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A129841
Antidiagonal sums of triangle T defined in A048594: T(j,k) = k! * Stirling1(j,k), 1<= k <= j.
3
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.
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
KEYWORD
sign
AUTHOR
Paul Curtz, May 22 2007
EXTENSIONS
Edited and extended by Klaus Brockhaus, Jun 03 2007
STATUS
approved