A129841 Antidiagonal sums of triangle T defined in A048594: T(j,k) = k! * Stirling1(j,k), 1<= k <= j.

1, -1, 4, -12, 52, -256, 1502, -10158, 78360, -680280, 6574872, -70075416, 816909816, -10342968456, 141357740736, -2074340369088, 32530886655168, -542971977209760, 9610316495698416, -179788450082431536, 3544714566466060032

Offset

1,3

References

P. Curtz, Integration numerique des systemes differentiels a conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, 1969, 135 pages, p. 61. Available from Centre d'Electronique de L'Armement, 35170 Bruz, France, or INRIA, Projets Algorithmes, 78150 Rocquencourt.

P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.

P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, no. 55, pp. 67-78.

Links

Table of n, a(n) for n=1..21.

Formula

E.g.f. for k-th column (k>=1): log(1+x)^k. For further formulas see the references.

Example

First seven rows of T are
[    1 ]
[   -1,      2 ]
[    2,     -6,      6 ]
[   -6,     22,    -36,     24 ]
[   24,   -100,    210,   -240,    120 ]
[ -120,    548,  -1350,   2040,  -1800,    720 ]
[  720,  -3528,   9744, -17640,  21000, -15120,   5040 ]

Mathematica

m = 21; t[j_, k_] := k!*StirlingS1[j, k]; Total /@ Table[ t[j-k+1, k], {j, 1, m}, {k, 1, Quotient[j+1, 2]}] (* Jean-François Alcover, Aug 13 2012, translated from Klaus Brockhaus's Magma program *)

Prog

(Magma) m:=21; T:=[ [ Factorial(k)*StirlingFirst(j, k): k in [1..j] ]: j in [1..m] ]; [ &+[ T[j-k+1][k]: k in [1..(j+1) div 2] ]: j in [1..m] ]; // Klaus Brockhaus, Jun 03 2007

Crossrefs

Cf. A048594 (T read by rows), A075181 (T unsigned with rows read backwards), A006252 (row sums of T), A000142 (main diagonal of T), A001286 (unsigned first subdiagonal of T). Unsigned values of second through sixth column of T are in A052517, A052748, A052753, A052767, A052779 resp.

Sequence in context: A124006 A215524 A034716 * A277431 A065525 A317281

Adjacent sequences: A129838 A129839 A129840 * A129842 A129843 A129844

Keyword

sign

Author

Paul Curtz, May 22 2007

Extensions

Edited and extended by Klaus Brockhaus, Jun 03 2007

Status

approved