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 1-5 of 5 results.

A337085 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} j^k/j!.

Original entry on oeis.org

1, 0, 2, 0, 1, 5, 0, 1, 4, 16, 0, 1, 6, 15, 65, 0, 1, 10, 27, 64, 326, 0, 1, 18, 57, 124, 325, 1957, 0, 1, 34, 135, 292, 645, 1956, 13700, 0, 1, 66, 345, 796, 1585, 3906, 13699, 109601, 0, 1, 130, 927, 2404, 4605, 9726, 27391, 109600, 986410, 0, 1, 258, 2577, 7804, 15145, 28926, 68425, 219192, 986409, 9864101
Offset: 0

Views

Author

Seiichi Manyama, Aug 14 2020

Keywords

Examples

			Square array begins:
     1,    0,    0,    0,     0,     0,      0, ...
     2,    1,    1,    1,     1,     1,      1, ...
     5,    4,    6,   10,    18,    34,     66, ...
    16,   15,   27,   57,   135,   345,    927, ...
    65,   64,  124,  292,   796,  2404,   7804, ...
   326,  325,  645, 1585,  4605, 15145,  54645, ...
  1957, 1956, 3906, 9726, 28926, 98646, 374526, ...

Links

Crossrefs

Columns k=0..5 give A000522, A007526, A030297, A337001, A337002, A368719.
Main diagonal gives A256016.
Cf. A368724.

Programs

  • Mathematica
    T[n_, k_] := n! * Sum[If[j == k == 0, 1, j^k]/j!, {j, 0, n}]; Table[T[k, n-k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 29 2021 *)

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. - Seiichi Manyama, Jan 04 2024

A368718 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^5 / k!.

Original entry on oeis.org

0, 1, 30, 153, 412, 1065, 1386, 7105, -24072, 275697, -2656970, 29387721, -352403820, 4581620953, -64142155518, 962133092145, -15394128425744, 261700184657505, -4710603321945522, 89501463119441017, -1790029262385620340, 37590614510102111241
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2024

Keywords

Comments

In general, for m >=0, Sum_{k=0..n} (-1)^(n-k) * k^m / k! ~ A000587(m) * (-1)^n * exp(-1). - Vaclav Kotesovec, Jul 18 2025

Links

Crossrefs

Column k=5 of A368724.
Cf. A048993, A368587, A368719.

Programs

  • Maple
    f:= proc(n) option remember;
  - n*procname(n-1)+n^5
end proc:
f(0):= 0:
seq(f(i),i=0..21); # Robert Israel, May 13 2025
  • Mathematica
    Table[-5*n + 3*n^3 + n^4 - 2*(-1)^n*n*Subfactorial[n-1], {n, 0, 20}] (* Vaclav Kotesovec, Jul 18 2025 *)
  • 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) ~ -2*(-1)^n * exp(-1) * n!. - Vaclav Kotesovec, Jul 18 2025

A368716 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^3 / k!.

Original entry on oeis.org

0, 1, 6, 9, 28, -15, 306, -1799, 14904, -133407, 1335070, -14684439, 176214996, -2290792751, 32071101258, -481066515495, 7697064252016, -130850092279359, 2355301661034294, -44750731559644727, 895014631192902540, -18795307255050944079
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2024

Keywords

Comments

abs(a(n))/n is prime for n = 2, 3, 4, 5, 7, 13, 19, 28, 643 and no others up to n = 2000. - Robert Israel, May 13 2025

Links

Crossrefs

Column k=3 of A368724.
Cf. A048993, A337001, A368585.

Programs

  • Maple
    f:= proc(n) option remember;
  - n*procname(n-1)+n^3
end proc:
f(0):= 0:
seq(f(i),i=0..30); # Robert Israel, May 13 2025
  • Mathematica
    Table[n + n^2 + (-1)^n*n*Subfactorial[n-1], {n, 0, 20}] (* Vaclav Kotesovec, Jul 18 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, 3, stirling(3, k, 2)*x^k)*exp(x)/(1+x))))

Formula

a(0) = 0; a(n) = -n*a(n-1) + n^3.
E.g.f.: B_3(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.
a(n) ~ (-1)^n * exp(-1) * n!. - Vaclav Kotesovec, Jul 18 2025

A368717 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^4 / k!.

Original entry on oeis.org

0, 1, 14, 39, 100, 125, 546, -1421, 15464, -132615, 1336150, -14683009, 176216844, -2290790411, 32071104170, -481066511925, 7697064256336, -130850092274191, 2355301661040414, -44750731559637545, 895014631192910900, -18795307255050934419
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2024

Keywords

Links

Crossrefs

Column k=4 of A368724.
Cf. A048993, A337002, A368586.

Programs

  • Maple
    f:= proc(n) option remember;
  - n*procname(n-1)+n^4
end proc:
f(0):= 0:
seq(f(i),i=0..30); # Robert Israel, May 13 2025
  • Mathematica
    Table[-n + 2*n^2 + n^3 + (-1)^n*n*Subfactorial[n-1], {n, 0, 20}] (* Vaclav Kotesovec, Jul 18 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, 4, stirling(4, k, 2)*x^k)*exp(x)/(1+x))))

Formula

a(0) = 0; a(n) = -n*a(n-1) + n^4.
E.g.f.: B_4(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.
a(n) ~ (-1)^n * exp(-1) * n!. - Vaclav Kotesovec, Jul 18 2025

A368725 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^n / k!.

Original entry on oeis.org

1, 1, 2, 9, 100, 1065, 10626, 224161, 4598504, 46288017, 2509940710, 84061763841, -1602021820596, 164372205860473, 5216105126641514, -883395389739028095, 79008645559978113616, -1023235751229436800735, -651030746777115881959602, 113943411938145511923004513
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2024

Keywords

Links

Crossrefs

Main diagonal of A368724.
Cf. A000587, A256016.

Programs

  • Mathematica
    Join[{1}, Table[n!*Sum[(-1)^(n-k)*k^n/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jul 18 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, (-1)^(n-k)*k^n/k!);

Formula

E.g.f.: Sum_{k>=0} (k * x)^k / (k! * (1 + k * x)).
a(n) = n! * [x^n] B_n(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.
a(n) ~ A000587(n) * (-1)^n * exp(-1) * n!. - Vaclav Kotesovec, Jul 18 2025
Showing 1-5 of 5 results.

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