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A106522 A Pascal type matrix based on the tribonacci numbers.

%I A106522 #10 Aug 07 2021 01:13:19
%S A106522 1,1,1,2,2,1,4,4,3,1,7,8,7,4,1,13,15,15,11,5,1,24,28,30,26,16,6,1,44,
%T A106522 52,58,56,42,22,7,1,81,96,110,114,98,64,29,8,1,149,177,206,224,212,
%U A106522 162,93,37,9,1,274,326,383,430,436,374,255,130,46,10,1,504,600,709,813,866,810,629,385,176,56,11,1
%N A106522 A Pascal type matrix based on the tribonacci numbers.
%C A106522 Row sums of A106522 mod 2 are A106524.
%H A106522 G. C. Greubel, <a href="/A106522/b106522.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A106522 Riordan array (1/(1-x-x^2-x^3), x/(1-x)).
%F A106522 Number triangle T(n, 0) = A000073(n+2), T(n, k) = T(n-1, k-1) + T(n-1, k).
%F A106522 Sum_{k=0..n} T(n,k) = A001590(n+3).
%F A106522 Sum_{k=0..floor(n/2)} T(n-k, k) = A106523(n).
%e A106522 Triangle begins:
%e A106522    1;
%e A106522    1,  1;
%e A106522    2,  2,  1;
%e A106522    4,  4,  3,  1;
%e A106522    7,  8,  7,  4, 1;
%e A106522   13, 13, 15, 11, 5, 1;
%t A106522 b[n_]:= b[n]= If[n<2, 0, If[n==2, 1, b[n-1] +b[n-2] +b[n-3]]]; (* A000073 *)
%t A106522 T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, b[n+2], T[n-1, k-1] +T[n-1, k]]];
%t A106522 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Aug 06 2021 *)
%o A106522 (Sage)
%o A106522 @CachedFunction
%o A106522 def b(n): return 0 if (n<2) else 1 if (n==2) else b(n-1) +b(n-2) +b(n-3)
%o A106522 def T(n,k):
%o A106522     if (k<0 or k>n): return 0
%o A106522     elif (k==0): return b(n+2)
%o A106522     else: return T(n-1, k) + T(n-1, k-1)
%o A106522 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Aug 06 2021
%Y A106522 Cf. A000073, A001590 (row sums), A106523 (diagonal sums).
%K A106522 easy,nonn,tabl
%O A106522 0,4
%A A106522 _Paul Barry_, May 06 2005