A176668 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial sum_{k=0..infinity} (2*k+1)^n*binomial(x,k) / 2^x.

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 8, 21, 10, 1, 1, 5, 45, 55, 15, 1, 1, 7, 30, 185, 120, 21, 1, 1, 70, -77, 245, 595, 231, 28, 1, 1, 72, 490, -756, 1435, 1596, 406, 36, 1, 1, -1311, 3762, -546, -2625, 6111, 3738, 666, 45, 1, 1, -1309, -11325, 35130, -20895, -1743, 20685

Offset: 0

Author

Roger L. Bagula, Apr 23 2010

Keywords

• sign
• tabl

Comments

Row sums are A007051(n).

Exponential Riordan array [exp(x), log((exp(2x)+1)/2)]=[exp(x),x+log(cosh(x))]. - Paul Barry Jan 10 2011

Examples

			1;
1, 1;
1, 3, 1;
1, 6, 6, 1;
1, 8, 21, 10, 1;
1, 5, 45, 55, 15, 1;
1, 7, 30, 185, 120, 21, 1;
1, 70, -77, 245, 595, 231, 28, 1;
1, 72, 490, -756, 1435, 1596, 406, 36, 1;
1, -1311, 3762, -546, -2625, 6111, 3738, 666, 45, 1;
1, -1309, -11325, 35130, -20895, -1743, 20685, 7890, 1035, 55, 1;
Production matrix begins
1, 1,
0, 2, 1,
0, -1, 3, 1,
0, 1, -3, 4, 1,
0, -1, 4, -6, 5, 1,
0, 1, -5, 10, -10, 6, 1,
0, -1, 6, -15, 20, -15, 7, 1,
0, 1, -7, 21, -35, 35, -21, 8, 1,
0, -1, 8, -28, 56, -70, 56, -28, 9, 1
- _Paul Barry_ Jan 10 2011
		

Crossrefs

Cf. A007051, A009393.

Programs

Maple

A176668 := proc(n,k) sum( (2*l+1)^n*binomial(x,l),l=0..infinity) ; simplify(%/2^x) ; coeftayl(%,x=0,k) ; end proc: # R. J. Mathar, Jan 15 2011

Mathematica

p[x_, n_] = Sum[(2*k + 1)^n*Binomial[x, k], {k, 0, Infinity}]/2^x ;
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]

Formula

From Peter Bala, Mar 16 2012. (Start)

The row polynomials of this triangle may be obtained by applying the operator x*d/dx repeatedly to x*(1+x^2)^n = sum {k = 0..n} binomial(n,k)*x^(2*k+1) and evaluating the result at x = 1. The first few results are:

sum {k = 0..n} (2*k+1)*binomial(n,k) = (n+1)*2^n

sum {k = 0..n} (2*k+1)^2*binomial(n,k) = (n^2+3*n^+1)*2^n

sum {k = 0..n} (2*k+1)^3*binomial(n,k) = (n^3+6*n^2+6*n+1)*2^n.

Compare with A209849. (End)