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A124959 Triangle read by rows: T(n,k) = a(k)*binomial(n,k) (0 <= k <= n), where a(0)=1, a(1)=2, a(k) = a(k-1) + 3*a(k-2) for k >= 2 (a(k) = A006138(k)).

%I A124959 #12 Sep 08 2022 08:45:28
%S A124959 1,1,2,1,4,5,1,6,15,11,1,8,30,44,26,1,10,50,110,130,59,1,12,75,220,
%T A124959 390,354,137,1,14,105,385,910,1239,959,314,1,16,140,616,1820,3304,
%U A124959 3836,2512,725,1,18,180,924,3276,7434,11508,11304,6525,1667,1,20,225,1320,5460,14868,28770,37680,32625,16670,3842
%N A124959 Triangle read by rows: T(n,k) = a(k)*binomial(n,k) (0 <= k <= n), where a(0)=1, a(1)=2, a(k) = a(k-1) + 3*a(k-2) for k >= 2 (a(k) = A006138(k)).
%C A124959 Sum of entries in row n = A006190(n+1).
%H A124959 G. C. Greubel, <a href="/A124959/b124959.txt">Rows n = 0..100 of triangle, flattened</a>
%e A124959 First few rows of the triangle:
%e A124959   1;
%e A124959   1,   2;
%e A124959   1,   4,   5;
%e A124959   1,   6,  15,  11;
%e A124959   1,   8,  30,  44,  26;
%e A124959   1,  10,  50, 110, 130,  59;
%e A124959   ...
%p A124959 a:=proc(n) if n=0 then 1 elif n=1 then 2 else a(n-1)+3*a(n-2) fi end: T:=(n,k)->a(k)*binomial(n,k): for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t A124959 T[n_, k_]:= T[n, k]= Simplify[(I*Sqrt[3])^(k-1)*Binomial[n,k]*(I*Sqrt[3]* ChebyshevU[k, 1/(2*I*Sqrt[3])] + ChebyshevU[k-1, 1/(2*I*Sqrt[3])])];
%t A124959 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 19 2019 *)
%o A124959 (PARI)
%o A124959 b(k) = if(k<2, k+1, b(k-1) + 3*b(k-2));
%o A124959 T(n,k) = binomial(n,k)*b(k);
%o A124959 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Nov 19 2019
%o A124959 (Magma)
%o A124959 function b(k)
%o A124959   if k lt 2 then return k+1;
%o A124959   else return b(k-1) + 3*b(k-2);
%o A124959   end if;
%o A124959   return b;
%o A124959 end function;
%o A124959 [Binomial(n,k)*b(k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 19 2019
%o A124959 (Sage)
%o A124959 @CachedFunction
%o A124959 def b(k):
%o A124959     if (k<2): return k+1
%o A124959     else: return b(k-1) + 3*b(k-2)
%o A124959 [[binomial(n, k)*b(k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 19 2019
%Y A124959 Cf. A006138, A006190.
%K A124959 nonn,tabl
%O A124959 0,3
%A A124959 _Gary W. Adamson_, Nov 13 2006
%E A124959 Edited by _N. J. A. Sloane_, Dec 03 2006