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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A155744 Triangle T(n, k) = (-1)^(n-k)*StirlingS1(n, k) + (-1)^k*StirlingS1(n, n-k) + (-1)^n*StirlingS1(n, k)*StirlingS1(n, n-k), read by rows.

Original entry on oeis.org

3, 1, 1, 1, 3, 1, 1, 11, 11, 1, 1, 48, 143, 48, 1, 1, 274, 1835, 1835, 274, 1, 1, 1935, 23649, 51075, 23649, 1935, 1, 1, 15861, 310639, 1195999, 1195999, 310639, 15861, 1, 1, 146188, 4221286, 25753812, 45832899, 25753812, 4221286, 146188, 1, 1, 1491876, 59942994, 535933124, 1510548249, 1510548249, 535933124, 59942994, 1491876, 1
Offset: 0

Views

Author

Roger L. Bagula, Jan 26 2009

Keywords

Examples

			  3;
  1,      1;
  1,      3,       1;
  1,     11,      11,        1;
  1,     48,     143,       48,        1;
  1,    274,    1835,     1835,      274,        1;
  1,   1935,   23649,    51075,    23649,     1935,       1;
  1,  15861,  310639,  1195999,  1195999,   310639,   15861,      1;
  1, 146188, 4221286, 25753812, 45832899, 25753812, 4221286, 146188, 1;

Links

Crossrefs

Cf. A000142, A001263, A048994, A342111.

Programs

  • Magma
    A155744:= func< n,k | (-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k) + (-1)^k*StirlingFirst(n, n-k) + (-1)^(n-k)*StirlingFirst(n, k) >;
[A155744(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 05 2021
  • Mathematica
    T[n_, k_] = (-1)^(n-k)*StirlingS1[n, k] + (-1)^k*StirlingS1[n, n-k] + (-1)^n*StirlingS1[n, k]*StirlingS1[n, n-k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 05 2021 *)
  • Sage
    def A155744(n,k): return stirling_number1(n, k)*stirling_number1(n, n-k) + stirling_number1(n, k) + stirling_number1(n, n-k)
flatten([[A155744(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 05 2021

Formula

T(n, k) = (-1)^(n-k)*StirlingS1(n, k) + (-1)^k*StirlingS1(n, n-k) + (-1)^n*StirlingS1(n, k)*StirlingS1(n, n-k).
Sum_{k=0..n} T(n, k) = 2*n! + A342111(n). - G. C. Greubel, Jun 05 2021

Extensions

Edited by G. C. Greubel, Jun 05 2021

A155868 Triangle T(n, k) = (-1)^n*StirlingS1(n, j)*StirlingS1(n, n-j), with T(n, 0) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 6, 6, 1, 1, 36, 121, 36, 1, 1, 240, 1750, 1750, 240, 1, 1, 1800, 23290, 50625, 23290, 1800, 1, 1, 15120, 308700, 1193640, 1193640, 308700, 15120, 1, 1, 141120, 4207896, 25738720, 45819361, 25738720, 4207896, 141120, 1, 1, 1451520, 59832864, 535810464, 1510458516, 1510458516, 535810464, 59832864, 1451520, 1
Offset: 0

Views

Author

Roger L. Bagula, Jan 29 2009

Keywords

Comments

Row sums are:
{1, 2, 3, 14, 195, 3982, 100807, 3034922, 105994835, 4215106730, 188097696347,...}

Examples

			Triangle begins as:
  1;
  1,      1;
  1,      1,       1;
  1,      6,       6,        1;
  1,     36,     121,       36,        1;
  1,    240,    1750,     1750,      240,        1;
  1,   1800,   23290,    50625,    23290,     1800,       1;
  1,  15120,  308700,  1193640,  1193640,   308700,   15120,      1;
  1, 141120, 4207896, 25738720, 45819361, 25738720, 4207896, 141120, 1;

Links

Crossrefs

Cf. A048994, A342111.

Programs

  • Magma
    A155868:= func< n,k | k eq 0 or k eq n select 1 else (-1)^n*StirlingFirst(n,k)* StirlingFirst(n,n-k) >;
[A155868(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 04 2021
  • Mathematica
    (* First program *)
p[n_, x_]:= If[n==0, 1, 1 +x^n +(-1)^n*Sum[StirlingS1[n, j]*StirlingS1[n, n-j]*x^j, {j,0,n}]];
Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten (* modified by G. C. Greubel, Jun 04 2021 *)
(* Second program *)
T[n_, k_]:=  If[k==0 || k==n, 1, (-1)^n*StirlingS1[n, k]*StirlingS1[n, n-k]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 04 2021 *)
  • Sage
    def A155868(n,k): return 1 if (k==0 or k==n) else stirling_number1(n,k)*stirling_number1(n,n-k)
flatten([[A155868(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021

Formula

T(n, k) = coefficients of p(n, x), where p(n, x) = 1 + x^n + (-1)^n*Sum_{j=0..n} StirlingS1(n, j)*StirlingS1(n, n-j)*x^k and p(0, x) = 1.
From G. C. Greubel, Jun 04 2021: (Start)
T(n, k) = (-1)^n*StirlingS1(n, j)*StirlingS1(n, n-j), with T(n, 0) = T(n, n) = 1.
Sum_{k=0..n} T(n, k) = 2 + A342111(n) - 2*[n==0]. (End)

Extensions

Edited by G. C. Greubel, Jun 04 2021

A384089 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^n.

Original entry on oeis.org

1, 0, 1, 63, 7206, 1357300, 384271700, 153027592116, 81648987014364, 56259916067074896, 48646018448463951450, 51584263505394472459750, 65833976467770842558152992, 99553004175105699906002335098, 176031670802373999913671973955080, 359870756416991348769957239299854000
Offset: 0

Views

Author

Seiichi Manyama, May 19 2025

Keywords

Crossrefs

Cf. A342111, A384018, A384029.
Cf. A351507.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 + k*x)^n, {k, 0, n-1}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 19 2025 *)
  • PARI
    a(n) = polcoef(prod(k=0, n-1, 1+k*x)^n, n);

Formula

a(n) = Sum_{0 <= x_1, x_2,..., x_n <= n and x_1 + x_2 + ... + x_n = (n-1)*n} Product_{k=1..n} |Stirling1(n,x_k)|.
a(n) ~ exp(n - 5/3) * n^(2*n+1) / (sqrt(Pi) * n^(3/2) * 2^(n + 1/2)). - Vaclav Kotesovec, May 19 2025
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