0,3

Equals the main diagonal of symmetric square array A101515 shift right.

The binomial transform of the rows of the Hadamard 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).

a:= proc(n) option remember; local k; if n<=0 then 1 else add (binomial(n-1, k)^2 *a(k), k=0..n-1) fi end: seq (a(n), n=0..18); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 05 2008]

(PARI) a(n)=if(n==0, 1, sum(k=0, n-1, binomial(n-1, k)^2*a(k)))

Cf. A101515, A101516, A008459.

Adjacent sequences: A101511 A101512 A101513 this_sequence A101515 A101516 A101517

Sequence in context: A000154 A003713 A058129 this_sequence A111908 A060814 A129261

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 06 2004