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
Examples
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;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
-
Magma
m:= 40; R
:=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 -
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 *) -
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
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
Extensions
Edited by G. C. Greubel, Apr 02 2021