*sum(j=0, k, 3^(j-1)*j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = (1+x)^(1/4) * A(x/(1+x))^(3/4).
G.f.: exp(Sum_{k>=1} (-1)^(k-1) * A050352(k) * x^k/k).
G.f.: 1/B(-x), where B(x) is the g.f. of A090353.

A384344 Expansion of Product_{k>=1} (1 + k*x)^((1/6) * (2/3)^k).

Original entry on oeis.org

1, 1, -2, 10, -77, 787, -9972, 150552, -2637729, 52615903, -1177590290, 29228602546, -796945212035, 23681656958269, -761803800466856, 26376749702235900, -978091742247376932, 38674335439691203644, -1624351949069462807480, 72221688529265896447384
Offset: 0

Views

Author

Seiichi Manyama, May 26 2025

Keywords

Crossrefs

Cf. A381890, A384343, A384345.
Cf. A050351, A090351.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, (-1)^(k-1)*sum(j=0, k, 2^(j-1)*j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = (1+x)^(1/3) * A(x/(1+x))^(2/3).
G.f.: exp(Sum_{k>=1} (-1)^(k-1) * A050351(k) * x^k/k).
G.f.: 1/B(-x), where B(x) is the g.f. of A090351.

A384343 Expansion of Product_{k>=1} (1 + k*x)^((1/2)^(k+1)).

Original entry on oeis.org

1, 1, -1, 3, -14, 86, -650, 5822, -60287, 708873, -9334633, 136142011, -2179136696, 37987580268, -716513806824, 14540745561432, -315936103907094, 7318039354370826, -180020739049731594, 4687207255550122014, -128782014195949550724, 3723598212075752653284, -113023054997369519314572
Offset: 0

Views

Author

Seiichi Manyama, May 26 2025

Keywords

Crossrefs

Cf. A381890, A384344, A384345.
Cf. A000670, A084784.

Programs

  • Mathematica
    terms = 25; A[] = 1; Do[A[x] = -A[x] + 2*((1 + x)*A[x/(1 + x)])^(1/2) + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Vaclav Kotesovec, May 29 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(exp(sum(k=1, N, (-1)^(k-1)*sum(j=0, k, j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = (1+x)^(1/2) * A(x/(1+x))^(1/2).
G.f.: exp(Sum_{k>=1} (-1)^(k-1) * A000670(k) * x^k/k).
G.f.: 1/B(-x), where B(x) is the g.f. of A084784.
a(n) ~ (-1)^(n+1) * (n-1)! / (2*log(2)^(n+1)). - Vaclav Kotesovec, May 29 2025

A384334 Expansion of Product_{k>=1} (1 + k*x)^((4/5)^k).

Original entry on oeis.org

1, 20, 110, 340, -1995, 53904, -1534600, 49159600, -1758057650, 69662897000, -3037327435860, 144787947993000, -7502235351828450, 420296374337607600, -25335189019626256200, 1636008982452733508400, -112721505676611504401025, 8256863266451569604835900
Offset: 0

Views

Author

Seiichi Manyama, May 26 2025

Keywords

Crossrefs

Cf. A116603, A384332, A384333.
Cf. A094417, A384326, A384345.

Programs

  • Mathematica
    terms = 20; A[] = 1; Do[A[x] = -4*A[x] + 5*A[x/(1+x)]^(4/5) * (1+x)^4 + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Vaclav Kotesovec, May 27 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(exp(5*sum(k=1, N, (-1)^(k-1)*sum(j=0, k, 4^j*j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = (1+x)^4 * A(x/(1+x))^(4/5).
G.f.: exp(5 * Sum_{k>=1} (-1)^(k-1) * A094417(k) * x^k/k).
G.f.: 1/B(-x), where B(x) is the g.f. of A384326.
G.f.: B(x)^20, where B(x) is the g.f. of A384345.
a(n) ~ (-1)^(n+1) * (n-1)! / log(5/4)^(n+1). - Vaclav Kotesovec, May 27 2025
Showing 1-4 of 4 results.

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