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A058399 Triangle of partial row sums of partition triangle A008284.

%I A058399 #54 Apr 29 2023 00:06:35
%S A058399 1,2,1,3,2,1,5,4,2,1,7,6,4,2,1,11,10,7,4,2,1,15,14,11,7,4,2,1,22,21,
%T A058399 17,12,7,4,2,1,30,29,25,18,12,7,4,2,1,42,41,36,28,19,12,7,4,2,1,56,55,
%U A058399 50,40,29,19,12,7,4,2,1,77,76,70,58,43,30,19,12,7,4,2,1,101,100,94,80,62
%N A058399 Triangle of partial row sums of partition triangle A008284.
%C A058399 T(n,m) is also the number of m-th largest elements in all partitions of n. - _Omar E. Pol_, Feb 14 2012
%C A058399 It appears that reversed rows converge to A000070. - _Omar E. Pol_, Mar 10 2012
%C A058399 The row sums give A006128. - _Omar E. Pol_, Mar 26 2012
%C A058399 T(n,m) is also the number of regions traversed by the m-th column of the section model of partitions with n sections (Cf. A135010, A206437). - _Omar E. Pol_, Apr 20 2012
%H A058399 Alois P. Heinz, <a href="/A058399/b058399.txt">Rows n = 1..141, flattened</a>
%F A058399 T(n, m) = Sum_{k=m..n} A008284(n, k).
%F A058399 G.f. for m-th column: Sum_{n>=1} x^(n)/Product_{k=1..n+m-1} (1 - x^k).
%F A058399 T(n, m) = Sum_{k=1..n} A207379(k, m). - _Omar E. Pol_, Apr 22 2012
%e A058399 From _Omar E. Pol_, Mar 10 2012: (Start)
%e A058399 Triangle begins:
%e A058399    1;
%e A058399    2,  1;
%e A058399    3,  2,  1;
%e A058399    5,  4,  2,  1;
%e A058399    7,  6,  4,  2,  1;
%e A058399   11, 10,  7,  4,  2,  1;
%e A058399   15, 14, 11,  7,  4,  2,  1;
%e A058399   22, 21, 17, 12,  7,  4,  2,  1;
%e A058399   30, 29, 25, 18, 12,  7,  4,  2,  1;
%e A058399   42, 41, 36, 28, 19, 12,  7,  4,  2,  1;
%e A058399   56, 55, 50, 40, 29, 19, 12,  7,  4,  2,  1;
%e A058399   77, 76, 70, 58, 43, 30, 19, 12,  7,  4,  2,  1;
%e A058399 (End)
%p A058399 b:= proc(n, k) option remember;
%p A058399       `if`(n=0, 1, `if`(k<1, 0, add(b(n-j*k, k-1), j=0..n/k)))
%p A058399     end:
%p A058399 T:= (n, m)-> b(n,n) -b(n,m-1):
%p A058399 seq (seq (T(n, m), m=1..n), n=1..15);  # _Alois P. Heinz_, Apr 20 2012
%t A058399 t[n_, m_] := Sum[ IntegerPartitions[n, {k}] // Length, {k, m, n}]; Table[t[n, m], {n, 1, 13}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 21 2013 *)
%Y A058399 Columns 1-5: A000041(n), A000065(n+1), A004250(n+2), A035300(n-1), A035301(n-1), n >= 1.
%Y A058399 Cf. A008284.
%K A058399 nonn,tabl
%O A058399 1,2
%A A058399 _Wolfdieter Lang_, Dec 11 2000