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.

A384345 Expansion of Product_{k>=1} (1 + k*x)^((1/20) * (4/5)^k).

Original entry on oeis.org

1, 1, -4, 36, -494, 9026, -205284, 5581276, -176518189, 6366839811, -257967985400, 11601382088720, -573484266103260, 30909105184132900, -1804012437852543160, 113356419526025564808, -7629831521445348113927, 547688013439312943707673, -41765446604358525581076812
Offset: 0

Views

Author

Seiichi Manyama, May 26 2025

Keywords

Crossrefs

Cf. A381890, A384343, A384344.
Cf. A050353, A090356.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, (-1)^(k-1)*sum(j=0, k, 4^(j-1)*j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = (1+x)^(1/5) * A(x/(1+x))^(4/5).
G.f.: exp(Sum_{k>=1} (-1)^(k-1) * A050353(k) * x^k/k).
G.f.: 1/B(-x), where B(x) is the g.f. of A090356.

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

A384332 Expansion of Product_{k>=1} (1 + k*x)^((2/3)^k).

Original entry on oeis.org

1, 6, 3, 20, -207, 2538, -36381, 599760, -11210229, 234779146, -5455240455, 139445920452, -3892724842549, 117916363928070, -3854035833235839, 135241405277665656, -5072575747811807052, 202559732310632082120, -8581116791103001216108
Offset: 0

Views

Author

Seiichi Manyama, May 26 2025

Keywords

Crossrefs

Cf. A116603, A384333, A384334.
Cf. A004123, A384324, A384344.

Programs

  • Mathematica
    terms = 20; A[] = 1; Do[A[x] = -2*A[x] + 3*A[x/(1+x)]^(2/3) * (1+x)^2 + 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(3*sum(k=1, N, (-1)^(k-1)*sum(j=0, k, 2^j*j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = (1+x)^2 * A(x/(1+x))^(2/3).
G.f.: exp(3 * Sum_{k>=1} (-1)^(k-1) * A004123(k+1) * x^k/k).
G.f.: 1/B(-x), where B(x) is the g.f. of A384324.
G.f.: B(x)^6, where B(x) is the g.f. of A384344.
a(n) ~ (-1)^(n+1) * (n-1)! / log(3/2)^(n+1). - Vaclav Kotesovec, May 27 2025
Showing 1-4 of 4 results.

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