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A079308 For a partition P of a positive integer, let f(P) be the product of k+1, over all parts k in P. Let a(n,r) be the sum of f(P) over all partitions P of n with smallest part r. Sequence gives table of a(n,r) for 1 <= r <= n, in the order a(1,1); a(2,1), a(2,2); a(3,1), a(3,2), a(3,3); ...

%I A079308 #13 May 22 2015 14:35:10
%S A079308 2,4,3,14,0,4,36,9,0,5,100,12,0,0,6,236,42,16,0,0,7,602,54,20,0,0,0,8,
%T A079308 1368,195,24,25,0,0,0,9,3242,246,92,30,0,0,0,0,10,7240,759,112,35,36,
%U A079308 0,0,0,0,11,16386,1134,232,40,42,0,0,0,0,0,12,35692,2859,528,170,48,49,0,0,0,0,0,13
%N A079308 For a partition P of a positive integer, let f(P) be the product of k+1, over all parts k in P. Let a(n,r) be the sum of f(P) over all partitions P of n with smallest part r. Sequence gives table of a(n,r) for 1 <= r <= n, in the order a(1,1); a(2,1), a(2,2); a(3,1), a(3,2), a(3,3); ...
%H A079308 Alois P. Heinz, <a href="/A079308/b079308.txt">Rows n = 1..141, flattened</a>
%e A079308 The partitions with minimal part 3 begin 3, 3+3, 4+3, 5+3, 6+3, 3+3+3, ... which yield the following values of f: 4, 16, 20, 24, 28, 64, ... therefore the 3rd column of our table begins 4,0,0,16,20,24,(28+64)=92,...
%e A079308 Triangle a(n,r) begins:
%e A079308 :    2;
%e A079308 :    4,   3;
%e A079308 :   14,   0,   4;
%e A079308 :   36,   9,   0,  5;
%e A079308 :  100,  12,   0,  0,  6;
%e A079308 :  236,  42,  16,  0,  0, 7;
%e A079308 :  602,  54,  20,  0,  0, 0, 8;
%e A079308 : 1368, 195,  24, 25,  0, 0, 0, 9;
%e A079308 : 3242, 246,  92, 30,  0, 0, 0, 0, 10;
%e A079308 : 7240, 759, 112, 35, 36, 0, 0, 0,  0, 11;
%p A079308 b:= proc(n, k) option remember; `if`(n=0, 1,
%p A079308       `if`(k>n, 0, b(n, k+1) +(k+1)*b(n-k, k)))
%p A079308     end:
%p A079308 a:= (n, k)-> b(n,k)-b(n, k+1):
%p A079308 seq(seq(a(n, k), k=1..n), n=1..12);  # _Alois P. Heinz_, May 22 2015
%t A079308 a[n_, r_] := Which[r>n, 0, r==n, n+1, True, a[n, r]=(r+1)Sum[a[n-r, s], {s, r, n-r}]]; Flatten[Table[a[n, r], {n, 1, 12}, {r, 1, n}]]
%Y A079308 Cf. A074139, A074141 (row sums).
%K A079308 easy,nonn,tabl
%O A079308 1,1
%A A079308 _Alford Arnold_, Feb 09 2003
%E A079308 Edited by _Dean Hickerson_, Feb 11 2003
%E A079308 Offset changed to 1 by _Alois P. Heinz_, May 22 2015