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

1, 1, 3, 1, 5, -2, 1, 10, 15, -10, 1, 14, 31, -46, 16, 1, 21, 105, 35, -210, 112, 1, 27, 183, 97, -832, 860, -272, 1, 36, 378, 1008, -1575, -2436, 5292, -2448, 1, 44, 586, 2144, -3719, -10876, 31036, -26896, 7936, 1, 55, 990, 6270, 3465, -51513, 27720, 135300, -208560

Offset: 2

Author

Eric W. Weisstein, Jan 15 2005

Keywords

• sign
• tabl

Comments

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

Examples

			Triangle begins:
  1;
  1, 3;
  1, 5, -2;
  1, 10, 15, -10;
  1, 14, 31, -46, 16;
  ...
E.g. Sum_{k=0..n} binomial(n,k) * k^4 = 2^(n-4) * n * (n+1) * (n^2 + 5*n - 2).
		

References

• 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

Links

• Eric Weisstein's World of Mathematics, Binomial Sums

Crossrefs

Cf. A209849.