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 21-27 of 27 results.

A365567 Expansion of e.g.f. 1 / (5 - 4 * exp(x))^(3/4).

Original entry on oeis.org

1, 3, 24, 297, 5001, 106578, 2748399, 83182347, 2890153626, 113364686403, 4954547485149, 238734066994272, 12573018414279501, 718498413957515103, 44278797576715884024, 2927171415480872824197, 206625238881832412874501, 15511299587628626891270178
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2023

Keywords

Crossrefs

Cf. A094417, A346983, A354242.

Programs

  • Mathematica
    a[n_] := Sum[Product[4*j + 3, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 11 2023 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 4*j+3)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (4*j+3)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (4 - k/n) * binomial(n,k) * a(n-k).
a(n) ~ Gamma(1/4) * n^(n + 1/4) / (5^(3/4) * sqrt(Pi) * exp(n) * log(5/4)^(n + 3/4)). - Vaclav Kotesovec, Nov 11 2023
a(0) = 1; a(n) = 3*a(n-1) - 5*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

A238466 Generalized ordered Bell numbers Bo(9,n).

Original entry on oeis.org

1, 9, 171, 4869, 184851, 8772309, 499559571, 33190014069, 2520110222451, 215270320769109, 20431783142389971, 2133148392099721269, 242954208655633344051, 29977118969127060357909, 3983272698956314883956371, 567091857051921058649396469
Offset: 0

Views

Author

Vincenzo Librandi, Mar 18 2014

Keywords

Comments

Row 9 of array A094416, which has more information.

Links

Crossrefs

Cf. A000670, A004123, A032033, A094417, A094418, A094419, A238464.

Programs

  • Magma
    m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(10 - 9*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]];
  • Mathematica
    t=30; Range[0, t]! CoefficientList[Series[1/(10 - 9 Exp[x]), {x, 0, t}], x]

Formula

E.g.f.: 1/(10 - 9*exp(x)).
a(n) ~ n! / (10*(log(10/9))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 9*a(n-1) - 10*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A238467 Generalized ordered Bell numbers Bo(10,n).

Original entry on oeis.org

1, 10, 210, 6610, 277410, 14553010, 916146210, 67285818610, 5647734061410, 533307215001010, 55954905981282210, 6457903731351210610, 813080459351919805410, 110901542660769629769010, 16290196917457939734258210
Offset: 0

Views

Author

Vincenzo Librandi, Mar 18 2014

Keywords

Comments

Row 10 of array A094416, which has more information.

Links

Crossrefs

Cf. A000670, A004123, A032033, A094417, A094418, A094419, A238464.

Programs

  • Magma
    m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(11 - 10*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]];
  • Mathematica
    t=30; Range[0, t]! CoefficientList[Series[1/(11 - 10 Exp[x]), {x, 0, t}], x]

Formula

E.g.f.: 1/(11 - 10*exp(x)).
a(n) ~ n! / (11*(log(11/10))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 10*a(n-1) - 11*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

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

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

A382753 Expansion of e.g.f. 3/(5 - 2*exp(3*x)).

Original entry on oeis.org

1, 2, 14, 138, 1806, 29562, 580734, 13309578, 348611886, 10272416922, 336326121054, 12112707922218, 475894244100366, 20255443904321082, 928448378212678974, 45597074777924954058, 2388608236671667179246, 132947999835258872046042, 7835059049893316949502494
Offset: 0

Views

Author

Seiichi Manyama, Jun 03 2025

Keywords

Links

Crossrefs

Cf. A094417, A326324, A384435.
Cf. A004123, A216794, A384521.
Cf. A201367.

Programs

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

Formula

a(n) = (-3)^(n+1)/5 * Li_{-n}(5/2), where Li_{n}(x) is the polylogarithm function.
a(n) = 3^(n+1)/5 * Sum_{k>=0} k^n * (2/5)^k.
a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * k! * Stirling2(n,k).
a(n) = (2/5) * A201367(n) = (2/5) * Sum_{k=0..n} 5^k * (-3)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 2 * Sum_{k=1..n} 3^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 2 * a(n-1) + 5 * Sum_{k=1..n-1} (-3)^(k-1) * binomial(n-1,k) * a(n-k).

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 21-27 of 27 results.

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