A007042 Left diagonal of partition triangle A047812.

0, 1, 3, 5, 9, 13, 20, 28, 40, 54, 75, 99, 133, 174, 229, 295, 383, 488, 625, 790, 1000, 1253, 1573, 1956, 2434, 3008, 3716, 4563, 5602, 6840, 8347, 10141, 12308, 14881, 17975, 21635, 26013, 31183, 37336, 44581, 53172, 63259, 75173, 89132, 105556, 124752

Offset: 1

Author

N. J. A. Sloane, R. K. Guy

Keywords

• nonn
• easy
• nice

Comments

For n > 2, a(n) is also the number of partitions of n into parts <= n-2: a(n) = A026820(n+1, n-1). - Reinhard Zumkeller, Jan 21 2010

Also, the number of partitions of 2*n in which n-1 is the maximal part; see the Mathematica section. - Clark Kimberling, Mar 13 2012

This is column 2 of the matrix A in Sect. 2.3 of the Govindarajan preprint, cf. references and A096651. - M. F. Hasler, Apr 12 2012

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• S. Govindarajan, Notes on higher-dimensional partitions, arXiv:1203.4419 [math.CO], 2012.
• R. K. Guy, Letter to N. J. A. Sloane, Aug. 1992.
• R. K. Guy, Parker's permutation problem involves the Catalan numbers, Preprint, 1992. (Annotated scanned copy)
• R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.

Crossrefs

Cf. A000041, A007044, A007045, A026820, A047812, A051643, A096651.

Column k = 2 of A081719.

Programs

Julia

using Nemo
function A007042List(len)
    R, z = PolynomialRing(ZZ, "z")
    e = eta_qexp(-1, len+2, z)
    [coeff(e, j) - 2 for j in 2:len+1] end
A007042List(45) |> println # Peter Luschny, May 30 2020

Mathematica

f[n_]:= Length[Select[IntegerPartitions[2 n], First[#]==n-1 &]]; Table[f[n], {n, 1, 24}] (* Clark Kimberling, Mar 13 2012 *)
a[n_]:= PartitionsP[n+1]-2; Table[a[n], {n,1,50}] (* Jean-François Alcover, Jan 28 2015, after M. F. Hasler *)

PARI

A007042(n)=numbpart(n+1)-2  \\ M. F. Hasler, Apr 12 2012

Formula

a(n) = A000041(n+1) - 2. - Vladeta Jovovic, Oct 06 2001

Extensions

More terms from James Sellers

Name edited by Petros Hadjicostas, May 31 2020