A212214 - OEIS

%I #17 Dec 06 2014 21:44:48

%S 1,1,2,3,6,8,14,18,29,39,57,74,109,138,192,247,335,421,565,703,926,

%T 1151,1484,1828,2349,2868,3624,4423,5538,6706,8345,10048,12394,14895,

%U 18219,21789,26549,31596,38226,45415,54656,64654,77501,91368,109003,128244,152279

%N Number of representations of n as a sum of products of pairs of positive integers, n = Sum_{k=1..m} i_k*j_k with i_k<=j_k, i_k<=i_{k+1}, j_k<=j_{k+1}, i_k*j_k<=i_{k+1}*j_{k+1}.

%H Alois P. Heinz, <a href="/A212214/b212214.txt">Table of n, a(n) for n = 0..200</a>

%e a(0) = 1: 0 = the empty sum.

%e a(1) = 1: 1 = 1*1.

%e a(2) = 2: 2 = 1*1 + 1*1 = 1*2.

%e a(3) = 3: 3 = 1*1 + 1*1 + 1*1 = 1*1 + 1*2 = 1*3.

%e a(7) = 18 = A182269(7)-1, one of the 19 sums counted by A182269(7) is not allowed: 7 = 1*3 + 2*2.

%p with(numtheory):

%p b:= proc(n, m, i, j) option remember;

%p       `if`(n=0, 1, `if`(m<1, 0, b(n, m-1, i, j) +`if`(m>n, 0,

%p         add(b(n-m, m, min(i, k), min(j, m/k)), k=select(x->

%p          is(x<=min(sqrt(m), i) and m<=j*x), divisors(m))))))

%p     end:

%p a:= n-> b(n$4):

%p seq(a(n), n=0..30);

%t b[n_, m_, i_, j_] := b[n, m, i, j] = If[n == 0, 1, If[m<1, 0, b[n, m-1, i, j] + If[m>n, 0, Sum[b[n-m, m, Min[i, k], Min[j, m/k]], {k, Select[Divisors[m], # <= Min[Sqrt[m], i] && m <= j*# &]}]]]]; a[n_] := b[n, n, n, n]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Dec 03 2014, after _Alois P. Heinz_ *)

%Y Cf. A066739, A182269, A182270, A211856, A211857, A212215, A212216, A212217, A212218, A212219.

%K nonn

%O 0,3

%A _Alois P. Heinz_, May 06 2012