A007044 Left diagonal of partition triangle A047812.

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

Author

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

Keywords

• nonn

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.

Crossrefs

Cf. A007042, A007045, A047812, A051643.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(n<0
      or t*i 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 *)

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

Extensions

Name edited by Petros Hadjicostas, May 31 2020