A000714 - OEIS

%I M2777 N1117 #35 Feb 04 2022 02:01:48

%S 1,3,9,21,47,95,186,344,620,1078,1835,3045,4967,7947,12534,19470,

%T 29879,45285,67924,100820,148301,216199,312690,448738,639464,905024,

%U 1272837,1779237,2473065,3418655,4701611,6434015,8763676

%N Number of partitions of n, with three kinds of 1 and 2 and two kinds of 3,4,5,....

%C Convolution of A000712 and A008619. - _Vaclav Kotesovec_, Aug 18 2015

%D H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vaclav Kotesovec, <a href="/A000714/b000714.txt">Table of n, a(n) for n = 0..2000</a>

%H T. Doslic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Doslic/doslic3.html">Kepler-Bouwkamp Radius of Combinatorial Sequences</a>, Journal of Integer Sequences, Vol. 17, 2014, #14.11.3.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F EULER transform of 3, 3, 2, 2, 2, 2, 2, 2, ...

%F G.f.: 1/((1-x)*(1-x^2)*Product_{k>=1} (1 - x^k)^2). - _Emeric Deutsch_, Apr 17 2006

%F a(n) ~ 3^(1/4) * exp(2*Pi*sqrt(n/3)) / (8 * Pi^2 * n^(1/4)). - _Vaclav Kotesovec_, Aug 18 2015

%e a(2)=9 because we have 2, 2', 2", 1+1, 1'+1', 1"+1", 1+1', 1+1", 1'+1".

%p g:=1/((1-x)*(1-x^2)*product((1-x^k)^2,k=1..40)): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..32); # _Emeric Deutsch_, Apr 17 2006

%t p=Product[1/(1-x^i),{i,1,20}];CoefficientList[Series[p^2/(1 - x)/(1 - x^2), {x, 0, 20}], x] (* _Geoffrey Critzer_, Nov 28 2011 *)

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E Extended with formula from _Christian G. Bower_, Apr 15 1998