A362379 Convolution triangle of A052547(n).

1, 0, 1, 2, 0, 1, 1, 4, 0, 1, 5, 2, 6, 0, 1, 5, 14, 3, 8, 0, 1, 14, 14, 27, 4, 10, 0, 1, 19, 49, 27, 44, 5, 12, 0, 1, 42, 68, 113, 44, 65, 6, 14, 0, 1, 66, 175, 159, 214, 65, 90, 7, 16, 0, 1, 131, 286, 465, 304, 360, 90, 119, 8, 18, 0, 1

Offset: 0

Author

Philippe Deléham, Apr 20 2023

Keywords

• nonn
• easy
• tabl

Examples

			Triangle begins, for n>=0, 0<=k<=n :
   1 ;
   0,  1 ;
   2,  0,   1 ;
   1,  4,   0,  1 ;
   5,  2,   6,  0,  1 ;
   5, 14,   3,  8,  0,  1 ;
  14, 14,  27,  4, 10,  0,  1 ;
  19, 49,  27, 44,  5, 12,  0, 1 ;
  42, 68, 113, 44, 65,  6, 14, 0, 1 ;
  ...
		

Crossrefs

Cf. A052547, A077998 (row sums), A052964 (diagonal sums).

Formula

T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k) - T(n-2,k-1) - T(n-3,k) ; T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,1) = 0, T(2,0) = 2, T(n,k) = 0 if k<0 or if k>n .

Sum_{k = 0..n} T(n,k)*x^k = A052547(n), A077998(n), A052536(n), A052941(n) for x = 0, 1, 2, 3 respectively.

Sum_{k = 0..n} T(n,k)*2^(n-k) = A139818(n+1) = A001045(n+1)^2.