login
A316591
a(n) equals the coefficient of x^n in Sum_{m>=0} (x^m + 1 + 1/x^m)^m for n >= 1.
8
1, 2, 6, 17, 45, 129, 357, 1026, 2908, 8380, 24068, 69670, 201643, 585956, 1704525, 4969937, 14508939, 42424376, 124191258, 363992450, 1067892560, 3136066153, 9217554129, 27114308763, 79818194926, 235128636132, 693085100406, 2044218141126, 6032675068061, 17812285026651, 52618949607541, 155512377987938, 459807788455686, 1360083815164974, 4024604728349478, 11913518538574370
OFFSET
1,2
COMMENTS
The coefficient of 1/x^n in Sum_{m>=0} (x^m + 1 + 1/x^m)^m equals a(n) for n > 0, while the constant term in the sum increases without limit.
a(n) = Sum_{k=0..n-1} A316590(n,k) for n >= 1; equals row sums of triangle A316590.
LINKS
FORMULA
a(n) ~ 3^(n + 1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Jul 10 2018
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 6*x^3 + 17*x^4 + 45*x^5 + 129*x^6 + 357*x^7 + 1026*x^8 + 2908*x^9 + 8380*x^10 + 24068*x^11 + 69670*x^12 + ...
such that Sum_{m>=0} (x^m + 1 + 1/x^m)^m = A(x) + A(1/x) + (infinity)*x^0.
PROG
(PARI) {a(n) = polcoeff( sum(m=1, n, (x^-m + 1 + x^m)^m +x*O(x^n)), n, x)}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 08 2018
STATUS
approved