A208475 -id:A208475 - cp's OEIS frontend

A210948 Triangle read by rows: T(n,k) = sum of all parts <= k of all partitions of n.

1, 2, 4, 4, 6, 9, 7, 13, 16, 20, 12, 20, 26, 30, 35, 19, 35, 47, 55, 60, 66, 30, 52, 70, 82, 92, 98, 105, 45, 83, 110, 134, 149, 161, 168, 176, 67, 119, 164, 196, 221, 239, 253, 261, 270, 97, 179, 242, 294, 334, 364, 385, 401, 410, 420

Offset: 1

Omar E. Pol, May 01 2012

Row n lists the partial sums of row n of triangle A138785.

			Triangle begins:
1;
2,    4;
4,    6,   9;
7,   13,  16,  20;
12,  20,  26,  30,  35;
19,  35,  47,  55,  60,  66;
30,  52,  70,  82,  92,  98, 105;
45,  83, 110, 134, 149, 161, 168, 176;
67, 119, 164, 196, 221, 239, 253, 261, 270;

Alois P. Heinz, Rows n = 1..141, flattened

Column 1 is A000070(n-1). Right border gives A066186.

Cf. A181187, A138785, A206561, A210948, A208475, A210956.

p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
b:= proc(n, i) option remember; local f, g;
      if n=0 then [1]
    elif i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0], b(n-i, i));
         p (p (f, g), [0$i, g[1]*i])
      fi
    end:
T:= proc(n, k) option remember;
       b(n, n)[k+1] +`if`(k<2, 0, T(n, k-1))
    end:
seq (seq (T(n,k), k=1..n), n=1..12);  # Alois P. Heinz, May 02 2012

p[f_, g_] := With[{m = Max[Length[f], Length[g]]}, PadRight[f, m, 0] + PadRight[g, m, 0]]; b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1}, i == 1, {1, n}, True, f = b[n, i-1]; g = If[i>n, {0}, b[n-i, i]]; p[p[f, g], Append[Array[0&, i], i*g[[1]]]]]]; T[n_, k_] := T[n, k] = b[n, n][[k+1]] + If[k<2, 0, T[n, k-1]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12 }] // Flatten (* Jean-François Alcover, Mar 11 2015, after Alois P. Heinz *)

T(n,k) = sum_{j=1..k} A138785(n,j).

A208476 Triangle read by rows: T(n,k) = total sum of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

Original entry on oeis.org

1, 1, 2, 5, 0, 3, 3, 8, 0, 4, 13, 2, 8, 0, 5, 13, 18, 6, 10, 0, 6

Offset: 1

Omar E. Pol, Feb 28 2012

Essentially this sequence is related to A206562 in the same way as A207032 is related to A207031 and also in the same way as A206563 is related to A181187. See the calculation in the example section of A206563.

			Triangle begins:
1;
1,   2;
5,   0,  3;
3,   8,  0,  4;
13,  2,  8,  0,  5;
13, 18,  6, 10,  0,  6;

Right border is A000027.

Cf. A135010, A182703, A206562, A207031, A207032, A207383, A208475.

cp's OEIS Frontend

A210948 Triangle read by rows: T(n,k) = sum of all parts <= k of all partitions of n.

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