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

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

Offset: 0

Author

Roger L. Bagula, Feb 20 2009

Keywords

• nonn
• tabl

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.

Examples

			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;
		

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

Crossrefs

Cf. A000166.

Programs

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

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

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

Formula

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

Extensions

Edited by G. C. Greubel, Mar 09 2021