A106580 Triangle T(n, k) = T(n, k-1) + Sum_{i >= 1} T(n-2i, k-i) with T(n,0)=1, read by rows.

1, 1, 1, 1, 2, 2, 1, 2, 3, 3, 1, 2, 5, 7, 7, 1, 2, 5, 9, 12, 12, 1, 2, 5, 13, 22, 29, 29, 1, 2, 5, 13, 26, 41, 53, 53, 1, 2, 5, 13, 34, 65, 101, 130, 130, 1, 2, 5, 13, 34, 73, 129, 194, 247, 247, 1, 2, 5, 13, 34, 89, 185, 322, 481, 611, 611, 1, 2, 5, 13, 34, 89, 201, 386, 645, 945, 1192, 1192, 1, 2, 5, 13, 34, 89, 233, 514, 973, 1613, 2354, 2965, 2965

Offset

0,5

Comments

Next term is previous term + terms directly above you on a vertical line.

An intermingling of two independent triangles, A106595 and A106596.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = T(n, k-1) + Sum_{i>=1} T(n-2*i, k-i), with T(n, 0) = 1.

Example

Triangle begins as:
  1;
  1, 1;
  1, 2, 2;
  1, 2, 3,  3;
  1, 2, 5,  7,  7;
  1, 2, 5,  9, 12, 12;
  1, 2, 5, 13, 22, 29,  29;
  1, 2, 5, 13, 26, 41,  53,  53;
  1, 2, 5, 13, 34, 65, 101, 130, 130;

Maple

A106580:= proc(n, k) option remember; if k=0 then 1; else A106580(n, k-1) + add(A106580(n-2*i, k-i), i=1..min(k, floor(n/2), n-k)); fi ; end: for n from 0 to 12 do for k from 0 to n do printf("%d, ", A106580(n, k)); od; od; # R. J. Mathar, May 02 2007

Mathematica

t[_, 0]= 1; t[n_, k_]:= t[n, k] = t[n, k-1] + Sum[t[n-2j, k-j], {j, 1, Min[k, Floor[n/2], n-k]}]; Table[t[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jan 08 2014, after Maple *)

Prog

(Sage)
@CachedFunction
def T(n, k):
    if (k<0): return 0
    elif (k==0): return 1
    else: return T(n, k-1) + sum( T(n-2*j, k-j) for j in (1..min(k, n//2, n-k)))
flatten([[T(n, k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Sep 07 2021

Crossrefs

Cf. A106595, A106596.

Sequence in context: A193738 A230494 A368175 * A355080 A165915 A269783

Adjacent sequences: A106577 A106578 A106579 * A106581 A106582 A106583

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, May 30 2005

Extensions

More terms from R. J. Mathar, May 02 2007

Status

approved