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A142472
Triangle T(n, k) = binomial(n, k) * Sum_{j=k..n} StirlingS1(n, j)*StirlingS1(j, k), read by rows.
1
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, ....
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
Sequence in context: A159841 A202550 A364760 * A360089 A299445 A135049
KEYWORD
sign,tabl
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, Sep 26 2008
STATUS
approved