Search: id:A101514
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%I A101514
%S A101514 1,1,2,7,35,236,2037,21695,277966,4198635,73558135,1475177880,
%T A101514 33495959399,853167955357,24182881926558,757554068775721,
%U A101514 26068954296880361,980202973852646786,40079727064364154465
%N A101514 Shifts one place left under the square binomial transform (A008459): 
               a(0) = 1, a(n+1) = Sum_{k=0..n-1} C(n-1,k)^2*a(k).
%C A101514 Equals the main diagonal of symmetric square array A101515 shift right.
%e A101514 The binomial transform of the rows of the Hadamard product of this sequence
%e A101514 with the rows of Pascal's triangle produces the symmetric square array
%e A101514 A101515, in which the main diagonal equals this sequence shift left:
%e A101514 BINOMIAL[1*1] = [_1,1,1,1,1,1,1,1,1,...],
%e A101514 BINOMIAL[1*1,1*1] = [1,_2,3,4,5,6,7,8,9,...],
%e A101514 BINOMIAL[1*1,1*2,2*1] = [1,3,_7,13,21,31,43,57,73,...],
%e A101514 BINOMIAL[1*1,1*3,2*3,7*1] = [1,4,13,_35,77,146,249,393,...],
%e A101514 BINOMIAL[1*1,1*4,2*6,7*4,35*1] = [1,5,21,77,_236,596,1290,...],
%e A101514 BINOMIAL[1*1,1*5,2*10,7*10,35*5,236*1] = [1,6,31,146,596,_2037,...],...
%e A101514 Thus the square binomial transform shifts this sequence one place left:
%e A101514 a(5) = 236 = 1^2*(1) + 4^2*(1) + 6^2*(2) + 4^2*(7) + 1^2*(35),
%e A101514 a(6) = 2037 = 1^2*(1) + 5^2*(1) + 10^2*(2) + 10^2*(7) + 5^2*(35) + 1^2*(236).
%p A101514 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]
%o A101514 (PARI) a(n)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)^2*a(k)))
%Y A101514 Cf. A101515, A101516, A008459.
%Y A101514 Sequence in context: A000154 A003713 A058129 this_sequence A111908 A060814 
               A129261
%Y A101514 Adjacent sequences: A101511 A101512 A101513 this_sequence A101515 A101516 
               A101517
%K A101514 nonn
%O A101514 0,3
%A A101514 Paul D. Hanna (pauldhanna(AT)juno.com), Dec 06 2004

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