A335323 First lower diagonal of Parker's triangle A047812.

0, 1, 3, 7, 11, 18, 26, 38, 52, 73, 97, 131, 172, 227, 293, 381, 486, 623, 788, 998, 1251, 1571, 1954, 2432, 3006, 3714, 4561, 5600, 6838, 8345, 10139, 12306, 14879, 17973, 21633, 26011, 31181, 37334, 44579, 53170, 63257, 75171, 89130, 105554, 124750, 147269

Offset

1,3

Comments

Apparently, this sequence was originally intended to be A7043 (now A007043), but for some reason it was crossed out on p. 4 of the annotated copy of Guy's 1992 preprint.

a(n) is the number of partitions of (n-2)*(n+1) into at most n parts each no bigger than n. Thus, a(n) is the coefficient of q^((n-2)*(n+1)) in the q-binomial coefficient [2*n, n].

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.

Example

a(1) = 0 because it does not make sense to talk about the partitions of (1-2)*(1+1) = -2.
a(2) = 1 because we have only the empty partition for (2-2)*(2+1) = 0.
a(3) = 3 because we have the following partitions of (3-2)*(3+1) = 4 into no more than 3 parts each no bigger than 3: 1+3 = 1+1+2 = 2+2.
a(4) = 7 because we have the following partitions of (4-2)*(4+1) = 10 into no more than 4 parts each no bigger than 4: 2+4+4 = 3+3+4 = 1+1+4+4 = 1+2+3+4 = 1+3+3+3 = 2+2+2+4 = 2+2+3+3.
The PARI function partitions((n-2)*(n+1), n, n) can generate these partitions.

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-2)*(n+1), 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[(n-2)(n+1), n, n];
Array[a, 50] (* Jean-François Alcover, Nov 27 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, 43, print1(T(n, n-2), ", "))

Crossrefs

Cf. A007042, A007043, A007044, A007045, A047812, A051643.

Sequence in context: A062851 A178464 A071979 * A097748 A255000 A244570

Adjacent sequences: A335320 A335321 A335322 * A335324 A335325 A335326

Keyword

nonn

Author

Petros Hadjicostas, May 31 2020

Status

approved