OFFSET
0,2
COMMENTS
In general, if m > 0 and g.f. = 1/(Product_{k=0..m-1} (1 - (m*k+1)*x))^(1/m), then a(n) ~ (m*(m-1) + 1)^(n + 1 - 1/m) / (Gamma(1/m) * Gamma(m+1)^(1/m) * m^(1 - 2/m) * n^(1 - 1/m)). - Vaclav Kotesovec, Aug 18 2025
LINKS
Robert Israel, Table of n, a(n) for n = 0..488
FORMULA
a(n) ~ 3^(n + 6/11) * 37^(n + 10/11) / (Gamma(1/11) * 2^(8/11) * 5^(2/11) * 7^(1/11) * 11^(10/11) * n^(10/11)). - Vaclav Kotesovec, May 12 2025
D-finite with recurrence: (122087424094272000 + 122087424094272000*n)*a(n) + (-292345718334707520 - 153133471508656320*n)*a(n + 1) + (96689928639812256 + 34309329517352736*n)*a(n + 2) + (-12869797460508636 - 3452872489404756*n)*a(n + 3) + (910607674046280 + 196405576755080*n)*a(n + 4) + (-38678676132205 - 6974843236955*n)*a(n + 5) + (1046517405528 + 162136499448*n)*a(n + 6) + (-18475647453 - 2509038543*n)*a(n + 7) + (211968120 + 25622520*n)*a(n + 8) + (-1522575 - 165825*n)*a(n + 9) + (6216 + 616*n)*a(n + 10) + (-n - 11)*a(n + 11) = 0. - Robert Israel, Mar 12 2026
MAPLE
f:= gfun:-rectoproc({(122087424094272000 + 122087424094272000*n)*a(n) + (-292345718334707520 - 153133471508656320*n)*a(n + 1) + (96689928639812256 + 34309329517352736*n)*a(n + 2) + (-12869797460508636 - 3452872489404756*n)*a(n + 3) + (910607674046280 + 196405576755080*n)*a(n + 4) + (-38678676132205 - 6974843236955*n)*a(n + 5) + (1046517405528 + 162136499448*n)*a(n + 6) + (-18475647453 - 2509038543*n)*a(n + 7) + (211968120 + 25622520*n)*a(n + 8) + (-1522575 - 165825*n)*a(n + 9) + (6216 + 616*n)*a(n + 10) + (-n - 11)*a(n + 11), a(0) = 1, a(1) = 56, a(2) = 3741, a(3) = 277256, a(4) = 22052713, a(5) = 1846878936, a(6) = 160878051401, a(7) = 14454374710216, a(8) = 1331486959280259, a(9) = 125190717874655720, a(10) = 11973642784650273211}, a(n), remember):
map(f, [$0..20]); # Robert Israel, Mar 12 2026
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(1/prod(k=0, 10, 1-(11*k+1)*x)^(1/11))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 03 2025
STATUS
approved