A050165 Triangle read by rows: T(n,k) = M(2n+1,k,-1), 0 <= k <= n, n >= 0, array M as in A050144.

1, 1, 1, 1, 3, 2, 1, 5, 9, 5, 1, 7, 20, 28, 14, 1, 9, 35, 75, 90, 42, 1, 11, 54, 154, 275, 297, 132, 1, 13, 77, 273, 637, 1001, 1001, 429, 1, 15, 104, 440, 1260, 2548, 3640, 3432, 1430, 1, 17, 135, 663, 2244, 5508, 9996, 13260, 11934

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

T is a mirror image of the array in A039599.

Examples

			Triangle begins:
  1;
  1,  1;
  1,  3,  2;
  1,  5,  9,   5;
  1,  7, 20,  28,  14;
  1,  9, 35,  75,  90,  42;
  1, 11, 54, 154, 275, 297, 132;
		

Links

• M. W. Coffey, M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015. See Section 4.

Crossrefs

Cf. A039599, A084938.

Formula

Triangle T(n, k) read by rows; given by [1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938. T(n, k) = C(2n, k)*(2n-2k+1)/(2n-k+1). - Philippe Deléham, Dec 07 2003

Sum_{k=0..min(m, n)} T(m, m-k)*T(n, n-k) = A000108(m+n); A000108: Catalan numbers. - Philippe Deléham, Dec 30 2003

T(n, k) = 0 if n < k, T(n, n)= A000108(n) and for n > k: T(n, k) = Sum_{j=0..k} T(n-1-j, k-j)*A000108(j+1). - Philippe Deléham, Feb 03 2004

T(n,k)= Sum_{j>=0} (-1)^(n-j)*A094385(n,j)*binomial(j,k). - Philippe Deléham, May 05 2007

T(2n,n) = A126596(n). - Philippe Deléham, Nov 23 2011