A106595 -id:A106595 - cp's OEIS frontend

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

N. J. A. Sloane, May 30 2005

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

An intermingling of two independent triangles, A106595 and A106596.

			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;

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

Cf. A106595, A106596.

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

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 *)

@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

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

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

A106585 Triangle read by rows: even-numbered rows of A106580.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 5, 7, 7, 1, 2, 5, 13, 22, 29, 29, 1, 2, 5, 13, 34, 65, 101, 130, 130, 1, 2, 5, 13, 34, 89, 185, 322, 481, 611, 611, 1, 2, 5, 13, 34, 89, 233, 514, 973, 1613, 2354, 2965, 2965, 1, 2, 5, 13, 34, 89, 233, 610, 1405, 2837, 5090, 8185, 11761, 14726, 14726

Offset: 0

N. J. A. Sloane, May 30 2005

			Irregular triangle begins as:
  1;
  1, 2, 2;
  1, 2, 5,  7,  7;
  1, 2, 5, 13, 22, 29,  29;
  1, 2, 5, 13, 34, 65, 101, 130,  130;
  1, 2, 5, 13, 34, 89, 185, 322,  481,  611,  611;
  1, 2, 5, 13, 34, 89, 233, 514,  973, 1613, 2354, 2965, 2965;
  1, 2, 5, 13, 34, 89, 233, 610, 1405, 2837, 5090, 8185, 11761, 14726, 14726;

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

Cf. A106580, A106595.

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 18 by 2 do for k from 0 to n do printf("%d, ",A106580(n,k)); od; od; # R. J. Mathar, Aug 10 2007

T[n_, k_]:= T[n, k]= If[k==0, 1, T[n, k-1] + Sum[T[n-2*j, k-j], {j, 1, Min[k, Floor[n/2], n-k]}]];  (* T(n, k) = A106580; T(2*n, k) = A106585 *)
Table[T[2*n, k], {n,0,12}, {k,0,2*n}]//Flatten (* G. C. Greubel, Sep 07 2021 *)

@CachedFunction
def T(n, k): # T(n, k) = A106580; T(2*n, k) = A106585
    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(2*n, k) for k in (0..2*n)] for n in (0..10)]) # G. C. Greubel, Sep 07 2021

T(n, k) = A106580(2*n, k).

More terms from R. J. Mathar, Aug 10 2007

cp's OEIS Frontend

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.

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