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.

Previous Showing 31-37 of 37 results.

A384333 Expansion of Product_{k>=1} (1 + k*x)^((3/4)^k).

Original entry on oeis.org

1, 12, 30, 76, -819, 15120, -320568, 7719984, -208986462, 6300545128, -209806494828, 7660698340008, -304718887446110, 13127557400200944, -609336227455254936, 30330991088734345200, -1612305658103085757467, 91179423240593288760396, -5467060640706775435713298
Offset: 0

Views

Author

Seiichi Manyama, May 26 2025

Keywords

Crossrefs

Cf. A116603, A384332, A384334.
Cf. A032033, A381890, A384325.

Programs

  • Mathematica
    terms = 20; A[] = 1; Do[A[x] = -3*A[x] + 4*A[x/(1+x)]^(3/4) * (1+x)^3 + 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(4*sum(k=1, N, (-1)^(k-1)*sum(j=0, k, 3^j*j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = (1+x)^3 * A(x/(1+x))^(3/4).
G.f.: exp(4 * Sum_{k>=1} (-1)^(k-1) * A032033(k) * x^k/k).
G.f.: 1/B(-x), where B(x) is the g.f. of A384325.
G.f.: B(x)^12, where B(x) is the g.f. of A381890.
a(n) ~ (-1)^(n+1) * (n-1)! / log(4/3)^(n+1). - Vaclav Kotesovec, May 27 2025

A365863 a(0) = 1; thereafter a(n) = n*Sum_{k = 0..n-1} binomial(n, k)*(-1)^(1+n+k)*a(k).

Original entry on oeis.org

1, 1, 2, 12, 156, 3380, 108930, 4876242, 289111032, 21916777752, 2067208751790, 237380181141950, 32601704893973556, 5276471519805880836, 993835167745129599162, 215520207875112312124890, 53311353846240820033325040, 14919977169758349265112350256, 4690364757880376663319746737926
Offset: 0

Views

Author

Thomas Scheuerle, Nov 09 2023

Keywords

Comments

Let P_k(x) be the polynomial of order k which satisfies a(m) = P_k(m) for m = 0..k, then a(k+1) = k * P_k(k+1).
This sequence is a member of a family of sequences with related properties. Here are some examples:
With b(k+1) = 1 + P_k(k+1) we get b(k) = A000079(k).
With b(k+1) = 2 + P_k(k+1) we get b(k) = A000225(k).
With b(k+1) = 3 + P_k(k+1) we get b(k) = A033484(k).
With b(k+1) = 2 * P_k(k+1) we get b(k) = A000629(k).
With b(k+1) = 1 + 2 * P_k(k+1) we get b(k) = A007047(k).
With b(k+1) = 3 * P_k(k+1) we get b(k) = A201339(k).
With b(k+1) = 5 * P_k(k+1) we get b(k) = A201365(k).
With b(k+1) = -1 * P_k(k+1) we get b(k) = A000670(k)*(-1)^k.
With b(k+1) = -2 * P_k(k+1) we get b(k) = A004123(k+1)*(-1)^k.
With b(k+1) = -3 * P_k(k+1) we get b(k) = A032033(k)*(-1)^k.
With b(k+1) = -4 * P_k(k+1) we get b(k) = A094417(k)*(-1)^k.
With b(k+1) = -m * P_k(k+1) we get b(k) = Bo(m, k)*(-1)^k, Bo(m, k) are Generalized ordered Bell numbers.

Crossrefs

Cf. A000079, A000225, A000629, A000670.
Cf. A004123, A007047, A032033, A033484.
Cf. A094417, A201339, A201365, A367308.

Programs

  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, n*Sum[Binomial[n, k]*(-1)^(1 + n + k)*a[k], {k, 0, n - 1}]]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Nov 12 2023 *)
  • PARI
    a(n) = if(n == 0, 1,sum(k = 0,n-1, n*binomial(n, k)*(-1)^(1+n+k)*a(k)))

Formula

a(n) ~ c * n^(2*n + 1/2) / exp(2*n), where c = 2.9711739498821842863440481701659942323709511474486414... - Vaclav Kotesovec, Nov 12 2023

A367488 Expansion of e.g.f. 1/(4 - 3*exp(x))^x.

Original entry on oeis.org

1, 0, 6, 36, 444, 6540, 119520, 2593164, 65233392, 1867289868, 59939612040, 2132540249532, 83293357351248, 3543242182036284, 163062595422642552, 8071964230348189260, 427682380939864204224, 24149065480351703398572, 1447640087400503974386504
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A032033, A367472, A367473.
Cf. A354413, A367486.
Cf. A367490.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*sum(k=1, j-1, 3^k*(k-1)!*stirling(j-1, k, 2))*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A367490(k) * binomial(n-1,k-1) * a(n-k).

A367925 Expansion of e.g.f. 1/(4 - x - 3*exp(x)).

Original entry on oeis.org

1, 4, 35, 459, 8025, 175383, 4599507, 140728437, 4920898317, 193579534155, 8461200381111, 406815231899409, 21337866382711521, 1212458502624643719, 74193773349948903483, 4864422156647044661949, 340191752483516373189621, 25278147388666498256368323
Offset: 0

Views

Author

Seiichi Manyama, Dec 05 2023

Keywords

Crossrefs

Cf. A006155, A367924.
Cf. A032033, A078940, A367831, A367836, A367923.

Programs

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

Formula

a(0) = 1; a(n) = n * a(n-1) + 3 * Sum_{k=1..n} binomial(n,k) * a(n-k).

A375992 Expansion of e.g.f. (4 - 3 * exp(x))^(4/3).

Original entry on oeis.org

1, -4, 0, 16, 112, 976, 11760, 184656, 3566192, 81556176, 2152839920, 64389871696, 2151410517872, 79406805184976, 3208188040810480, 140812644820877136, 6671575179144279152, 339348322285418119376, 18443287953728909235440, 1066619199816333440144976
Offset: 0

Views

Author

Seiichi Manyama, Sep 05 2024

Keywords

Crossrefs

Cf. A032033, A346982, A365558, A375949, A375952, A375993.
Cf. A375988.

Programs

  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 3*j-4)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (3*j-4)) * Stirling2(n,k).

A375993 Expansion of e.g.f. (4 - 3 * exp(x))^(5/3).

Original entry on oeis.org

1, -5, 5, 35, 165, 1075, 10805, 152035, 2719365, 58547475, 1469512405, 42082036035, 1353220758565, 48264167285875, 1890433757030005, 80656857839376035, 3723074712045197765, 184851684577600696275, 9822823990059902723605, 556226222504163445932035
Offset: 0

Views

Author

Seiichi Manyama, Sep 05 2024

Keywords

Crossrefs

Cf. A032033, A346982, A365558, A375949, A375952, A375992.

Programs

  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 3*j-5)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (3*j-5)) * Stirling2(n,k).
a(n) ~ 5 * sqrt(Pi) * 2^(29/6) * n^(n - 13/6) / (9 * Gamma(1/3) * exp(n) * log(4/3)^(n - 5/3)). - Vaclav Kotesovec, Sep 06 2024

A384435 Expansion of e.g.f. 2/(5 - 3*exp(2*x)).

Original entry on oeis.org

1, 3, 24, 282, 4416, 86448, 2030784, 55656912, 1743277056, 61427981568, 2405046994944, 103579443604992, 4866448609591296, 247692476576575488, 13576823521525653504, 797345878311609526272, 49948684871884896731136, 3324530341927517641310208, 234293439367907438337982464
Offset: 0

Views

Author

Seiichi Manyama, Jun 03 2025

Keywords

Links

Crossrefs

Cf. A094417, A326324, A382753.
Cf. A032033, A328182, A384522.
Cf. A201366.

Programs

  • PARI
    a(n) = (-2)^(n+1)*polylog(-n, 5/3)/5;

Formula

a(n) = (-2)^(n+1)/5 * Li_{-n}(5/3), where Li_{n}(x) is the polylogarithm function.
a(n) = 2^(n+1)/5 * Sum_{k>=0} k^n * (3/5)^k.
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * k! * Stirling2(n,k).
a(n) = (3/5) * A201366(n) = (3/5) * Sum_{k=0..n} 5^k * (-2)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} 2^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3 * a(n-1) + 5 * Sum_{k=1..n-1} (-2)^(k-1) * binomial(n-1,k) * a(n-k).
Previous Showing 31-37 of 37 results.

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