A101514 - OEIS

%I #29 Jul 07 2017 08:11:32

%S 1,1,2,7,35,236,2037,21695,277966,4198635,73558135,1475177880,

%T 33495959399,853167955357,24182881926558,757554068775721,

%U 26068954296880361,980202973852646786,40079727064364154465,1774594774575753650941,84756211791797266285252

%N Shifts one place left under the square binomial transform (A008459): a(0) = 1, a(n) = Sum_{k=0..n-1} C(n-1,k)^2*a(k).

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

%C Empirical: a(n) = sum((number of standard immaculate tableaux of shape m)^2, m|=n), where this sum is over all compositions m of n > 0. - _John M. Campbell_, Jul 07 2017

%H Vaclav Kotesovec, <a href="/A101514/b101514.txt">Table of n, a(n) for n = 0..330</a>

%F E.g.f. satisfies: B(x)/A(x) = Sum_{n>=0} x^n/n!^2 where A(x) = Sum_{n>=0} a(n)*x^n/n!^2 and B(x) = Sum_{n>=0} a(n+1)*x^n/n!^2. - _Paul D. Hanna_, Oct 10 2014

%e 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:

%e BINOMIAL[1*1] = [(1),1,1,1,1,1,1,1,1,...],

%e BINOMIAL[1*1,1*1] = [1,(2),3,4,5,6,7,8,9,...],

%e BINOMIAL[1*1,1*2,2*1] = [1,3,(7),13,21,31,43,57,73,...],

%e BINOMIAL[1*1,1*3,2*3,7*1] = [1,4,13,(35),77,146,249,393,...],

%e BINOMIAL[1*1,1*4,2*6,7*4,35*1] = [1,5,21,77,(236),596,1290,...],

%e BINOMIAL[1*1,1*5,2*10,7*10,35*5,236*1] = [1,6,31,146,596,(2037),...],...

%e Thus the square binomial transform shifts this sequence one place left:

%e a(5) = 236 = 1^2*(1) + 4^2*(1) + 6^2*(2) + 4^2*(7) + 1^2*(35),

%e a(6) = 2037 = 1^2*(1) + 5^2*(1) + 10^2*(2) + 10^2*(7) + 5^2*(35) + 1^2*(236).

%p a:= proc(n) option remember; if n<=0 then 1 else

%p       add(binomial(n-1,k)^2 *a(k), k=0..n-1) fi

%p     end:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 05 2008

%t a[0] = 1;

%t a[n_] := Sum[Binomial[n - 1, k]^2 a[k], {k, 0, n - 1}];

%t Table[a[i], {i, 0, 20}] (* _Philip B. Zhang_, Oct 10 2014 *)

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

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A008459, A101515, A101516.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 06 2004

%E Typo in definition corrected by _Philip B. Zhang_, Oct 10 2014