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A220504 Triangle read by rows: T(n,k) is the total number of appearances of k as the smallest part in all partitions of n.

%I A220504 #48 Dec 18 2023 14:48:57
%S A220504 1,2,1,4,0,1,7,2,0,1,12,1,0,0,1,19,4,2,0,0,1,30,3,1,0,0,0,1,45,8,1,2,
%T A220504 0,0,0,1,67,7,4,1,0,0,0,0,1,97,15,3,1,2,0,0,0,0,1,139,15,4,1,1,0,0,0,
%U A220504 0,0,1,195,27,8,4,1,2,0,0,0,0,0,1,272,29,8,3,1,1,0,0,0,0,0,0,1
%N A220504 Triangle read by rows: T(n,k) is the total number of appearances of k as the smallest part in all partitions of n.
%C A220504 In other words, T(n,k) is the total number of appearances of k in all partitions of n whose smallest part is k.
%C A220504 The sum of row n equals spt(n), the smallest part partition function (see A092269).
%C A220504 T(n,k) is also the sum of row k in the slice n of tetrahedron A209314.
%H A220504 Alois P. Heinz, <a href="/A220504/b220504.txt">Rows n = 1..141, flattened</a>
%e A220504 Triangle begins:
%e A220504     1;
%e A220504     2,  1;
%e A220504     4,  0, 1;
%e A220504     7,  2, 0, 1;
%e A220504    12,  1, 0, 0, 1;
%e A220504    19,  4, 2, 0, 0, 1;
%e A220504    30,  3, 1, 0, 0, 0, 1;
%e A220504    45,  8, 1, 2, 0, 0, 0, 1;
%e A220504    67,  7, 4, 1, 0, 0, 0, 0, 1;
%e A220504    97, 15, 3, 1, 2, 0, 0, 0, 0, 1;
%e A220504   139, 15, 4, 1, 1, 0, 0, 0, 0, 0, 1;
%e A220504   195, 27, 8, 4, 1, 2, 0, 0, 0, 0, 0, 1;
%e A220504   272, 29, 8, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1;
%e A220504   ...
%e A220504 The partitions of 6 with the smallest part in brackets are
%e A220504 ..........................
%e A220504 .                      [6]
%e A220504 ..........................
%e A220504 .                  [3]+[3]
%e A220504 ..........................
%e A220504 .                   4 +[2]
%e A220504 .              [2]+[2]+[2]
%e A220504 ..........................
%e A220504 .                   5 +[1]
%e A220504 .               3 + 2 +[1]
%e A220504 .               4 +[1]+[1]
%e A220504 .           2 + 2 +[1]+[1]
%e A220504 .           3 +[1]+[1]+[1]
%e A220504 .       2 +[1]+[1]+[1]+[1]
%e A220504 .  [1]+[1]+[1]+[1]+[1]+[1]
%e A220504 ..........................
%e A220504 There are 19 smallest parts of size 1. Also there are four smallest parts of size 2. Also there are two smallest parts of size 3. There are no smallest part of size 4 or 5. Finally there is only one smallest part of size 6. So row 6 gives 19, 4, 2, 0, 0, 1. The sum of row 6 is 19+4+2+0+0+1 = A092269(6) = 26.
%p A220504 b:= proc(n, i) option remember; local j, r; if n=0 or i<1 then 0
%p A220504       else `if`(irem(n, i, 'r')=0, [0$(i-1), r], []); for j from 0
%p A220504       to n/i do zip((x, y)->x+y, %, [b(n-i*j, i-1)], 0) od; %[] fi
%p A220504     end:
%p A220504 T:= n-> b(n, n):
%p A220504 seq(T(n), n=1..20);  # _Alois P. Heinz_, Jan 20 2013
%t A220504 b[n_, i_] := b[n, i] = Module[{j, q, r, pc}, If [n == 0 || i<1, 0, {q, r} = QuotientRemainder[n, i]; pc = If[r == 0, Append[Array[0&, i-1], q], {}]; For[j = 0, j <= n/i, j++, pc = Plus @@ PadRight[{pc, b[n-i*j, i-1]}]]; pc]]; T[n_] := b[n, n]; Table[T[n], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Jan 30 2014, after _Alois P. Heinz_ *)
%Y A220504 Columns 1-3: A000070, A087787, A174455.
%Y A220504 Row sums give A092269.
%Y A220504 Cf. A026794, A182703, A209314.
%K A220504 nonn,tabl
%O A220504 1,2
%A A220504 _Omar E. Pol_, Jan 19 2013