A000715 Number of partitions of n, with three kinds of 1,2 and 3 and two kinds of 4,5,6,.... (Formerly M2786 N1121)

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

0,2

Comments

Convolution of A000712 and A001399. - Vaclav Kotesovec, Aug 18 2015

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

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

Example

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

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 *)

Crossrefs

Sequence in context: A001937 A086817 A247188 * A260545 A034505 A143099

Adjacent sequences: A000712 A000713 A000714 * A000716 A000717 A000718

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved