A000715 Number of partitions of n, with three kinds of 1,2 and 3 and two kinds of 4,5,6,....

1, 3, 9, 22, 50, 104, 208, 394, 724, 1286, 2229, 3769, 6253, 10176, 16303, 25723, 40055, 61588, 93647, 140875, 209889, 309846, 453565, 658627, 949310, 1358589, 1931464, 2728547, 3831654, 5350119, 7430158, 10265669, 14113795, 19313168, 26309405, 35685523

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Convolution of A000712 and A001399. - 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..10000
• N. J. A. Sloane, Transforms

Programs

Maple

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
# second Maple program
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

Mathematica

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 *)

Formula

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

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

a(n) ~ exp(2*Pi*sqrt(n/3)) * n^(1/4) / (8 * 3^(1/4) * Pi^3). - Vaclav Kotesovec, Aug 18 2015

Extensions

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