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A101514 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). 6
1, 1, 2, 7, 35, 236, 2037, 21695, 277966, 4198635, 73558135, 1475177880, 33495959399, 853167955357, 24182881926558, 757554068775721, 26068954296880361, 980202973852646786, 40079727064364154465, 1774594774575753650941, 84756211791797266285252 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

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

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..330

FORMULA

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

EXAMPLE

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).

MAPLE

a:= proc(n) option remember; 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..25);  # Alois P. Heinz, Sep 05 2008

MATHEMATICA

a[0] = 1;

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

Table[a[i], {i, 0, 20}] (* Philip B. Zhang, Oct 10 2014 *)

PROG

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

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

CROSSREFS

Cf. A008459, A101515, A101516.

Sequence in context: A000154 A003713 A058129 * A247240 A266356 A196857

Adjacent sequences:  A101511 A101512 A101513 * A101515 A101516 A101517

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 06 2004

EXTENSIONS

Typo in definition corrected by Philip B. Zhang, Oct 10 2014

STATUS

approved

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Last modified December 2 07:33 EST 2021. Contains 349437 sequences. (Running on oeis4.)