A124959 - OEIS

%I #12 Sep 08 2022 08:45:28

%S 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 390,354,137,1,14,105,385,910,1239,959,314,1,16,140,616,1820,3304,

%U 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 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 Sum of entries in row n = A006190(n+1).

%H G. C. Greubel, <a href="/A124959/b124959.txt">Rows n = 0..100 of triangle, flattened</a>

%e First few rows of the triangle:

%e   1;

%e   1,   2;

%e   1,   4,   5;

%e   1,   6,  15,  11;

%e   1,   8,  30,  44,  26;

%e   1,  10,  50, 110, 130,  59;

%e   ...

%p 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 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 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 19 2019 *)

%o (PARI)

%o b(k) = if(k<2, k+1, b(k-1) + 3*b(k-2));

%o T(n,k) = binomial(n,k)*b(k);

%o for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Nov 19 2019

%o (Magma)

%o function b(k)

%o   if k lt 2 then return k+1;

%o   else return b(k-1) + 3*b(k-2);

%o   end if;

%o   return b;

%o end function;

%o [Binomial(n,k)*b(k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 19 2019

%o (Sage)

%o @CachedFunction

%o def b(k):

%o     if (k<2): return k+1

%o     else: return b(k-1) + 3*b(k-2)

%o [[binomial(n, k)*b(k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 19 2019

%Y Cf. A006138, A006190.

%K nonn,tabl

%O 0,3

%A _Gary W. Adamson_, Nov 13 2006

%E Edited by _N. J. A. Sloane_, Dec 03 2006