OFFSET
0,15
COMMENTS
Number of partitions into distinct parts of the form 4*k+1.
In general, if a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} (1 + x^(a*k + b)), then a(n) ~ exp(Pi*sqrt(n/(3*a))) / (2^(1 + b/a) * (3*a)^(1/4) * n^(3/4)) [Meinardus, 1954]. - Vaclav Kotesovec, Aug 26 2015
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Günter Meinardus, Über Partitionen mit Differenzenbedingungen, Mathematische Zeitschrift (1954/55), Volume: 61, page 289-302
FORMULA
G.f.: Sum_{n>=0} (x^(2*n^2 - n) / Product_{k=1..n} (1 - x^(4*k))). - Joerg Arndt, Mar 10 2011
G.f.: G(0)/x where G(k) = 1 - 1/(1 - 1/(1 - 1/(1+(x)^(4*k+1))/G(k+1) )); (recursively defined continued fraction, see A006950). - Sergei N. Gladkovskii, Jan 28 2013
a(n) ~ exp(Pi*sqrt(n)/(2*sqrt(3))) / (2^(7/4) * 3^(1/4) * n^(3/4)) * (1 - (3*sqrt(3)/(4*Pi) + Pi/(192*sqrt(3))) / sqrt(n)). - Vaclav Kotesovec, Aug 26 2015, extended Jan 18 2017
MATHEMATICA
nmax = 200; CoefficientList[Series[Product[(1 + x^(4*k+1)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 4] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Aug 29 2010
STATUS
approved