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

0,3

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

Cf. A000166.

Sequence in context: A179972 A085826 A112477 * A181621 A307070 A084268

Adjacent sequences: A156981 A156982 A156983 * A156985 A156986 A156987

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 20 2009

Extensions

Edited by G. C. Greubel, Mar 09 2021

Status

approved