A013613 - OEIS

%I #39 Aug 19 2021 01:01:26

%S 1,1,6,1,12,36,1,18,108,216,1,24,216,864,1296,1,30,360,2160,6480,7776,

%T 1,36,540,4320,19440,46656,46656,1,42,756,7560,45360,163296,326592,

%U 279936,1,48,1008,12096,90720,435456,1306368,2239488,1679616

%N Triangle of coefficients in expansion of (1+6x)^n.

%C T(n,k) equals the number of n-length words on {0,1,...,6} having n-k zeros. - _Milan Janjic_, Jul 24 2015

%H Michael De Vlieger and Reinhard Zumkeller, <a href="/A013613/b013613.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened, rows 0..125 from Reinhard Zumkeller)

%H Ömür Deveci and Anthony G. Shannon, <a href="https://doi.org/10.20948/mathmontis-2021-50-4">Some aspects of Neyman triangles and Delannoy arrays</a>, Mathematica Montisnigri (2021) Vol. L, 36-43.

%F G.f.: 1 / (1 - x(1+6y)).

%F T(n,k) = 6^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*5^(n-i). Row sums are 7^n = A000420. - _Mircea Merca_, Apr 28 2012

%F T(n,k) = A007318(n,k)*A000400(k), 0 <= k <= n. - _Reinhard Zumkeller_, Nov 21 2013

%e Triangle begins:

%e 1;

%e 1, 6;

%e 1, 12, 36;

%e 1, 18, 108, 216;

%e 1, 24, 216, 864, 1296;

%e ...

%o (Haskell)

%o import Data.List (inits)

%o a013613 n k = a013613_tabl !! n !! k

%o a013613_row n = a013613_tabl !! n

%o a013613_tabl = zipWith (zipWith (*))

%o                (tail $ inits a000400_list) a007318_tabl

%o -- _Reinhard Zumkeller_, Nov 21 2013

%Y Cf. A038255 (mirrored).

%K tabl,nonn,easy

%O 0,3

%A _N. J. A. Sloane_