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%I A258323 #30 Nov 01 2018 18:00:01
%S A258323 1,0,2,0,3,0,5,6,0,7,10,0,11,29,0,13,43,30,0,17,94,42,0,19,128,136,0,
%T A258323 23,231,293,0,29,279,551,210,0,31,484,892,330,0,37,584,1765,852,0,41,
%U A258323 903,2570,1826,0,43,1051,4273,4207,0,47,1552,6747,6595,2310
%N A258323 Sum T(n,k) over all partitions lambda of n into k distinct parts of Product_{i:lambda} prime(i); triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.
%H A258323 Alois P. Heinz, <a href="/A258323/b258323.txt">Rows n = 0..500, flattened</a>
%e A258323 T(6,2) = 43 because the partitions of 6 into 2 distinct parts are {[5,1], [4,2]} and prime(5)*prime(1) + prime(4)*prime(2) = 11*2 + 7*3 = 22 + 21 = 43.
%e A258323 Triangle T(n,k) begins:
%e A258323   1
%e A258323   0,  2;
%e A258323   0,  3;
%e A258323   0,  5,   6;
%e A258323   0,  7,  10;
%e A258323   0, 11,  29;
%e A258323   0, 13,  43,  30;
%e A258323   0, 17,  94,  42;
%e A258323   0, 19, 128, 136;
%e A258323   0, 23, 231, 293;
%e A258323   0, 29, 279, 551, 210;
%p A258323 g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, expand(
%p A258323       add(g(n-i*j, i-1)*(ithprime(i)*x)^j, j=0..min(1, n/i)))))
%p A258323     end:
%p A258323 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n$2)):
%p A258323 seq(T(n), n=0..20);
%t A258323 g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, Expand[Sum[g[n-i*j, i-1] * (Prime[i]*x)^j, {j, 0, Min[1, n/i]}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][g[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Jan 06 2017, after _Alois P. Heinz_ *)
%Y A258323 Columns k=0-10 give: A000007, A000040, A025129(n+1), A258358, A258359, A258360, A258361, A258362, A258363, A258364, A258365.
%Y A258323 Row sums give A147655.
%Y A258323 T(n*(n+1)/2,n) = A002110(n).
%Y A258323 T(n^2,n) = A321267(n).
%Y A258323 Cf. A000217, A003056, A145518, A246867.
%K A258323 nonn,tabf
%O A258323 0,3
%A A258323 _Alois P. Heinz_, May 26 2015

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