A145519 - OEIS

%I #37 Apr 11 2020 04:38:29

%S 1,2,7,19,54,134,354,838,2057,4794,11232,25412,58075,128670,286152,

%T 625829,1365653,2941088,6331146,13474533,28642325,60404681,127082128,

%U 265712673,554608226,1151374963,2385950536,4924685252,10145267212,20831428273,42708248451

%N a(n) = Sum_{k=1..n} A145518(n,k).

%C Row sums of A145518.

%C Also row sums of A129129, A215366.

%C a(n) = sum of the Heinz numbers of the partitions of n. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by _Alois P. Heinz_ in A215366 as an "encoding" of a partition). For example, for the 3 partitions of 3, namely [3], [1,2], and [1,1,1] we get 5, 2*3=6, and 2*2*2=8, respectively; their sum is a(3) = 19. - _Emeric Deutsch_, Jun 09 2015

%H Alois P. Heinz, <a href="/A145519/b145519.txt">Table of n, a(n) for n = 0..1000</a>

%H More terms in <a href="http://www.tilman-neumann.de/docs/A145518.txt">A145518 and A145519</a>

%F G.f.: 1/Product_{i>=1}(1-prime(i)*x^i). - _Vladeta Jovovic_, Nov 09 2008

%F a(n) ~ c * 2^n, where c = Product_{k>=2} 1/(1 - prime(k)/2^k) = 50.412394245500690832088704444961002125578414895935257436317... . - _Vaclav Kotesovec_, Sep 10 2014, updated Apr 11 2020

%p b:= proc(n, i) option remember; `if`(n=0 or i<2, 2^n,

%p       add(b(n-i*j, i-1)*ithprime(i)^j, j=0..iquo(n, i)))

%p     end:

%p a:= n-> b(n, n):

%p seq(a(n), n=0..35);  # _Alois P. Heinz_, Feb 19 2013

%t b[n_, i_] := b[n, i] = If[n == 0 || i < 2, 2^n, Sum[b[n-i*j, i-1]*Prime[i]^j, {j, 0, Quotient[n, i]}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Feb 24 2015, after _Alois P. Heinz_ *)

%Y Cf. A129129, A145518, A215366.

%K nonn

%O 0,2

%A _Tilman Neumann_, Oct 12 2008

%E a(0) inserted by _Alois P. Heinz_, Feb 19 2013