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

%I A106585 #13 Sep 08 2021 01:27:34
%S A106585 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 A106585 13,34,89,185,322,481,611,611,1,2,5,13,34,89,233,514,973,1613,2354,
%U A106585 2965,2965,1,2,5,13,34,89,233,610,1405,2837,5090,8185,11761,14726,14726
%N A106585 Triangle read by rows: even-numbered rows of A106580.
%H A106585 G. C. Greubel, <a href="/A106585/b106585.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%F A106585 T(n, k) = A106580(2*n, k).
%e A106585 Irregular triangle begins as:
%e A106585   1;
%e A106585   1, 2, 2;
%e A106585   1, 2, 5,  7,  7;
%e A106585   1, 2, 5, 13, 22, 29,  29;
%e A106585   1, 2, 5, 13, 34, 65, 101, 130,  130;
%e A106585   1, 2, 5, 13, 34, 89, 185, 322,  481,  611,  611;
%e A106585   1, 2, 5, 13, 34, 89, 233, 514,  973, 1613, 2354, 2965, 2965;
%e A106585   1, 2, 5, 13, 34, 89, 233, 610, 1405, 2837, 5090, 8185, 11761, 14726, 14726;
%p A106585 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 A106585 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 A106585 Table[T[2*n, k], {n,0,12}, {k,0,2*n}]//Flatten (* _G. C. Greubel_, Sep 07 2021 *)
%o A106585 (Sage)
%o A106585 @CachedFunction
%o A106585 def T(n, k): # T(n, k) = A106580; T(2*n, k) = A106585
%o A106585     if (k<0): return 0
%o A106585     elif (k==0): return 1
%o A106585     else: return T(n, k-1) + sum( T(n-2*j, k-j) for j in (1..min(k, n//2, n-k)))
%o A106585 flatten([[T(2*n, k) for k in (0..2*n)] for n in (0..10)]) # _G. C. Greubel_, Sep 07 2021
%Y A106585 Cf. A106580, A106595.
%K A106585 nonn,tabf,easy
%O A106585 0,3
%A A106585 _N. J. A. Sloane_, May 30 2005
%E A106585 More terms from _R. J. Mathar_, Aug 10 2007