A007045 Second (lower) diagonal of partition triangle A047812. (Formerly M3896)

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

2,3

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.

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(n<0
      or t*i<n, 0, b(n, i-1, t)+b(n-i, min(i, n-i), t-1)))
    end:
a:= n-> 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 *)

Prog

(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

Crossrefs

Cf. A007042, A007044, A047812, A051643.

Sequence in context: A062158 A034133 A006504 * A102227 A173034 A006010

Adjacent sequences: A007042 A007043 A007044 * A007046 A007047 A007048

Keyword

nonn,nice

Author

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

Status

approved