A007045 Second (lower) diagonal of partition triangle A047812.

0, 1, 5, 20, 51, 112, 221, 411, 720, 1221, 2003, 3206, 5021, 7728, 11698, 17472, 25766, 37580, 54254, 77617, 110087, 154942, 216488, 300456, 414365, 568113, 774571, 1050572, 1417868, 1904641, 2547152, 3392042, 4498948, 5944158, 7824703, 10263932, 13418043, 17484554

Offset: 2

Author

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

Keywords

• nonn
• nice

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..500 (terms n=2..53 from Vincenzo Librandi, n=54..103 from Bruno Berselli)
• 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. A007042, A007044, A047812, A051643.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(n<0
      or t*i b((n-3)*(n+1), n$2):
seq(a(n), n=2..40);  # Alois P. Heinz, May 31 2020

Mathematica

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 *)

PARI

T(n, k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
for(n=3, 33, print1(T(n, n-3), ", ")) \\ Petros Hadjicostas, May 31 2020