A000711 Number of partitions of n, with three kinds of 1,2,3 and 4 and two kinds of 5,6,7,...

1, 3, 9, 22, 51, 107, 217, 416, 775, 1393, 2446, 4185, 7028, 11569, 18749, 29908, 47083, 73157, 112396, 170783, 256972, 383003, 565961, 829410, 1206282, 1741592, 2497425, 3557957, 5037936, 7091711, 9927583, 13823626, 19151731, 26404879, 36236988, 49509149

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Convolution of A000712 and A001400. - Vaclav Kotesovec, Aug 18 2015

Examples

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

References

• H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
• N. J. A. Sloane, Transforms

Programs

Maple

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

Mathematica

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 *)

Formula

EULER transform of 3, 3, 3, 3, 2, 2, 2, 2, ...

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

a(n) ~ exp(2*Pi*sqrt(n/3)) * 3^(1/4) * n^(3/4) / (32*Pi^4). - Vaclav Kotesovec, Aug 18 2015

Extensions

Extended with formula from Christian G. Bower, Apr 15 1998

Edited by Emeric Deutsch, Mar 22 2005