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A101515
Symmetric square array, read by antidiagonals, such that the inverse binomial transform of row n forms the sequence: {C(n,k)*A101514(k), 0<=k<=n}, where A101514 equals the main diagonal shift right.
2
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 21, 35, 21, 6, 1, 1, 7, 31, 77, 77, 31, 7, 1, 1, 8, 43, 146, 236, 146, 43, 8, 1, 1, 9, 57, 249, 596, 596, 249, 57, 9, 1, 1, 10, 73, 393, 1290, 2037, 1290, 393, 73, 10, 1, 1, 11, 91, 585, 2486, 5772, 5772, 2486, 585
OFFSET
0,5
COMMENTS
The main diagonal equals A101514 shift one place left. The antidiagonal sums form A101516.
EXAMPLE
Rows begin:
[_1,1,1,1,1,1,1,1,1,...],
[1,_2,3,4,5,6,7,8,9,...],
[1,3,_7,13,21,31,43,57,73,...],
[1,4,13,_35,77,146,249,393,585,...],
[1,5,21,77,_236,596,1290,2486,4387,...],
[1,6,31,146,596,_2037,5772,13987,29987,...],
[1,7,43,249,1290,5772,_21695,67943,181811,...],
[1,8,57,393,2486,13987,67943,_277966,951051,...],
[1,9,73,585,4387,29987,181811,951051,_4198635,...],...
The inverse binomial transform of the rows of this array are generated
from the products of the main diagonal with rows of Pascal's triangle:
BINOMIAL[1*1] = [_1,1,1,1,1,1,1,1,1,...],
BINOMIAL[1*1,1*1] = [1,_2,3,4,5,6,7,8,9,...],
BINOMIAL[1*1,1*2,2*1] = [1,3,_7,13,21,31,43,57,73,...],
BINOMIAL[1*1,1*3,2*3,7*1] = [1,4,13,_35,77,146,249,393,...],
BINOMIAL[1*1,1*4,2*6,7*4,35*1] = [1,5,21,77,_236,596,1290,...],
BINOMIAL[1*1,1*5,2*10,7*10,35*5,236*1] = [1,6,31,146,596,_2037,...],...
PROG
(PARI) T(n, k)=if(n<0 || k<0, 0, if(n==0 || k==0, 1, if(n>k, T(k, n), 1+sum(j=1, k, binomial(k, j)*binomial(n, j)*T(j-1, j-1)); )))
CROSSREFS
Sequence in context: A086617 A094526 A088699 * A327913 A028657 A053534
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Dec 06 2004
STATUS
approved