A007044 Left diagonal of partition triangle A047812. (Formerly M4370)

0, 0, 1, 7, 20, 48, 100, 194, 352, 615, 1034, 1693, 2705, 4239, 6522, 9889, 14786, 21844, 31913, 46165, 66162, 94035, 132600, 185637, 258128, 356674, 489906, 669173, 909212, 1229217, 1653993, 2215597, 2955192, 3925659, 5194520, 6847963, 8995524, 11776227

Offset

1,4

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 = 1..500

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(2*n+2, n$2):
seq(a(n), n=1..50);  # Alois P. Heinz, May 31 2020

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[n < 0 || t i < n, 0, b[n, i - 1, t] + b[n - i, Min[i, n - i], t - 1]]];
a[n_] := b[2n+2, n, n];
Array[a, 50] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

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=1, 40, print1(T(n, 2), ", ")) \\ Petros Hadjicostas, May 31 2020

Crossrefs

Cf. A007042, A007045, A047812, A051643.

Sequence in context: A100206 A298288 A299384 * A047862 A264879 A320681

Adjacent sequences: A007041 A007042 A007043 * A007045 A007046 A007047

Keyword

nonn

Author

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

Extensions

Name edited by Petros Hadjicostas, May 31 2020

Status

approved