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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. 1

%I

%S 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 2,139,113,20,3,195,168,31,5,272,234,43,7,373,334,62,11,508,460,85,15,

%U 684,635,120,23,1,915,857,161,31,1,1212,1165,221,44,2,1597

%N 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 Alois P. Heinz, <a href="/A252866/b252866.txt">Rows n = 1..2048, flattened</a>

%F T(2^k,k) = 1.

%e Triangle T(n,k) begins:

%e 01: 1;

%e 02: 2, 1;

%e 03: 4, 2;

%e 04: 7, 4, 1;

%e 05: 12, 7, 1;

%e 06: 19, 14, 2;

%e 07: 30, 21, 3;

%e 08: 45, 34, 6, 1;

%e 09: 67, 51, 9, 1;

%e 10: 97, 79, 14, 2;

%e 11: 139, 113, 20, 3;

%e 12: 195, 168, 31, 5;

%e 13: 272, 234, 43, 7;

%e 14: 373, 334, 62, 11;

%e 15: 508, 460, 85, 15;

%e 16: 684, 635, 120, 23, 1;

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,

%p add((p-> p+[0, p[1]*j*x^igcd(seq(h[2], h=ifactors(i)[2]))]

%p )(b(n-i*j, i-1)), j=0..n/i)))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)[2]):

%p seq(T(n), n=1..25);

%Y Column k=0 gives A000070(n-1).

%Y Row sums give: A006128.

%Y Cf. A000523, A052409.

%K nonn,tabf,look

%O 1,2

%A _Alois P. Heinz_, Dec 23 2014

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Last modified January 18 16:27 EST 2020. Contains 331011 sequences. (Running on oeis4.)