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.

A367471 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^3.

Original entry on oeis.org

1, 6, 54, 630, 8982, 150966, 2918934, 63772470, 1552910742, 41690570166, 1223096629014, 38924237638710, 1335418262833302, 49129420920630966, 1929262811804022294, 80540656071983191350, 3561781875173605408662, 166331104582900651581366
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A004123, A367470.
Cf. A226515, A367473.

Programs

  • PARI
    a(n) = sum(k=0, n, 2^k*(k+2)!*stirling(n, k, 2))/2;

Formula

a(n) = (1/2) * Sum_{k=0..n} 2^k * (k+2)! * Stirling2(n,k).
a(0) = 1; a(n) = 2*Sum_{k=1..n} (2*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 6*a(n-1) - 3*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k).

A367472 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^2.

Original entry on oeis.org

1, 6, 60, 816, 13992, 289176, 6990360, 193432056, 6028092312, 208891033656, 7966989308760, 331618933474296, 14958464943057432, 726825458489514936, 37846457287387667160, 2102428978611587164536, 124109778776508893651352, 7758254465575303379273016
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A032033, A367473.
Cf. A005649, A367470.

Programs

  • PARI
    a(n) = sum(k=0, n, 3^k*(k+1)!*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} 3^k * (k+1)! * Stirling2(n,k).
a(0) = 1; a(n) = 3*Sum_{k=1..n} (k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 6*a(n-1) - 4*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k).

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).

A375722 Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^3.

Original entry on oeis.org

1, 9, 117, 1962, 40122, 966276, 26755812, 836862192, 29167596504, 1120629465432, 47044646845848, 2142210019297680, 105154320625284240, 5534780654854980000, 310945503593770489440, 18570787974013838515200, 1174884522886771261079040
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2024

Keywords

Crossrefs

Cf. A354263, A375721.
Cf. A354122, A367475.
Cf. A367473.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*log(1-x))^3))
  • PARI
    a(n) = sum(k=0, n, 3^k*(k+2)!*abs(stirling(n, k, 1)))/2;

Formula

a(n) = (1/2) * Sum_{k=0..n} 3^k * (k+2)! * |Stirling1(n,k)|.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} (2*k/n + 1) * (k-1)! * binomial(n,k) * a(n-k).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.