A106585 - OEIS

%I #13 Sep 08 2021 01:27:34

%S 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,

%T 13,34,89,185,322,481,611,611,1,2,5,13,34,89,233,514,973,1613,2354,

%U 2965,2965,1,2,5,13,34,89,233,610,1405,2837,5090,8185,11761,14726,14726

%N Triangle read by rows: even-numbered rows of A106580.

%H G. C. Greubel, <a href="/A106585/b106585.txt">Rows n = 0..50 of the irregular triangle, flattened</a>

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

%e Irregular triangle begins as:

%e   1;

%e   1, 2, 2;

%e   1, 2, 5,  7,  7;

%e   1, 2, 5, 13, 22, 29,  29;

%e   1, 2, 5, 13, 34, 65, 101, 130,  130;

%e   1, 2, 5, 13, 34, 89, 185, 322,  481,  611,  611;

%e   1, 2, 5, 13, 34, 89, 233, 514,  973, 1613, 2354, 2965, 2965;

%e   1, 2, 5, 13, 34, 89, 233, 610, 1405, 2837, 5090, 8185, 11761, 14726, 14726;

%p 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 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 *)

%t Table[T[2*n, k], {n,0,12}, {k,0,2*n}]//Flatten (* _G. C. Greubel_, Sep 07 2021 *)

%o (Sage)

%o @CachedFunction

%o def T(n, k): # T(n, k) = A106580; T(2*n, k) = A106585

%o     if (k<0): return 0

%o     elif (k==0): return 1

%o     else: return T(n, k-1) + sum( T(n-2*j, k-j) for j in (1..min(k, n//2, n-k)))

%o flatten([[T(2*n, k) for k in (0..2*n)] for n in (0..10)]) # _G. C. Greubel_, Sep 07 2021

%Y Cf. A106580, A106595.

%K nonn,tabf,easy

%O 0,3

%A _N. J. A. Sloane_, May 30 2005

%E More terms from _R. J. Mathar_, Aug 10 2007