A178126 Triangle T(n, k) = coefficients of (n+1)!*(binomial(x+n+1, n+1) - binomial(x, n+1)), read by rows.

1, 2, 4, 6, 9, 9, 24, 56, 24, 16, 120, 250, 275, 50, 25, 720, 1884, 1350, 960, 90, 36, 5040, 12348, 14896, 5145, 2695, 147, 49, 40320, 114624, 105056, 80416, 15680, 6496, 224, 64, 362880, 986256, 1282284, 605556, 336609, 40824, 13986, 324, 81

Offset

0,2

References

Brendan Hassett, Introduction to algebraic Geometry, Cambridge University Press, New York, 2007, page 214

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = coefficients of n!*(binomial(x+n+1, n+1) - binomial(x, n+1)).

From G. C. Greubel, Apr 14 2021: (Start)

T(n, k) = coefficients of Sum_{j=0..n+1} Stirling1(n+1, j)*( (x+n+1)^j - x^j ).

T(n, 0) = (n+1)!.

T(n, n) = (n+1)^2.

Sum_{k=0..n} T(n,k) = (n+2)! - [n=0]. (End)

Example

Triangle begins as:
        1;
        2,        4;
        6,        9,        9;
       24,       56,       24,      16;
      120,      250,      275,      50,      25;
      720,     1884,     1350,     960,      90,      36;
     5040,    12348,    14896,    5145,    2695,     147,    49;
    40320,   114624,   105056,   80416,   15680,    6496,   224,    64;
   362880,   986256,  1282284,  605556,  336609,   40824, 13986,   324,  81;
  3628800, 10991520, 11727000, 9582200, 2693250, 1171380, 94500, 27600, 450, 100;

Mathematica

T[n_, k_]:= SeriesCoefficient[Series[Sum[StirlingS1[n+1, j]*((x+n+1)^j -x^j), {j, 0, n+1}], {x, 0, n+1}], k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 14 2021 *)

Prog

(Sage)
def T(n, k): return ( sum((-1)^(n+j+1)*stirling_number1(n+1, j)*((x+n+1)^j - x^j) for j in (0..n+1)) ).series(x, n+1).list()[k]
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 14 2021

Crossrefs

Cf. A048994, A139167.

Sequence in context: A084407 A114526 A333387 * A359515 A162202 A210380

Adjacent sequences: A178123 A178124 A178125 * A178127 A178128 A178129

Keyword

nonn,tabl

Author

Roger L. Bagula, May 20 2010

Extensions

Edited by G. C. Greubel, Apr 14 2021

Status

approved