A137681 -id:A137681 - cp's OEIS frontend

A137682 Left border of triangle A137680.

1, 1, 3, 7, 17, 40, 96, 228, 544, 1296, 3089, 7361, 17544, 41810, 99643, 237471, 565946, 1348773, 3214424, 7660679, 18257085, 43510652, 103695461, 247129108, 588963062, 1403628615, 3345155947, 7972242937, 18999609718, 45280252031

Offset: 1

Gary W. Adamson, Feb 05 2008

nonn

Each term in the sequence (n > 1) = sum of previous terms of triangle A137680 = partial sums of sequence A137681: (1, 2, 4, 10, 23, ...).

Starting (1, 3, 7, ...) = INVERT transform of A160096. - Gary W. Adamson, May 01 2009

			First few rows of triangle A137680 =
  1;
  1, 1;
  3, 0, 1;
  7, 2, 0, 1;
  ...
a(5) = 17 is the sum of 1 through 4 row terms of triangle A137680: (1 + 2 + 4 + 10); where (1, 2, 4, 10, 23, ...) = A137681 = row sums of triangle A137680 = first difference row of A137682, n > 1.

Cf. A137680, A137681.

Cf. A160096.

A137682 := proc(n)
    A137680(n,1) ;
end proc:
seq(A137682(n),n=1..30) ; # R. J. Mathar, Aug 12 2012

T[n_, k_] := T[n, k] = Which[k < 1 || k > n, 0, n == 1, 1, k == 1, Sum[T[r, j], {r, 1, n-1}, {j, 1, r}], True, T[n-1, k-1] - T[n-k, k-1]];
a[n_] := T[n, 1];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Sep 19 2024, after R. J. Mathar in A137680 *)

Partial sums of sequence A137681 prefaced with a 1. a(n) is the sum of all terms in rows 1 through (n-1) in triangle A137680.

A137680 Triangle read by rows, T(n,k) = T(n-1, k-1) - T(n-k, k-1); with leftmost term in each row = sum of all previous terms.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 7, 2, 0, 1, 17, 4, 1, 0, 1, 40, 10, 4, 1, 0, 1, 96, 23, 8, 3, 1, 0, 1, 228, 56, 19, 8, 3, 1, 0, 1, 544, 132, 46, 18, 7, 3, 1, 0, 1, 1296, 316, 109, 42, 18, 7, 3, 1, 0, 1, 3089, 752, 260, 101, 41, 17, 7, 3, 1, 0, 1, 7361, 1793, 620, 241, 98, 41, 17, 7, 3, 1, 0, 1, 17544, 4272, 1477, 574, 233, 97, 40, 17, 7, 3, 1, 0, 1

Offset: 1

Gary W. Adamson, Feb 05 2008

A variation of the same sequence = column 2 of the triangle: (1, 0, 2, 4, 10, 23, 56, 132, ...) = first difference row of column 1. Left border of the triangle = A137682.

Left column starting (1, 3, ...) = INVERT transform of A160096. - Gary W. Adamson, May 01 2009

			First few rows of the triangle:
     1;
     1,   1;
     3,   0,   1;
     7,   2,   0,   1;
    17,   4,   1,   0,  1;
    40,  10,   4,   1,  0,  1;
    96,  23,   8,   3,  1,  0, 1;
   228,  56,  19,   8,  3,  1, 0, 1;
   544, 132,  46,  18,  7,  3, 1, 0, 1;
  1296, 316, 109,  42, 18,  7, 3, 1, 0, 1;
  3089, 752, 260, 101, 41, 17, 7, 3, 1, 0, 1;
  ...

Cf. A137681 (row sums), A137682.

Cf. A160096. - Gary W. Adamson, May 01 2009

A137680 := proc(n,k)
    if k < 1 or k > n then
        0 ;
    elif n = 1 then
        1;
    elif k = 1 then
        add(add(procname(r,j),j=1..r),r=1..n-1) ;
    else
        procname(n-1,k-1)-procname(n-k,k-1) ;
    end if;
end proc: # R. J. Mathar, Aug 12 2012

T[n_, k_] := T[n, k] = Which[k < 1 || k > n, 0, n == 1, 1, k == 1, Sum[T[r, j], {r, 1, n-1}, {j, 1, r}], True, T[n-1, k-1] - T[n-k, k-1]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 02 2024, after R. J. Mathar *)

cp's OEIS Frontend

A137682 Left border of triangle A137680.

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