OFFSET
0,3
COMMENTS
Row sums are: {1, 2, 8, 37, 194, 1136, 7426, 54251, 442526, 4014940, ...}.
The first column gives the subfactorials, or rencontres, numbers A000166. See Riordan's p_n(k) equation 17 for further reference.
REFERENCES
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 57-65
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n, k) = n!*Sum_{j=k..n} (-1)^(j+k)*binomial(k+j, j)/j!.
EXAMPLE
Triangle begins as:
1;
0, 2;
1, 1, 6;
2, 7, 8, 20;
9, 23, 47, 45, 70;
44, 121, 214, 281, 224, 252;
265, 719, 1312, 1602, 1554, 1050, 924;
1854, 5041, 9148, 11334, 10548, 8142, 4752, 3432;
14833, 40319, 73229, 90507, 84879, 63849, 41019, 21021, 12870;
133496, 362881, 659006, 814783, 763196, 576643, 364166, 200629, 91520, 48620;
MAPLE
A156984:= (n, k) -> add( (-1)^(j+k)*binomial(k+j, j)*(n!/j!), j=k..n );
seq(seq(A156984(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 09 2021
MATHEMATICA
Table[n!*Sum[(-1)^(j-k)*Binomial[k+j, j]/j!, {j, k, n}], {n, 0, 12}, {k, 0, n}]//Flatten
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 20 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 09 2021
STATUS
approved