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A064580 Triangle associated with rooted trees with a degree constraint (A036765).

%I A064580 #50 Aug 18 2024 16:42:19
%S A064580 1,1,1,1,2,2,1,3,5,5,1,4,9,14,13,1,5,14,28,40,36,1,6,20,48,87,118,104,
%T A064580 1,7,27,75,161,273,357,309,1,8,35,110,270,536,866,1100,939,1,9,44,154,
%U A064580 423,951,1782,2772,3441,2905,1,10,54,208,630,1572,3310,5928,8946,10900,9118
%N A064580 Triangle associated with rooted trees with a degree constraint (A036765).
%C A064580 Main diagonal is A036765. - _Paul D. Hanna_, Nov 18 2016
%F A064580 a(n, k) = a(n-1, k) + a(n-1, k-1) + a(n-1, k-2) + a(n-1, k-3) with a(0, 0)=1 and a(n, k)=0 if n < k or k < 0.
%e A064580 Triangle begins:
%e A064580   1;
%e A064580   1,  1;
%e A064580   1,  2,  2;
%e A064580   1,  3,  5,  5;
%e A064580   1,  4,  9, 14, 13;
%e A064580   1,  5, 14, 28, 40, 36;
%e A064580   ...
%t A064580 a[n_, k_] /; 0 <= k <= n = a[n, k] = a[n - 1, k] + a[n - 1, k - 1] + a[n - 1, k - 2] + a[n - 1, k - 3]; a[0, 0] = 1; a[_, _] = 0;
%t A064580 Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 30 2018 *)
%o A064580 (Sage) # uses[riordan_array from A256893]
%o A064580 M = riordan_array(1, x/(1+x+x^2+x^3), 12).inverse()
%o A064580 for m in M[1:]:
%o A064580     print([r for r in reversed(list(m)) if r != 0]) # _Peter Luschny_, Aug 17 2016
%Y A064580 Columns include A000012, A000027, A000096.
%Y A064580 Main diagonal is A036765.
%Y A064580 The sequence of triangles A010054 (triangle indicator), A007318 (Pascal), A026300 (Motzkin), A064580, ... converges to the triangle A009766 (Catalan).
%Y A064580 Row sums give A159772.
%K A064580 nonn,tabl
%O A064580 0,5
%A A064580 _Henry Bottomley_, Sep 21 2001
%E A064580 Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 17 2007