A101515 -id:A101515 - cp's OEIS frontend

A101516 Antidiagonal sums of symmetric square array A101515 and also equals the binomial transform of a sequence formed from terms of A101514 repeated twice.

1, 2, 4, 8, 17, 38, 91, 232, 632, 1824, 5571, 17892, 60355, 212898, 784416, 3008480, 11997341, 49612426, 212536067, 941213428, 4305049140, 20302469824, 98641434683, 493038167880, 2533414749409, 13366134856170, 72361098996208

Offset: 0

Paul D. Hanna, Dec 06 2004

nonn

A101514 equals the main diagonal of A101515 shift one place right and also A101514 shifts one place left under the square binomial transform (A008459): A101514(n+1) = Sum_{k=0..n-1} C(n-1,k)^2*A101514(k).

			Given A101514 = [1,1,2,7,35,236,2037,21695,277966,4198635,...],
the binomial transform of A101514 terms repeated twice returns this sequence:
BINOMIAL[1,1,1,1,2,2,7,7,35,35,...] = [1,2,4,8,17,38,91,232,632,1824,...].

Cf. A101514, A101515.

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

G.f.: A(x) = G101514(x^2/(1-x)^2)/(1-x)^2, where G101514(x)= g.f. of A101514. a(n) = Sum_{k=0..n} C(n, k)*A101514([k/2]).

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).

Original entry on oeis.org

1, 1, 2, 7, 35, 236, 2037, 21695, 277966, 4198635, 73558135, 1475177880, 33495959399, 853167955357, 24182881926558, 757554068775721, 26068954296880361, 980202973852646786, 40079727064364154465, 1774594774575753650941, 84756211791797266285252

Offset: 0

Paul D. Hanna, Dec 06 2004

nonn

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

			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).

Vaclav Kotesovec, Table of n, a(n) for n = 0..330
John M. Campbell, Semisimple algebras related to immaculate tableaux, arXiv:2507.02539 [math.CO], 2025. See p. 7.

Cf. A008459, A101515, A101516.

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

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 *)

{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),", "))

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

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

cp's OEIS Frontend

A101516 Antidiagonal sums of symmetric square array A101515 and also equals the binomial transform of a sequence formed from terms of A101514 repeated twice.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula