cp's OEIS Frontend

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

%I A050165 #26 Jan 24 2020 03:26:54
%S A050165 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,
%T A050165 132,1,13,77,273,637,1001,1001,429,1,15,104,440,1260,2548,3640,3432,
%U A050165 1430,1,17,135,663,2244,5508,9996,13260,11934
%N A050165 Triangle read by rows: T(n,k) = M(2n+1,k,-1), 0 <= k <= n, n >= 0, array M as in A050144.
%C A050165 T is a mirror image of the array in A039599.
%H A050165 M. W. Coffey, M. C. Lettington, <a href="http://arxiv.org/abs/1510.05402">On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat</a>, arXiv:1510.05402 [math.NT], 2015. See Section 4.
%F A050165 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
%F A050165 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
%F A050165 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
%F A050165 T(n,k)= Sum_{j>=0} (-1)^(n-j)*A094385(n,j)*binomial(j,k). - _Philippe Deléham_, May 05 2007
%F A050165 T(2n,n) = A126596(n). - _Philippe Deléham_, Nov 23 2011
%e A050165 Triangle begins:
%e A050165   1;
%e A050165   1,  1;
%e A050165   1,  3,  2;
%e A050165   1,  5,  9,   5;
%e A050165   1,  7, 20,  28,  14;
%e A050165   1,  9, 35,  75,  90,  42;
%e A050165   1, 11, 54, 154, 275, 297, 132;
%Y A050165 Cf. A039599, A084938.
%K A050165 nonn,tabl
%O A050165 0,5
%A A050165 _Clark Kimberling_