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A007042 Left diagonal of partition triangle A047812.

%I A007042 M2451 #60 Jul 02 2025 16:01:55
%S A007042 0,1,3,5,9,13,20,28,40,54,75,99,133,174,229,295,383,488,625,790,1000,
%T A007042 1253,1573,1956,2434,3008,3716,4563,5602,6840,8347,10141,12308,14881,
%U A007042 17975,21635,26013,31183,37336,44581,53172,63259,75173,89132,105556,124752
%N A007042 Left diagonal of partition triangle A047812.
%C A007042 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 A007042 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 A007042 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 A007042 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007042 S. Govindarajan, <a href="http://arxiv.org/abs/1203.4419">Notes on higher-dimensional partitions</a>, arXiv:1203.4419 [math.CO], 2012.
%H A007042 R. K. Guy, <a href="/A007042/a007042.pdf">Letter to N. J. A. Sloane, Aug. 1992</a>.
%H A007042 R. K. Guy, <a href="/A007042/a007042_1.pdf">Parker's permutation problem involves the Catalan numbers</a>, Preprint, 1992. (Annotated scanned copy)
%H A007042 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 A007042 a(n) = A000041(n+1) - 2. - _Vladeta Jovovic_, Oct 06 2001
%t A007042 f[n_]:= Length[Select[IntegerPartitions[2 n], First[#]==n-1 &]]; Table[f[n], {n, 1, 24}] (* _Clark Kimberling_, Mar 13 2012 *)
%t A007042 a[n_]:= PartitionsP[n+1]-2; Table[a[n], {n,1,50}] (* _Jean-François Alcover_, Jan 28 2015, after _M. F. Hasler_ *)
%o A007042 (PARI) A007042(n)=numbpart(n+1)-2  \\ _M. F. Hasler_, Apr 12 2012
%o A007042 (Julia)
%o A007042 using Nemo
%o A007042 function A007042List(len)
%o A007042     R, z = PolynomialRing(ZZ, "z")
%o A007042     e = eta_qexp(-1, len+2, z)
%o A007042     [coeff(e, j) - 2 for j in 2:len+1] end
%o A007042 A007042List(45) |> println # _Peter Luschny_, May 30 2020
%Y A007042 Cf. A000041, A007044, A007045, A026820, A047812, A051643, A096651.
%Y A007042 Column k = 2 of A081719.
%K A007042 nonn,easy,nice
%O A007042 1,3
%A A007042 _N. J. A. Sloane_, _R. K. Guy_
%E A007042 More terms from _James Sellers_
%E A007042 Name edited by _Petros Hadjicostas_, May 31 2020