A000711 - OEIS

%I M2787 N1122 #41 Feb 04 2022 02:01:44

%S 1,3,9,22,51,107,217,416,775,1393,2446,4185,7028,11569,18749,29908,

%T 47083,73157,112396,170783,256972,383003,565961,829410,1206282,

%U 1741592,2497425,3557957,5037936,7091711,9927583,13823626,19151731,26404879,36236988,49509149

%N Number of partitions of n, with three kinds of 1,2,3 and 4 and two kinds of 5,6,7,...

%C Convolution of A000712 and A001400. - _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="/A000711/b000711.txt">Table of n, a(n) for n = 0..1000</a>

%H M. A. Harrison, <a href="/A000711/a000711.pdf">On the number of classes of binary matrices</a>, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)

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

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

%F G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*Product_{k>=1} (1 - x^k)^2).

%F a(n) ~ exp(2*Pi*sqrt(n/3)) * 3^(1/4) * n^(3/4) / (32*Pi^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 with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> `if`(n<5,3,2)): seq(a(n), n=0..40); # _Alois P. Heinz_, Sep 08 2008

%t nn=31;CoefficientList[Series[1/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/Product[(1-x^i)^2,{i,1,nn}],{x,0,nn}],x] (* _Geoffrey Critzer_, Sep 28 2013 *)

%K nonn

%O 0,2

%A _N. J. A. Sloane_

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

%E Edited by _Emeric Deutsch_, Mar 22 2005