A154843 Triangular array, T(n,k) = s(n,k) + s(n,n-k), where s(n,k) are the Stirling numbers of the first kind.

2, 1, 1, 1, -2, 1, 1, -1, -1, 1, 1, -12, 22, -12, 1, 1, 14, -15, -15, 14, 1, 1, -135, 359, -450, 359, -135, 1, 1, 699, -1589, 889, 889, -1589, 699, 1, 1, -5068, 13390, -15092, 13538, -15092, 13390, -5068, 1, 1, 40284, -109038, 113588, -44835, -44835, 113588, -109038, 40284, 1

Offset

0,1

Comments

Except for the first two rows the row sums are zero.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n,k) = s(n, k) + s(n, n - k), where s(n,k) are the Stirling numbers of the first kind (A048994).

Example

Triangle begins as:
  2;
  1,     1;
  1,    -2,     1;
  1,    -1,    -1,      1;
  1,   -12,    22,    -12,     1;
  1,    14,   -15,    -15,    14,      1;
  1,  -135,   359,   -450,   359,   -135,     1;
  1,   699, -1589,    889,   889,  -1589,   699,     1;
  1, -5068, 13390, -15092, 13538, -15092, 13390, -5068, 1;

Mathematica

Table[StirlingS1[n, k] +StirlingS1[n, n-k], {n, 0, 10}, {k, 0, n} ]//Flatten (* modified by G. C. Greubel, Apr 07 2019 *)

Prog

(PARI) {T(n, k) = stirling(n, k, 1) + stirling(n, n-k, 1)}; \\ G. C. Greubel, Apr 07 2019
(Magma) [[StirlingFirst(n, k) + StirlingFirst(n, n-k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 07 2019
(Sage) [[(-1)^(n-k)*(stirling_number1(n, k) + (-1)^n*stirling_number1(n, n-k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 07 2019

Crossrefs

Cf. A048994.

Sequence in context: A299152 A332770 A113120 * A062557 A276438 A338274

Adjacent sequences: A154840 A154841 A154842 * A154844 A154845 A154846

Keyword

tabl,sign

Author

Roger L. Bagula, Jan 16 2009

Status

approved