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

0,2

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

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

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 *)

Crossrefs

Cf. A061260.

Sequence in context: A238830 A018018 A030009 * A335903 A291414 A098790

Adjacent sequences: A061258 A061259 A061260 * A061262 A061263 A061264

Keyword

easy,nonn

Author

Vladeta Jovovic, Apr 23 2001

Status

approved