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