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Number of partitions of n, with three kinds of 1,2 and 3 and two kinds of 4,5,6,....
(Formerly M2786 N1121)
2

%I M2786 N1121 #37 Feb 03 2019 14:47:53

%S 1,3,9,22,50,104,208,394,724,1286,2229,3769,6253,10176,16303,25723,

%T 40055,61588,93647,140875,209889,309846,453565,658627,949310,1358589,

%U 1931464,2728547,3831654,5350119,7430158,10265669,14113795,19313168,26309405,35685523

%N Number of partitions of n, with three kinds of 1,2 and 3 and two kinds of 4,5,6,....

%C Convolution of A000712 and A001399. - _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 Alois P. Heinz, <a href="/A000715/b000715.txt">Table of n, a(n) for n = 0..10000</a>

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

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

%F G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*Product_{k>=1}(1-x^k)^2). - _Emeric Deutsch_, Apr 17 2006

%F a(n) ~ exp(2*Pi*sqrt(n/3)) * n^(1/4) / (8 * 3^(1/4) * Pi^3). - _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)*(1-x^3)*product((1-x^k)^2,k=1..40)): gser:=series(g,x=0,40): seq(coeff(gser,x,n),n=0..31); # _Emeric Deutsch_, Apr 17 2006

%p # second Maple program

%p a:= proc(n) a(n):= `if`(n=0, 1, add(add(d*`if`(d<4, 3, 2), d=numtheory [divisors](j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..50); # _Alois P. Heinz_, Sep 25 2012

%t nn=25;p=Product[1/(1- x^i)^2,{i,1,nn}];CoefficientList[Series[p /(1-x)/(1-x^2)/(1-x^3),{x,0,nn}],x] (* _Geoffrey Critzer_, Sep 25 2012 *)

%K nonn

%O 0,2

%A _N. J. A. Sloane_

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