A100851 - OEIS

%I #14 Nov 11 2024 03:58:01

%S 1,2,6,4,12,36,8,24,72,216,16,48,144,432,1296,32,96,288,864,2592,7776,

%T 64,192,576,1728,5184,15552,46656,128,384,1152,3456,10368,31104,93312,

%U 279936,256,768,2304,6912,20736,62208,186624,559872,1679616,512,1536,4608,13824,41472,124416,373248,1119744,3359232,10077696

%N Triangle read by rows: T(n,k) = 2^n * 3^k, 0 <= k <= n, n >= 0.

%H G. C. Greubel, <a href="/A100851/b100851.txt">Rows n = 0..100 of the triangle, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>

%F T(n,0) = A000079(n).

%F T(n,1) = A007283(n) for n>0.

%F T(n,2) = A005010(n) for n>1.

%F T(n,n) = A000400(n) = A100852(n,n).

%F Sum_{k=0..n} T(n, k) = A016129(n).

%F T(2*n, n) = A001021(n). - _Reinhard Zumkeller_, Mar 04 2006

%F G.f.: 1/((1 - 2*x)*(1 - 6*x*y)). - _Stefano Spezia_, Apr 28 2024

%F From _G. C. Greubel_, Nov 11 2024: (Start)

%F Sum_{k=0..n} (-1)^k*T(n, k) = A053524(n+1).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*((1-(-1)^n)*A248337((n+1)/2) + (1 + (-1)^n)*A016149(n/2)).

%F Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1/2)*(-1)^floor(n/2)*( (1+(-1)^n) *A051958((n+2)/2) + 2*(1-(-1)^n)*A051958((n+1)/2)). (End)

%e From _Stefano Spezia_, Apr 28 2024: (Start)

%e Triangle begins:

%e    1;

%e    2,  6;

%e    4, 12,  36;

%e    8, 24,  72, 216;

%e   16, 48, 144, 432, 1296;

%e   32, 96, 288, 864, 2592, 7776;

%e   ...

%e (End)

%t A100851[n_, k_]= 2^n*3^k;

%t Table[A100851[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 11 2024 *)

%o (Magma)

%o A100851:= func< n,k | 2^n*3^k >;

%o [A100851(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 11 2024

%o (SageMath)

%o def A100851(n,k): return 2^n*3^k

%o flatten([[A100851(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Nov 11 2024

%Y Cf. A000079, A000400, A001021, A003586, A005010, A007283, A016129.

%Y Cf. A016149, A051958, A053524, A100852, A248337.

%Y Cf. A065333(T(n, k))=1.

%K nonn,tabl,easy

%O 0,2

%A _Reinhard Zumkeller_, Nov 20 2004