A260894 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 - x^(n+k))/(1 - x^k).

1, 1, 2, 2, 4, 5, 8, 10, 15, 20, 28, 36, 50, 64, 86, 110, 145, 184, 238, 300, 384, 481, 608, 756, 948, 1172, 1456, 1790, 2208, 2700, 3310, 4026, 4906, 5941, 7200, 8680, 10470, 12570, 15092, 18050, 21583, 25718, 30634, 36376, 43174, 51102, 60446, 71324, 84102, 98948

Offset: 0

Author

Paul D. Hanna, Aug 03 2015

Keywords

• nonn

Comments

Antidiagonal sums of irregular triangle A063746 yields this sequence at offset 1.

From Jeremy Lovejoy, Oct 13 2022: (Start)

a(n) is also the number of partitions of n whose rank is at least -1, the rank of a partition being the largest part minus the number of parts. For example, the 7 partitions of 5 are (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), and (1,1,1,1,1), having ranks 4,2,1,0,-1,-2, and -4, respectively, and so a(5) = 5.

a(n) is also the number of partitions of n such that the smallest positive multiple of 3 not occurring in the partition is not a multiple of 6. For example, in the 7 partitions of 5 listed above, the smallest positive multiples of 3 not occurring are 3,3,6,6,3,3, and 3, respectively, and so a(5) = 5. (End)

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 8*x^6 + 10*x^7 +...
where
A(x) = 1 + x*(1-x^2)/(1-x) + x^2*(1-x^3)*(1-x^4)/((1-x)*(1-x^2)) + x^3*(1-x^4)*(1-x^5)*(1-x^6)/((1-x)*(1-x^2)*(1-x^3)) + x^4*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^8)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) +...
		

Links

• George E. Andrews, David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
• Rupam Barman, Ajit Singh, On Mex-related partition functions of Andrews and Newman, arXiv:2009.11602 [math.NT], 2020.

Crossrefs

Cf. A064174, A063746.

Cf. A000041, A064173, A064174.

Programs

Mathematica

Table[SeriesCoefficient[Sum[x^k * Product[(1-x^(k+j))/(1-x^j), {j,1,k}], {k,0,n}], {x,0,n}], {n,0,50}] (* Vaclav Kotesovec, Aug 08 2015 *)

PARI

{a(n) = local(A=1); A = sum(m=0,n,x^m*prod(k=1,m,(1-x^(m+k))/(1-x^k +x*O(x^n)))); polcoeff(A,n)}
for(n=0,60,print1(a(n),", "))

Formula

G.f.: (1/Product_{n>=1}(1-x^n))*Sum_{n>=0}(-1)^n*x^((3*n^2+3*n)/2). - Jeremy Lovejoy, Oct 13 2022.