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A174789
Triangle read by rows: expansion of Sum_{k=0..n} binomial(n, k)*(Product_{j=0..n-k+1} (x + i)) * (-1)^k * x^(k-1).
2
1, 2, 2, 6, 7, 1, 24, 32, 8, 120, 178, 61, 3, 720, 1164, 494, 50, 5040, 8748, 4348, 655, 15, 40320, 74304, 41768, 8204, 420, 362880, 704016, 437148, 104272, 8365, 105, 3628800, 7362720, 4965912, 1376864, 149282, 4410, 39916800, 84255840, 60961176, 19079836, 2580550, 123795, 945
OFFSET
0,2
COMMENTS
Row sums are: {1, 4, 14, 64, 362, 2428, 18806, 165016, 1616786, 17487988, 206918942, ...}.
EXAMPLE
Triangle begins as:
1;
2, 2;
6, 7, 1;
24, 32, 8;
120, 178, 61, 3;
720, 1164, 494, 50;
5040, 8748, 4348, 655, 15;
40320, 74304, 41768, 8204, 420;
362880, 704016, 437148, 104272, 8365, 105;
3628800, 7362720, 4965912, 1376864, 149282, 4410;
MATHEMATICA
p[x, 0]:= 1; p[x_, n_]:= Sum[Binomial[n, k]* Product[x+j, {j, 0, n-k+1}] *(-x)^k, {k, 0, n}]/x; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]//Flatten
p[x_, 0]:= 1; p[x_, n_]:= (x+1)*Pochhammer[x+2, n]*Hypergeometric1F1[-n, -1-n-x, -x]; Table[CoefficientList[Series[p[x, n], {x, 0, 50}], x], {n, 0, 12}]//Flatten (* G. C. Greubel, Apr 22 2019 *)
CROSSREFS
Sequence in context: A186944 A247525 A305295 * A210865 A210861 A062073
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Mar 29 2010
EXTENSIONS
Edited by G. C. Greubel, Apr 22 2019
STATUS
approved