cp's OEIS Frontend

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

%I A376527 #14 Oct 11 2024 16:03:03
%S A376527 1,1,2,7,34,216,1610,13461,122254,1183568,12054498,127960158,
%T A376527 1405852350,15901061916,184381675404,2184565641269,26375002217314,
%U A376527 323767457670588,4033503712929478,50917059047932592,650430305337318538,8398511711996887848,109507259507469905574,1440631950110092280386
%N A376527 a(n) = Sum_{k=0..n*(n-1)/2} A227543(n,k)^2 for n >= 0.
%C A376527 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).
%C A376527 G.f. F(x,q) of triangle A227543 satisfies F(x,q) = 1 + x*F(x,q)*F(q*x,q).
%C A376527 Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0.
%H A376527 Paul D. Hanna, <a href="/A376527/b376527.txt">Table of n, a(n) for n = 0..301</a>
%F A376527 a(n) ~ c * 16^n / n^(9/2), where c = 0.430217025951475334005904244213062400539... - _Vaclav Kotesovec_, Oct 11 2024
%e A376527 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 + ...
%e A376527 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.
%e A376527 a(0) = 1^2 = 1;
%e A376527 a(1) = 1^2 = 1;
%e A376527 a(2) = 1^2 + 1^2 = 2;
%e A376527 a(3) = 1^2 + 2^2 + 1^2 + 1^2 = 7;
%e A376527 a(4) = 1^2 + 3^2 + 3^2 + 3^2 + 2^2 + 1^2 + 1^2 = 34;
%e A376527 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;
%e A376527 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;
%e A376527 ...
%o A376527 (PARI) \\ From g.f. of A227543, F(x, q) = 1 + x*F(q*x, q)*F(x, q)
%o A376527 {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)}
%o A376527 {a(n) = sum(k=0, n*(n-1)/2, A227543(n, k)^2)}
%o A376527 for(n=0,25, print1(a(n),", "))
%o A376527 (PARI) \\ faster (using program by Joerg Arndt in A227543)
%o A376527 N=30;
%o A376527 VP=vector(N+1); VP[1] = VP[2] = 1;  \\ one-based; memoization
%o A376527 P(n) = VP[n+1];
%o A376527 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,",") );
%o A376527 for(n=0,N, AV=Vec(P(n)); print1(AV*AV~,", "))
%Y A376527 Cf. A227543.
%K A376527 nonn
%O A376527 0,3
%A A376527 _Paul D. Hanna_, Oct 11 2024