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Left diagonal of partition triangle A047812.
(Formerly M2451)
20

%I M2451 #59 Jun 01 2020 13:09:34

%S 0,1,3,5,9,13,20,28,40,54,75,99,133,174,229,295,383,488,625,790,1000,

%T 1253,1573,1956,2434,3008,3716,4563,5602,6840,8347,10141,12308,14881,

%U 17975,21635,26013,31183,37336,44581,53172,63259,75173,89132,105556,124752

%N Left diagonal of partition triangle A047812.

%C 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

%C 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

%C 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

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

%H S. Govindarajan, <a href="http://arxiv.org/abs/1203.4419">Notes on higher-dimensional partitions</a>, arXiv:1203.4419 [math.CO], 2012.

%H R. K. Guy, <a href="/A007042/a007042.pdf">Letter to N. J. A. Sloane, Aug. 1992</a>.

%H R. K. Guy, <a href="/A007042/a007042_1.pdf">Parker's permutation problem involves the Catalan numbers</a>, Preprint, 1992. (Annotated scanned copy)

%H R. K. Guy, <a href="http://www.jstor.org/stable/2324467">Parker's permutation problem involves the Catalan numbers</a>, Amer. Math. Monthly 100 (1993), 287-289.

%F a(n) = A000041(n+1) - 2. - _Vladeta Jovovic_, Oct 06 2001

%t f[n_]:= Length[Select[IntegerPartitions[2 n], First[#]==n-1 &]]; Table[f[n], {n, 1, 24}] (* _Clark Kimberling_, Mar 13 2012 *)

%t a[n_]:= PartitionsP[n+1]-2; Table[a[n], {n,1,50}] (* _Jean-François Alcover_, Jan 28 2015, after _M. F. Hasler_ *)

%o (PARI) A007042(n)=numbpart(n+1)-2 \\ _M. F. Hasler_, Apr 12 2012

%o (Julia)

%o using Nemo

%o function A007042List(len)

%o R, z = PolynomialRing(ZZ, "z")

%o e = eta_qexp(-1, len+2, z)

%o [coeff(e, j) - 2 for j in 2:len+1] end

%o A007042List(45) |> println # _Peter Luschny_, May 30 2020

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

%Y Column k = 2 of A081719.

%K nonn,easy,nice

%O 1,3

%A _N. J. A. Sloane_, _R. K. Guy_

%E More terms from _James A. Sellers_

%E Name edited by _Petros Hadjicostas_, May 31 2020