A061261 Limits of diagonals in triangle defined in A061260.

1, 2, 6, 15, 37, 85, 194, 423, 912, 1917, 3974, 8096, 16302, 32382, 63668, 123851, 238756, 456190, 864821, 1627016, 3039845, 5641884, 10406924, 19083836, 34802782, 63135539, 113965033, 204739662, 366156396, 652001918, 1156200929, 2042173379, 3593341512

Offset: 0

Author

Vladeta Jovovic, Apr 23 2001

Keywords

• easy
• nonn

Comments

Terms 1, 2, 6, 15, 37, 85, ... are limits of diagonals in the triangle T(n,k) = A061260: 1 2, 1 3, 2, 1, 5, 6, 2, 1, 7, 11, 6, 2, 1, 11, 23, 15, 6, 2, 1, 15, 40, 32, 15, 6, 2, 1, 22, 73, 67, 37, 15, 6, 2, 1, 30, 120, 134, 79, 37, 15, 6, 2, 1, 42, 202, 255, 172, 85, 37, 15, 6, 2, 1, 56, 320, 470, 348, 187, 85, 37, 15, 6, 2, 1

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A061260.

Programs

Maple

with(numtheory): with(combinat):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 14 2017, revised Sep 19 2017

Mathematica

a[n_] := a[n] = If[n==0, 1, Sum[Sum[d PartitionsP[d+1], {d, Divisors[j]}] a[n-j], {j, 1, n}]/n];
a /@ Range[0, 40] (* Jean-François Alcover, Nov 10 2020, after Alois P. Heinz *)

Formula

G.f.: Product_{k >= 1} (1 - x^k)^( - numbpart(k + 1)), where numbpart(k) = number of partitions of k, cf. A000041. a(n) = 1/n*Sum_{k = 1..n} b(k)*a(n - k), n>0, a(0) = 1, where b(k) = Sum_{d|k} d*numbpart(d + 1).

a(n) = A061260(2n,n). - Alois P. Heinz, Oct 21 2018