A007042 Left diagonal of partition triangle A047812. (Formerly M2451)

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

1,3

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

Table of n, a(n) for n=1..46.

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.

Formula

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

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 *)

Prog

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

Crossrefs

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

Column k = 2 of A081719.

Sequence in context: A267262 A106607 A305082 * A178415 A249424 A076274

Adjacent sequences: A007039 A007040 A007041 * A007043 A007044 A007045

Keyword

nonn,easy,nice

Author

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

Extensions

More terms from James A. Sellers

Name edited by Petros Hadjicostas, May 31 2020

Status

approved