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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). 3

%I

%S 1,2,2,6,7,1,24,32,8,120,178,61,3,720,1164,494,50,5040,8748,4348,655,

%T 15,40320,74304,41768,8204,420,362880,704016,437148,104272,8365,105,

%U 3628800,7362720,4965912,1376864,149282,4410,39916800,84255840,60961176,19079836,2580550,123795,945

%N 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).

%C Row sums are: {1, 4, 14, 64, 362, 2428, 18806, 165016, 1616786, 17487988, 206918942, ...}.

%H G. C. Greubel, <a href="/A174789/b174789.txt">Rows n = 0..25 of triangle, flattened</a>

%e Triangle begins as:

%e 1;

%e 2, 2;

%e 6, 7, 1;

%e 24, 32, 8;

%e 120, 178, 61, 3;

%e 720, 1164, 494, 50;

%e 5040, 8748, 4348, 655, 15;

%e 40320, 74304, 41768, 8204, 420;

%e 362880, 704016, 437148, 104272, 8365, 105;

%e 3628800, 7362720, 4965912, 1376864, 149282, 4410;

%t 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

%t 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 *)

%K nonn,tabl

%O 0,2

%A _Roger L. Bagula_, Mar 29 2010

%E Edited by _G. C. Greubel_, Apr 22 2019

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Last modified December 7 07:00 EST 2022. Contains 358649 sequences. (Running on oeis4.)