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EXAMPLE
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The binomial transform of the rows of the term-wise product of this sequence with the rows of Pascal's triangle produces the symmetric square array A101515, in which the main diagonal equals this sequence shift left:
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),...],...
Thus the square binomial transform shifts this sequence one place left:
a(5) = 236 = 1^2*(1) + 4^2*(1) + 6^2*(2) + 4^2*(7) + 1^2*(35),
a(6) = 2037 = 1^2*(1) + 5^2*(1) + 10^2*(2) + 10^2*(7) + 5^2*(35) + 1^2*(236).
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