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A222656 Number T(n,k) of partitions of n using exactly k primes; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

%I A222656 #33 Aug 06 2019 19:10:56
%S A222656 1,1,1,1,1,2,2,2,1,2,3,2,3,4,3,1,3,6,4,2,5,7,6,3,1,6,9,8,5,2,8,11,12,
%T A222656 7,3,1,8,17,14,10,5,2,12,20,19,14,8,3,1,13,26,25,19,11,5,2,17,31,35,
%U A222656 24,16,8,3,1,19,41,42,34,21,12,5,2,26,47,56,44,29
%N A222656 Number T(n,k) of partitions of n using exactly k primes; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.
%H A222656 Alois P. Heinz, <a href="/A222656/b222656.txt">Rows n = 0..200, flattened</a>
%F A222656 Sum_{k=1..floor(n/2)} k * T(n,k) = A037032(n).
%F A222656 G.f.: G(t,x) = Product_{i>=1} (1 - x^prime(i))/((1 - x^i)*(1 - t*x^prime(i))). - _Emeric Deutsch_, Nov 11 2015
%e A222656 T(6,0) = 3: [6], [4,1,1], [1,1,1,1,1,1].
%e A222656 T(6,1) = 4: [5,1], [4,2], [3,1,1,1], [2,1,1,1,1].
%e A222656 T(6,2) = 3: [3,3], [3,2,1], [2,2,1,1].
%e A222656 T(6,3) = 1: [2,2,2].
%e A222656 Triangle T(n,k) begins:
%e A222656   1;
%e A222656   1;
%e A222656   1,  1;
%e A222656   1,  2;
%e A222656   2,  2,  1;
%e A222656   2,  3,  2;
%e A222656   3,  4,  3,  1;
%e A222656   3,  6,  4,  2;
%e A222656   5,  7,  6,  3, 1;
%e A222656   6,  9,  8,  5, 2;
%e A222656   8, 11, 12,  7, 3, 1;
%e A222656   8, 17, 14, 10, 5, 2;
%e A222656   ...
%p A222656 b:= proc(n, i) option remember; local j; if n=0 then 1 elif i<1 then 0
%p A222656       else []; for j from 0 to n/i do zip((x, y)->x+y, %,
%p A222656       [`if`(isprime(i), 0$j, NULL), b(n-i*j, i-1)], 0) od; %[] fi
%p A222656     end:
%p A222656 T:= n-> b(n$2):
%p A222656 seq(T(n), n=0..16);
%t A222656 zip[f_, x_List, y_List, z_] := With[{m = Max[Length[x], Length[y]]}, Thread[f[PadRight[x, m, z], PadRight[y, m, z]]]]; b[n_, i_] := b[n, i] = Module[{j, pc}, Which[n == 0, {1}, i<1, {0}, True, pc = {}; For[j = 0, j <= n/i, j++, pc = zip[Plus, pc, Join[If[PrimeQ[i], Array[0&, j], {}], b[n-i*j, i-1]], 0]]; pc]]; T[n_] := b[n, n]; Table[T[n], {n, 0, 16}] // Flatten (* _Jean-François Alcover_, Jan 29 2014, after _Alois P. Heinz_ *)
%Y A222656 Column k=0 gives: A002095.
%Y A222656 Row sums give: A000041.
%Y A222656 Cf. A000040, A004526, A224344.
%K A222656 nonn,tabf
%O A222656 0,6
%A A222656 _Alois P. Heinz_, May 29 2013