OFFSET
0,3
COMMENTS
Equals row sums of triangle A147767. - Gary W. Adamson, Nov 11 2008
Also number of partitions of n into parts of 2 kinds except for 1. - Reinhard Zumkeller, Nov 06 2012
Antidiagonal sums of triangle A093010.
REFERENCES
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 105.
L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.7).
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 (terms n = 1001..7000 from Vaclav Kotesovec)
M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273.
M. S. Cheema, Letter to N. J. A. Sloane, Jul 15 1970 [scanned copy]
P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116.
R. Newton and A. R. Camacho, Strangely dual orbifold equivalence I, arXiv preprint arXiv:1509.08069 [math.QA], 2015.
Steven Rayan, Aspects of the topology and combinatorics of Higgs bundle moduli spaces, arXiv:1809.05732 [math.AG], 2018.
FORMULA
G.f.: 1 / ( (1-x) * Product_{m>=2} (1-x^m)^2 ) = (1-x) / Product_{m>=1} (1-x^m)^2.
G.f.: exp( Sum_{n>=1} ((1+x^n)/(1-x^n))*x^n/n ). - Paul D. Hanna, Apr 22 2010
a(n) ~ Pi * exp(2*Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(7/4)). - Vaclav Kotesovec, Oct 28 2015
G.f.: exp(Sum_{k>=1} (2*sigma_1(k) - 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(min(i, 2)+j-1, j)*
b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..45); # Alois P. Heinz, Mar 15 2014
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[Min[i, 2]+j-1, j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
Flatten[{1, Differences[Table[Sum[PartitionsP[j]*PartitionsP[n-j], {j, 0, n}], {n, 0, 50}]]}] (* Vaclav Kotesovec, Oct 28 2015 *)
CoefficientList[(1-q)/QPochhammer[q]^2+O[q]^50, q] (* Jean-François Alcover, Nov 27 2015 *)
PROG
(PARI) a(n)=if(n<0, 0, polcoeff((1-x)/prod(k=1, n, 1-x^k, 1+x*O(x^n))^2, n)) /* Michael Somos, Jan 29 2005 */
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, ((1+x^m)/(1-x^m+x*O(x^n)))*x^m/m)), n)} \\ Paul D. Hanna, Apr 22 2010
(Haskell)
a000990 = p $ tail a008619_list where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 06 2012
(PARI) x='x+O('x^66); Vec((1-x)/eta(x)^2) \\ Joerg Arndt, May 01 2013
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(&*[1-x^j: j in [1..2*m]] )^2 )); // G. C. Greubel, Dec 06 2018
(Sage) s=((1-x)/prod(1-x^j for j in (1..60))^2).series(x, 50); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 06 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved