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.

A256016 a(n) = n! * Sum_{k=0..n} k^n/k!.

Original entry on oeis.org

1, 1, 6, 57, 796, 15145, 374526, 11669665, 447595800, 20733553809, 1141067915290, 73552752257281, 5484203261135028, 467864288815609465, 45236104846954021014, 4915818294874879570305, 596044703812665607374256, 80118478395137652912476449, 11870487496575403846760198322
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 01 2015

Keywords

Links

Crossrefs

Cf. A000110, A031971, A072034, A242446, A337001, A337002, A354436.
Main diagonal of A337085.

Programs

  • Mathematica
    Join[{1}, Table[n!*Sum[k^n/k!,{k,0,n}],{n,1,20}]]
  • PARI
    a(n) = n!*sum(k=0, n, k^n/k!); \\ Michel Marcus, Aug 15 2020
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x)^k/(k!*(1-k*x))))) \\ Seiichi Manyama, Aug 23 2022

Formula

a(n) ~ e*Bell(n)*n!, for the Bell numbers see A000110.
a(n) ~ sqrt(2*Pi) * n^(2*n+1/2) * exp(n/LambertW(n)-2*n) / (sqrt(1+LambertW(n)) * LambertW(n)^n).
E.g.f.: Sum_{k>=0} (k * x)^k / (k! * (1 - k * x)). - Seiichi Manyama, Aug 23 2022
a(n) = n! * [x^n] B_n(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials. - Seiichi Manyama, Jan 04 2024
a(n) = Sum_{k=0..n} k^n*(n-k)!*binomial(n,k). - Ridouane Oudra, Jun 16 2025

Extensions

a(0)=1 prepended by Seiichi Manyama, Aug 14 2020

A368724 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} (-1)^(n-j) * j^k/j!.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 1, 0, -2, 0, 1, 2, 3, 9, 0, 1, 6, 3, -8, -44, 0, 1, 14, 9, 4, 45, 265, 0, 1, 30, 39, 28, 5, -264, -1854, 0, 1, 62, 153, 100, -15, 6, 1855, 14833, 0, 1, 126, 543, 412, 125, 306, 7, -14832, -133496, 0, 1, 254, 1809, 1924, 1065, 546, -1799, 8, 133497, 1334961
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2024

Keywords

Examples

			Square array begins:
    1,    0, 0,   0,   0,    0,     0, ...
    0,    1, 1,   1,   1,    1,     1, ...
    1,    0, 2,   6,  14,   30,    62, ...
   -2,    3, 3,   9,  39,  153,   543, ...
    9,   -8, 4,  28, 100,  412,  1924, ...
  -44,   45, 5, -15, 125, 1065,  6005, ...
  265, -264, 6, 306, 546, 1386, 10626, ...

Links

Crossrefs

Columns k=0..5 give A182386, (-1)^(n-1) * A000240(n), A001477, A368716, A368717, A368718.
Main diagonal gives A368725.
Cf. A337085.

Programs

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

Formula

T(0,k) = 0^k and T(n,k) = n^k - n * T(n-1,k) for n>0.
E.g.f. of column k: B_k(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.

A368719 a(n) = n! * Sum_{k=0..n} k^5 / k!.

Original entry on oeis.org

0, 1, 34, 345, 2404, 15145, 98646, 707329, 5691400, 51281649, 512916490, 5642242441, 67707158124, 880193426905, 12322708514494, 184840628476785, 2957450056677136, 50276650964931169, 904979717370650610, 17194614630044837689, 343892292600899953780
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2024

Keywords

Links

Crossrefs

Column k=5 of A337085.
Cf. A048993, A368718.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, 5, stirling(5, k, 2)*x^k)*exp(x)/(1-x))))

Formula

a(0) = 0; a(n) = n*a(n-1) + n^5.
E.g.f.: B_5(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials.
a(n) ~ 52*exp(1)*n!. - Vaclav Kotesovec, Jan 13 2024

A368759 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * (1 + Sum_{j=0..n} j^k/j!).

Original entry on oeis.org

2, 1, 3, 1, 2, 7, 1, 2, 6, 22, 1, 2, 8, 21, 89, 1, 2, 12, 33, 88, 446, 1, 2, 20, 63, 148, 445, 2677, 1, 2, 36, 141, 316, 765, 2676, 18740, 1, 2, 68, 351, 820, 1705, 4626, 18739, 149921, 1, 2, 132, 933, 2428, 4725, 10446, 32431, 149920, 1349290, 1, 2, 260, 2583, 7828, 15265, 29646, 73465, 259512, 1349289, 13492901
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2024

Keywords

Examples

			Square array begins:
     2,    1,    1,     1,     1,     1,      1, ...
     3,    2,    2,     2,     2,     2,      2, ...
     7,    6,    8,    12,    20,    36,     68, ...
    22,   21,   33,    63,   141,   351,    933, ...
    89,   88,  148,   316,   820,  2428,   7828, ...
   446,  445,  765,  1705,  4725, 15265,  54765, ...
  2677, 2676, 4626, 10446, 29646, 99366, 375246, ...

Links

Crossrefs

Columns k=0..3 give A038159, A033540(n+1), A053817, A368760.
Cf. A337085.

Programs

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

Formula

T(0,k) = 1 + 0^k and T(n,k) = n^k + n * T(n-1,k) for n>0.
T(n,k) = n! + A337085(n,k).
E.g.f. of column k: (1+ B_k(x) * exp(x)) / (1-x), where B_n(x) = Bell polynomials.
Showing 1-4 of 4 results.

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