A211183 Triangle T(n,k), 0<=k<=n, read by rows, given by (0, 1, 1, 3, 3, 6, 6, 10, 10, 15, ...) DELTA (1, 0, 2, 0, 3, 0, 4, 0, 5, ...) where DELTA is the operator defined in A084938.

1, 0, 1, 0, 1, 1, 0, 2, 4, 1, 0, 7, 19, 11, 1, 0, 38, 123, 107, 26, 1, 0, 295, 1076, 1195, 474, 57, 1, 0, 3098, 12350, 16198, 8668, 1836, 120, 1, 0, 42271, 180729, 268015, 176091, 52831, 6549, 247, 1, 0, 726734, 3290353, 5369639, 4105015, 1564817, 287473, 22145

Offset: 0

Author

Philippe Deléham, Feb 02 2013

Keywords

• nonn
• tabl

Examples

			Triangle begins :
1;
0, 1;
0, 1, 1;
0, 2, 4, 1;
0, 7, 19, 11, 1;
0, 38, 123, 107, 26, 1;
0, 295, 1076, 1195, 474, 57, 1;
0, 3098, 12350, 16198, 8668, 1836, 120, 1;
0, 42271, 180729, 268015, 176091, 52831, 6549, 247, 1;
0, 726734, 3290353, 5369639, 4105015, 1564817, 287473, 22145, 502, 1; ...
		

Links

• Paul D. Hanna, Rows n = 0..31, flattened.

Crossrefs

Cf. A000366, A110501, A211194, A221972.

Programs

PARI

T(n,k)=polcoeff(polcoeff(sum(m=0, n, m!*x^m*prod(k=1, m, (y + (k-1)/2)/(1+(k*y+k*(k-1)/2)*x+x*O(x^n)))), n,x),k,y)
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Feb 03 2013

Formula

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k) = A000012(n), A000366(n+1), A110501(n+1), A211194(n), A221972(n) for x = 0, 1, 2, 3, 4 respectively.

T(n,n-1) = A000295(n).

T(n,1) = A000366(n).

G.f.: A(x,y) = Sum_{n>=0} n! * x^n * Product_{k=1..n} (y + (k-1)/2) / (1 + (k*y + k*(k-1)/2)*x). - Paul D. Hanna, Feb 03 2013