A156654 Triangle T(n, k) = coefficients of p(x,n), where p(x,n) = ((1-x)^(2*n+1)/x^n) * Sum_{j >= n} ( (2*j+1)^n * binomial(j, n) * x^j ), read by rows.

1, 3, 1, 25, 22, 1, 343, 515, 101, 1, 6561, 14156, 5766, 396, 1, 161051, 456197, 299342, 49642, 1447, 1, 4826809, 16985858, 15796159, 4592764, 371239, 5090, 1, 170859375, 719818759, 878976219, 383355555, 58474285, 2550165, 17481, 1, 6975757441, 34264190872, 52246537948, 31191262504, 7488334150, 660394024, 16574428, 59032, 1

Offset

0,2

Links

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

Formula

Define p(x,n) = ((1-x)^(2*n+1)/x^n) * Sum_{j >= n} ( (2*j+1)^n * binomial(j, n) * x^j ) then the triangle is defined by T(n, k) = coefficients of p(x,n) for row n and column k.

Sum_{k=0..n} T(n,k) = 2^(n-1) * n! * Catalan(n-1) = A144828(n) = A052714(n+1). - G. C. Greubel, Apr 02 2021

Example

Triangle begins as:
          1;
          3,         1;
         25,        22,         1;
        343,       515,       101,         1;
       6561,     14156,      5766,       396,        1;
     161051,    456197,    299342,     49642,     1447,       1;
    4826809,  16985858,  15796159,   4592764,   371239,    5090,     1;
  170859375, 719818759, 878976219, 383355555, 58474285, 2550165, 17481, 1;

Mathematica

p[x_, n_]:= ((1-x)^(2*n+1)/x^n)*Sum[(2*j+1)^n*Binomial[j, n]*x^j, {j, n, 2*n}];
Table[CoefficientList[Series[p[x, n], {x, 0, n}], x], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Apr 02 2021 *)

Prog

(Magma)
m:= 40; R<x>:=PowerSeriesRing(Rationals(), m);
T:= func< n | Coefficients(R!( ((1-x)^(2*n+1)/x^n)*(&+[ (2*j+1)^n*Binomial(j, n)*x^j: j in [n..m]] ) )) >;
[T(n): n in [0..12]]; // G. C. Greubel, Apr 02 2021
(Sage)
def p(n, x): return ((1-x)^(2*n+1)/x^n)*sum( (2*j+1)^n*binomial(j, n)*x^j for j in (n..2*n) )
flatten([[( p(n, x) ).series(x, n+1).list()[k] for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 02 2021

Crossrefs

Cf. A000108, A052714, A144828.

Sequence in context: A332411 A104033 A217472 * A087071 A363736 A098815

Adjacent sequences: A156651 A156652 A156653 * A156655 A156656 A156657

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 12 2009

Extensions

Edited by G. C. Greubel, Apr 02 2021

Status

approved