|
%I #37 Jan 07 2021 01:19:09
%S 1,1,0,0,0,1,1,0,0,1,1,0,0,1,2,1,0,1,2,1,0,1,3,2,0,1,3,3,1,1,4,4,1,1,
%T 4,5,2,1,5,7,3,1,5,8,5,2,6,10,6,2,6,12,9,3,7,14,11,4,7,16,15,6,8,19,
%U 18,8,9,21,23,11,10,24,27,14,11,27,34,19,13,30,39,24,15,33,47,31,18,37,54,38
%N Expansion of Product_{i>=0} (1 + x^(4*i+1)).
%C Number of partitions into distinct parts of the form 4*k+1.
%C 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
%C Convolution of A147599 and A169975 is A000700. - _Vaclav Kotesovec_, Jan 18 2017
%H Vaclav Kotesovec, <a href="/A169975/b169975.txt">Table of n, a(n) for n = 0..10000</a>
%H Günter Meinardus, <a href="https://eudml.org/doc/169463">Über Partitionen mit Differenzenbedingungen</a>, Mathematische Zeitschrift (1954/55), Volume: 61, page 289-302
%F G.f.: Sum_{n>=0} (x^(2*n^2 - n) / Product_{k=1..n} (1 - x^(4*k))). - _Joerg Arndt_, Mar 10 2011
%F 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
%F 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
%t nmax = 200; CoefficientList[Series[Product[(1 + x^(4*k+1)), {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 26 2015 *)
%t 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 *)
%Y Cf. A000700, A261612, A280454, A280456, A280457.
%Y Cf. A015128, A080054, A261610, A261611.
%Y Cf. A000041, A000009, A035382, A035451.
%Y Cf. A170966-A170975.
%Y Cf. A262928, A147599, A281243, A281244, A281245.
%K nonn
%O 0,15
%A _N. J. A. Sloane_, Aug 29 2010
|
