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.

A367470 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^2.

Original entry on oeis.org

1, 4, 28, 268, 3244, 47404, 810988, 15891628, 350851564, 8615761324, 232911898348, 6872755977388, 219799913877484, 7572909749244844, 279630706025296108, 11016315458773541548, 461211305514352065004, 20448268640012928321964
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A004123, A367471.
Cf. A005649, A367472.
Cf. A367474.

Programs

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

Formula

a(n) = Sum_{k=0..n} 2^k * (k+1)! * Stirling2(n,k).
a(0) = 1; a(n) = 2*Sum_{k=1..n} (k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 4*a(n-1) - 3*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n + 3/2) / (9 * log(3/2)^(n+2) * exp(n)). - Vaclav Kotesovec, May 20 2025

A367473 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^3.

Original entry on oeis.org

1, 9, 117, 1953, 39645, 946089, 25926597, 801869553, 27618402285, 1048096422009, 43444114011477, 1952712851250753, 94592798546953725, 4912513525545837129, 272265236648295312357, 16039329591716508497553, 1000809252891040145821965
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A032033, A367472.
Cf. A226515, A367471.

Programs

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

Formula

a(n) = (1/2) * Sum_{k=0..n} 3^k * (k+2)! * Stirling2(n,k).
a(0) = 1; a(n) = 3*Sum_{k=1..n} (2*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 9*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).

A375721 Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^2.

Original entry on oeis.org

1, 6, 60, 822, 14238, 297684, 7286076, 204251328, 6450932448, 226613038608, 8763294140064, 369900822475728, 16922169163019088, 833991953707934496, 44050579327333028448, 2482381132145285334912, 148660444826262311114880, 9427874254540824544312320
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2024

Keywords

Crossrefs

Cf. A354263, A375722.
Cf. A052801, A367474.
Cf. A367472.

Programs

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

Formula

a(n) = Sum_{k=0..n} 3^k * (k+1)! * |Stirling1(n,k)|.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n + 3/2) / (9 * exp(2*n/3) * (exp(1/3) - 1)^(n+2)). - Vaclav Kotesovec, Sep 06 2024
Showing 1-4 of 4 results.

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

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