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A156984 Triangle T(n, k) = n!*Sum_{j=k..n} (-1)^(j+k)*binomial(k+j, j)/j!, read by rows.

%I A156984 #13 Dec 17 2023 15:08:45
%S A156984 1,0,2,1,1,6,2,7,8,20,9,23,47,45,70,44,121,214,281,224,252,265,719,
%T A156984 1312,1602,1554,1050,924,1854,5041,9148,11334,10548,8142,4752,3432,
%U A156984 14833,40319,73229,90507,84879,63849,41019,21021,12870,133496,362881,659006,814783,763196,576643,364166,200629,91520,48620
%N A156984 Triangle T(n, k) = n!*Sum_{j=k..n} (-1)^(j+k)*binomial(k+j, j)/j!, read by rows.
%C A156984 Row sums are: {1, 2, 8, 37, 194, 1136, 7426, 54251, 442526, 4014940, ...}.
%C A156984 The first column gives the subfactorials, or rencontres, numbers A000166. See Riordan's p_n(k) equation 17 for further reference.
%D A156984 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 57-65
%H A156984 G. C. Greubel, <a href="/A156984/b156984.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A156984 T(n, k) = n!*Sum_{j=k..n} (-1)^(j+k)*binomial(k+j, j)/j!.
%e A156984 Triangle begins as:
%e A156984        1;
%e A156984        0,      2;
%e A156984        1,      1,      6;
%e A156984        2,      7,      8,     20;
%e A156984        9,     23,     47,     45,     70;
%e A156984       44,    121,    214,    281,    224,    252;
%e A156984      265,    719,   1312,   1602,   1554,   1050,    924;
%e A156984     1854,   5041,   9148,  11334,  10548,   8142,   4752,   3432;
%e A156984    14833,  40319,  73229,  90507,  84879,  63849,  41019,  21021, 12870;
%e A156984   133496, 362881, 659006, 814783, 763196, 576643, 364166, 200629, 91520, 48620;
%p A156984 A156984:= (n,k) -> add( (-1)^(j+k)*binomial(k+j,j)*(n!/j!), j=k..n );
%p A156984 seq(seq(A156984(n,k), k=0..n), n=0..12); # _G. C. Greubel_, Mar 09 2021
%t A156984 Table[n!*Sum[(-1)^(j-k)*Binomial[k+j, j]/j!, {j,k,n}], {n,0,12}, {k,0,n}]//Flatten
%o A156984 (Sage) flatten([[sum( (-1)^(j+k)*binomial(n,j)*binomial(k+j,j)*factorial(n-j) for j in (k..n) ) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 09 2021
%o A156984 (Magma) [(&+[ (-1)^(j+k)*Binomial(n,j)*Binomial(k+j,j)*Factorial(n-j): j in [k..n]]): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 09 2021
%Y A156984 Cf. A000166.
%K A156984 nonn,tabl
%O A156984 0,3
%A A156984 _Roger L. Bagula_, Feb 20 2009
%E A156984 Edited by _G. C. Greubel_, Mar 09 2021