cp's OEIS Frontend

A007045 Second (lower) diagonal of partition triangle A047812.

%I A007045 M3896 #53 May 31 2020 19:40:25
%S A007045 0,1,5,20,51,112,221,411,720,1221,2003,3206,5021,7728,11698,17472,
%T A007045 25766,37580,54254,77617,110087,154942,216488,300456,414365,568113,
%U A007045 774571,1050572,1417868,1904641,2547152,3392042,4498948,5944158,7824703,10263932,13418043,17484554
%N A007045 Second (lower) diagonal of partition triangle A047812.
%D A007045 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007045 Alois P. Heinz, <a href="/A007045/b007045.txt">Table of n, a(n) for n = 2..500</a> (terms n=2..53 from Vincenzo Librandi, n=54..103 from Bruno Berselli)
%H A007045 R. K. Guy, <a href="/A007042/a007042.pdf">Letter to N. J. A. Sloane, Aug. 1992</a>.
%H A007045 R. K. Guy, <a href="/A007042/a007042_1.pdf">Parker's permutation problem involves the Catalan numbers</a>, preprint, 1992. (Annotated scanned copy)
%H A007045 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.
%p A007045 b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(n<0
%p A007045       or t*i<n, 0, b(n, i-1, t)+b(n-i, min(i, n-i), t-1)))
%p A007045     end:
%p A007045 a:= n-> b((n-3)*(n+1), n$2):
%p A007045 seq(a(n), n=2..40);  # _Alois P. Heinz_, May 31 2020
%t A007045 s[n_] := s[n] = Series[Product[(1 - q^(2*n - k))/(1 - q^(k + 1)), {k, 0, n - 1}], {q, 0, n^2}]; t[n_, k_] := SeriesCoefficient[s[n], k*(n + 1)]; A007045 = Join[{0}, Table[t[n + 3, n], {n, 0, 25}] ] (* _Jean-François Alcover_, Apr 25 2012 *)
%o A007045 (PARI) T(n, k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
%o A007045 for(n=3, 33, print1(T(n, n-3), ", ")) \\ _Petros Hadjicostas_, May 31 2020
%Y A007045 Cf. A007042, A007044, A047812, A051643.
%K A007045 nonn,nice
%O A007045 2,3
%A A007045 _N. J. A. Sloane_ and _R. K. Guy_