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A102573 Triangle of coefficients of polynomials in Sum_{k=0..n} binomial(n,k)*k^r. 6

%I #12 Dec 16 2012 22:35:21

%S 1,1,3,1,5,-2,1,10,15,-10,1,14,31,-46,16,1,21,105,35,-210,112,1,27,

%T 183,97,-832,860,-272,1,36,378,1008,-1575,-2436,5292,-2448,1,44,586,

%U 2144,-3719,-10876,31036,-26896,7936,1,55,990,6270,3465,-51513,27720,135300,-208560

%N Triangle of coefficients of polynomials in Sum_{k=0..n} binomial(n,k)*k^r.

%C For a table of coefficients of these polynomials without factors removed see A209849. - Peter Bala, Mar 16 2012

%D E. Kilic, Y. T. Ulutas and N. Omur, Formulas for weighted binomial sums using the powers of terms of binary recurrences, Miskolc Mathematical Notes, Vol. 13 (2012), No. 1, pp. 53-65. - From _N. J. A. Sloane_, Dec 16 2012

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialSums.html">Binomial Sums</a>

%e 1;

%e 1, 3;

%e 1, 5, -2;

%e 1, 10, 15, -10;

%e 1, 14, 31, -46, 16;

%e ...

%e E.g. Sum[binomial[n,k]k^4,{k,0,n}] = 2^(-4 + n)*n*(1 + n)*(-2 + 5*n + n^2)

%Y A209849.

%K sign,tabl

%O 2,3

%A _Eric W. Weisstein_, Jan 15 2005

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