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A252866 Number T(n,k) of parts p in all partitions of n with largest integer power k (such that A052409(p)=k); triangle T(n,k), n>=1, 0<=k<=A000523(n), read by rows.

%I A252866 #14 Mar 30 2016 16:06:12
%S A252866 1,2,1,4,2,7,4,1,12,7,1,19,14,2,30,21,3,45,34,6,1,67,51,9,1,97,79,14,
%T A252866 2,139,113,20,3,195,168,31,5,272,234,43,7,373,334,62,11,508,460,85,15,
%U A252866 684,635,120,23,1,915,857,161,31,1,1212,1165,221,44,2,1597
%N A252866 Number T(n,k) of parts p in all partitions of n with largest integer power k (such that A052409(p)=k); triangle T(n,k), n>=1, 0<=k<=A000523(n), read by rows.
%H A252866 Alois P. Heinz, <a href="/A252866/b252866.txt">Rows n = 1..2048, flattened</a>
%F A252866 T(2^k,k) = 1.
%e A252866 Triangle T(n,k) begins:
%e A252866 01:    1;
%e A252866 02:    2,   1;
%e A252866 03:    4,   2;
%e A252866 04:    7,   4,   1;
%e A252866 05:   12,   7,   1;
%e A252866 06:   19,  14,   2;
%e A252866 07:   30,  21,   3;
%e A252866 08:   45,  34,   6,  1;
%e A252866 09:   67,  51,   9,  1;
%e A252866 10:   97,  79,  14,  2;
%e A252866 11:  139, 113,  20,  3;
%e A252866 12:  195, 168,  31,  5;
%e A252866 13:  272, 234,  43,  7;
%e A252866 14:  373, 334,  62, 11;
%e A252866 15:  508, 460,  85, 15;
%e A252866 16:  684, 635, 120, 23,  1;
%p A252866 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A252866       add((p-> p+[0, p[1]*j*x^igcd(seq(h[2], h=ifactors(i)[2]))]
%p A252866       )(b(n-i*j, i-1)), j=0..n/i)))
%p A252866     end:
%p A252866 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)[2]):
%p A252866 seq(T(n), n=1..25);
%Y A252866 Column k=0 gives A000070(n-1).
%Y A252866 Row sums give: A006128.
%Y A252866 Cf. A000523, A052409.
%K A252866 nonn,tabf,look
%O A252866 1,2
%A A252866 _Alois P. Heinz_, Dec 23 2014