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A352094
Column 2 of triangle A352093.
0
1, 8, 46, 256, 1286, 6400, 30348, 142352, 652486, 2958864, 13238308, 58745872, 258345884, 1128790784, 4900778456, 21168054432, 90993848134, 389548241904, 1661387607860, 7062104700176, 29927650910900, 126478906062144, 533174476617768, 2242449423806688
OFFSET
1,2
COMMENTS
This sequence gives the coefficients of x^n*y^2 in the g.f. of triangle A352093, for n >= 1; the g.f. of triangle A352093 equals lim_{N->infinity} (1 - P(N,x,y))/(2*x)^N, where P(0,x,y) = -y, and P(n+1,x,y) = sqrt(1 - 4*x + 4*x*P(n,x,y)) for n = 0..N-1.
The two preceding columns of triangle A352093 are A351509 and A351511.
FORMULA
Sum_{n>=1} a(n)/8^n = 1/2.
EXAMPLE
G.f. A(x) = x + 8*x^2 + 46*x^3 + 256*x^4 + 1286*x^5 + 6400*x^6 + 30348*x^7 + 142352*x^8 + 652486*x^9 + 2958864*x^10 + ...
Specific values.
A(1/8) = 1/2 = 1/8 + 8/8^2 + 46/8^3 + 256/8^4 + 1286/8^5 + 6400/8^6 + ...
PROG
(PARI) {a(n) = my(A, P = -y + x*O(x^(2*n+1)));
for(i=1, n+1, P = sqrt(1 - 4*x + 4*x*P +x*O(x^(2*n+1))); );
A = subst(derivn( (1 - P)/(2*x)^(n+1) , 2, y)/2, y, 0); polcoeff(A, n, x)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 05 2022
STATUS
approved