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).

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, ...}.

Links

G. C. Greubel, Rows n = 0..25 of triangle, flattened

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

Adjacent sequences: A174786 A174787 A174788 * A174790 A174791 A174792

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 29 2010

Extensions

Edited by G. C. Greubel, Apr 22 2019

Status

approved