cp's OEIS Frontend

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.

%I A101515 #5 Jun 13 2017 22:17:09
%S A101515 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,
%T A101515 77,77,31,7,1,1,8,43,146,236,146,43,8,1,1,9,57,249,596,596,249,57,9,1,
%U A101515 1,10,73,393,1290,2037,1290,393,73,10,1,1,11,91,585,2486,5772,5772,2486,585
%N 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.
%C A101515 The main diagonal equals A101514 shift one place left. The antidiagonal sums form A101516.
%e A101515 Rows begin:
%e A101515 [_1,1,1,1,1,1,1,1,1,...],
%e A101515 [1,_2,3,4,5,6,7,8,9,...],
%e A101515 [1,3,_7,13,21,31,43,57,73,...],
%e A101515 [1,4,13,_35,77,146,249,393,585,...],
%e A101515 [1,5,21,77,_236,596,1290,2486,4387,...],
%e A101515 [1,6,31,146,596,_2037,5772,13987,29987,...],
%e A101515 [1,7,43,249,1290,5772,_21695,67943,181811,...],
%e A101515 [1,8,57,393,2486,13987,67943,_277966,951051,...],
%e A101515 [1,9,73,585,4387,29987,181811,951051,_4198635,...],...
%e A101515 The inverse binomial transform of the rows of this array are generated
%e A101515 from the products of the main diagonal with rows of Pascal's triangle:
%e A101515 BINOMIAL[1*1] = [_1,1,1,1,1,1,1,1,1,...],
%e A101515 BINOMIAL[1*1,1*1] = [1,_2,3,4,5,6,7,8,9,...],
%e A101515 BINOMIAL[1*1,1*2,2*1] = [1,3,_7,13,21,31,43,57,73,...],
%e A101515 BINOMIAL[1*1,1*3,2*3,7*1] = [1,4,13,_35,77,146,249,393,...],
%e A101515 BINOMIAL[1*1,1*4,2*6,7*4,35*1] = [1,5,21,77,_236,596,1290,...],
%e A101515 BINOMIAL[1*1,1*5,2*10,7*10,35*5,236*1] = [1,6,31,146,596,_2037,...],...
%o A101515 (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));)))
%Y A101515 Cf. A101514, A101516.
%K A101515 nonn,tabl
%O A101515 0,5
%A A101515 _Paul D. Hanna_, Dec 06 2004