A142472 Triangle T(n, k) = binomial(n, k) * Sum_{j=k..n} StirlingS1(n, j)*StirlingS1(j, k), read by rows.

1, -4, 1, 21, -18, 1, -140, 240, -48, 1, 1140, -3150, 1300, -100, 1, -11004, 43620, -29700, 4800, -180, 1, 123074, -650769, 647780, -175175, 13965, -294, 1, -1566928, 10517108, -14190400, 5676160, -764400, 34496, -448, 1, 22390488, -184052520, 319680732, -175091112, 35160048, -2698920, 75600, -648, 1

Offset

1,2

Comments

Row sums are: 1, -3, 4, 53, -809, 7537, -41418, -294411, 15463669, -352665269, ....

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

T(n, k) = binomial(n, k) * Sum_{j=k..n} StirlingS1(n, j)*StirlingS1(j, k).

Example

The triangle begins as:
         1;
        -4,          1;
        21,        -18,         1;
      -140,        240,       -48,          1;
      1140,      -3150,      1300,       -100,        1;
    -11004,      43620,    -29700,       4800,     -180,        1;
    123074,    -650769,    647780,    -175175,    13965,     -294,     1;
  -1566928,   10517108, -14190400,    5676160,  -764400,    34496,  -448,    1;
  22390488, -184052520, 319680732, -175091112, 35160048, -2698920, 75600, -648, 1;

Maple

A142472:= (n, k)-> binomial(n, k)*add(Stirling1(n, j)*Stirling1(j, k), j=k..n);
seq(seq(A142472(n, k), k=1..n), n=1..12); # G. C. Greubel, Apr 02 2021

Mathematica

T[n_, k_]:= Binomial[n, k]*Sum[StirlingS1[n, j]*StirlingS1[j, k], {j, k, n}];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 02 2021 *)

Prog

(Magma)
A142472:= func< n, k | Binomial(n, k)*(&+[StirlingFirst(n, j)*StirlingFirst(j, k): j in [k..n]]) >;
[A142472(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 02 2021
(Sage)
def A142472(n, k): return (-1)^(n-k)*binomial(n, k)*sum( stirling_number1(n, j)*stirling_number1(j, k) for j in (k..n) )
flatten([[A142472(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 02 2021

Crossrefs

Cf. A008275, A039814.

Sequence in context: A159841 A202550 A364760 * A360089 A299445 A135049

Adjacent sequences: A142469 A142470 A142471 * A142473 A142474 A142475

Keyword

sign,tabl

Author

Roger L. Bagula and Gary W. Adamson, Sep 22 2008

Extensions

Edited by N. J. A. Sloane, Sep 26 2008

Status

approved