A376527 a(n) = Sum_{k=0..n*(n-1)/2} A227543(n,k)^2 for n >= 0.

1, 1, 2, 7, 34, 216, 1610, 13461, 122254, 1183568, 12054498, 127960158, 1405852350, 15901061916, 184381675404, 2184565641269, 26375002217314, 323767457670588, 4033503712929478, 50917059047932592, 650430305337318538, 8398511711996887848, 109507259507469905574, 1440631950110092280386

Offset: 0

Author

Paul D. Hanna, Oct 11 2024

Keywords

• nonn

Comments

Compare to binomial(2*n,n)/(n+1) = Sum_{k=0..n*(n-1)/2} A227543(n,k) for n >= 0; that is, the row sums of A227543 equals the Catalan numbers (A000108).

G.f. F(x,q) of triangle A227543 satisfies F(x,q) = 1 + x*F(x,q)*F(q*x,q).

Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 216*x^5 + 1610*x^6 + 13461*x^7 + 122254*x^8 + 1183568*x^9 + 12054498*x^10 + ...
where coefficient a(n) of x^n in A(x) equals the sum of the square of the terms in row n of triangle A227543, as follows.
a(0) = 1^2 = 1;
a(1) = 1^2 = 1;
a(2) = 1^2 + 1^2 = 2;
a(3) = 1^2 + 2^2 + 1^2 + 1^2 = 7;
a(4) = 1^2 + 3^2 + 3^2 + 3^2 + 2^2 + 1^2 + 1^2 = 34;
a(5) = 1^2 + 4^2 + 6^2 + 7^2 + 7^2 + 5^2 + 5^2 + 3^2 + 2^2 + 1^2 + 1^2 = 216;
a(6) = 1^2 + 5^2 + 10^2 + 14^2 + 17^2 + 16^2 + 16^2 + 14^2 + 11^2 + 9^2 + 7^2 + 5^2 + 3^2 + 2^2 + 1^2 + 1^2 = 1610;
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Links

• Paul D. Hanna, Table of n, a(n) for n = 0..301

Crossrefs

Cf. A227543.

Programs

PARI

\\ From g.f. of A227543, F(x, q) = 1 + x*F(q*x, q)*F(x, q)
{A227543(n, k) = my(F=1); for(i=1, n, F = 1 + x*F*subst(F, x, q*x) +x*O(x^n)); polcoef(polcoef(F, n, x), k, q)}
{a(n) = sum(k=0, n*(n-1)/2, A227543(n, k)^2)}
for(n=0,25, print1(a(n),", "))

PARI

\\ faster (using program by Joerg Arndt in A227543)
N=30;
VP=vector(N+1); VP[1] = VP[2] = 1;  \\ one-based; memoization
P(n) = VP[n+1];
for (n=2, N, VP[n+1] = sum( i=0, n-1, P(i) * P(n-1 -i) * x^((i+1)*(n-1-i)) );print1(n,",") );
for(n=0,N, AV=Vec(P(n)); print1(AV*AV~,", "))

Formula

a(n) ~ c * 16^n / n^(9/2), where c = 0.430217025951475334005904244213062400539... - Vaclav Kotesovec, Oct 11 2024