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
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.
Programs
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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
Comments