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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
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
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
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 12 2009
EXTENSIONS
Edited by G. C. Greubel, Apr 02 2021
STATUS
approved