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.

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A375591 Expansion of e.g.f. exp( x * (exp(x^2/2) - 1) ).

Original entry on oeis.org

1, 0, 0, 3, 0, 15, 90, 105, 2520, 8505, 66150, 634095, 3118500, 40675635, 285675390, 2896618725, 31556725200, 281774718225, 3691224687150, 37783760189175, 483465043561500, 6108282465360075, 76126660317858150, 1102221773079151725, 14598579860502838200
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2024

Keywords

Crossrefs

Cf. A052506, A375592.
Cf. A357966, A375586.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(x^2/2)-1))))
  • PARI
    a(n) = n!*sum(k=0, n\2, stirling(k, n-2*k, 2)/(2^k*k!));

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} Stirling2(k,n-2*k)/(2^k*k!).

A375592 Expansion of e.g.f. exp( x * (exp(x^3/6) - 1) ).

Original entry on oeis.org

1, 0, 0, 0, 4, 0, 0, 70, 560, 0, 2800, 92400, 369600, 200200, 19619600, 252252000, 695094400, 5717712000, 171531360000, 1307259553600, 4852431584000, 136882025280000, 2175699958432000, 12698089211808000, 150837908212992000, 3848134863773200000
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2024

Keywords

Crossrefs

Cf. A052506, A375591.
Cf. A357967, A375587.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(x^3/6)-1))))
  • PARI
    a(n) = n!*sum(k=0, n\3, stirling(k, n-3*k, 2)/(6^k*k!));

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(k,n-3*k)/(6^k*k!).

A355650 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k/k! * (exp(x) - 1)).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 2, 5, 1, 0, 0, 3, 15, 1, 0, 0, 3, 16, 52, 1, 0, 0, 0, 6, 65, 203, 1, 0, 0, 0, 4, 10, 336, 877, 1, 0, 0, 0, 0, 10, 105, 1897, 4140, 1, 0, 0, 0, 0, 5, 20, 651, 11824, 21147, 1, 0, 0, 0, 0, 0, 15, 35, 2968, 80145, 115975, 1, 0, 0, 0, 0, 0, 6, 35, 616, 18936, 586000, 678570
Offset: 0

Views

Author

Seiichi Manyama, Jul 12 2022

Keywords

Examples

			Square array begins:
    1,   1,   1,  1,  1, 1, 1, ...
    1,   0,   0,  0,  0, 0, 0, ...
    2,   2,   0,  0,  0, 0, 0, ...
    5,   3,   3,  0,  0, 0, 0, ...
   15,  16,   6,  4,  0, 0, 0, ...
   52,  65,  10, 10,  5, 0, 0, ...
  203, 336, 105, 20, 15, 6, 0, ...

Crossrefs

Columns k=0..3 give A000110, A052506, A354000, A354001.
Cf. A145460, A292892, A355610.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 2)/(k!^j*(n-k*j)!));

Formula

T(0,k) = 1 and T(n,k) = ((n-1)!/k!) * Sum_{j=k+1..n} (j/(j-k)!) * T(n-j,k)/(n-j)! for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling2(n-k*j,j)/(k!^j * (n-k*j)!).

A362839 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^(n-j) * Stirling2(n-j,j)/(n-j)!.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 12, 16, 0, 1, 0, 8, 27, 80, 65, 0, 1, 0, 10, 48, 216, 560, 336, 0, 1, 0, 12, 75, 448, 2025, 4512, 1897, 0, 1, 0, 14, 108, 800, 5120, 21708, 40768, 11824, 0, 1, 0, 16, 147, 1296, 10625, 67584, 260253, 407808, 80145, 0
Offset: 0

Views

Author

Seiichi Manyama, May 05 2023

Keywords

Examples

			Square array begins:
  1,  1,   1,    1,    1,     1, ...
  0,  0,   0,    0,    0,     0, ...
  0,  2,   4,    6,    8,    10, ...
  0,  3,  12,   27,   48,    75, ...
  0, 16,  80,  216,  448,   800, ...
  0, 65, 560, 2025, 5120, 10625, ...

Crossrefs

Columns k=0..3 give: A000007, A052506, A351736, A351737.
Main diagonal gives A356806.
Cf. A362652.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\2, k^(n-j)*stirling(n-j, j, 2)/(n-j)!);

Formula

E.g.f. of column k: exp(x * (exp(k * x) - 1)).
G.f. of column k: Sum_{j>=0} x^j / (1 - (k*j-1)*x)^(j+1).
T(n,k) = Sum_{j=0..n} (k*j-1)^(n-j) * binomial(n,j).
Previous Showing 21-24 of 24 results.

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