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

%I A129841 #12 Sep 08 2022 08:45:30
%S A129841 1,-1,4,-12,52,-256,1502,-10158,78360,-680280,6574872,-70075416,
%T A129841 816909816,-10342968456,141357740736,-2074340369088,32530886655168,
%U A129841 -542971977209760,9610316495698416,-179788450082431536,3544714566466060032
%N A129841 Antidiagonal sums of triangle T defined in A048594: T(j,k) = k! * Stirling1(j,k), 1<= k <= j.
%D A129841 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.
%D A129841 P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.
%D A129841 P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, no. 55, pp. 67-78.
%F A129841 E.g.f. for k-th column (k>=1): log(1+x)^k. For further formulas see the references.
%e A129841 First seven rows of T are
%e A129841 [    1 ]
%e A129841 [   -1,      2 ]
%e A129841 [    2,     -6,      6 ]
%e A129841 [   -6,     22,    -36,     24 ]
%e A129841 [   24,   -100,    210,   -240,    120 ]
%e A129841 [ -120,    548,  -1350,   2040,  -1800,    720 ]
%e A129841 [  720,  -3528,   9744, -17640,  21000, -15120,   5040 ]
%t A129841 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 *)
%o A129841 (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
%Y A129841 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.
%K A129841 sign
%O A129841 1,3
%A A129841 _Paul Curtz_, May 22 2007
%E A129841 Edited and extended by _Klaus Brockhaus_, Jun 03 2007