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A013613 Triangle of coefficients in expansion of (1+6x)^n.

%I A013613 #39 Aug 19 2021 01:01:26
%S A013613 1,1,6,1,12,36,1,18,108,216,1,24,216,864,1296,1,30,360,2160,6480,7776,
%T A013613 1,36,540,4320,19440,46656,46656,1,42,756,7560,45360,163296,326592,
%U A013613 279936,1,48,1008,12096,90720,435456,1306368,2239488,1679616
%N A013613 Triangle of coefficients in expansion of (1+6x)^n.
%C A013613 T(n,k) equals the number of n-length words on {0,1,...,6} having n-k zeros. - _Milan Janjic_, Jul 24 2015
%H A013613 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 A013613 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 A013613 G.f.: 1 / (1 - x(1+6y)).
%F A013613 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 A013613 T(n,k) = A007318(n,k)*A000400(k), 0 <= k <= n. - _Reinhard Zumkeller_, Nov 21 2013
%e A013613 Triangle begins:
%e A013613 1;
%e A013613 1, 6;
%e A013613 1, 12, 36;
%e A013613 1, 18, 108, 216;
%e A013613 1, 24, 216, 864, 1296;
%e A013613 ...
%o A013613 (Haskell)
%o A013613 import Data.List (inits)
%o A013613 a013613 n k = a013613_tabl !! n !! k
%o A013613 a013613_row n = a013613_tabl !! n
%o A013613 a013613_tabl = zipWith (zipWith (*))
%o A013613                (tail $ inits a000400_list) a007318_tabl
%o A013613 -- _Reinhard Zumkeller_, Nov 21 2013
%Y A013613 Cf. A038255 (mirrored).
%K A013613 tabl,nonn,easy
%O A013613 0,3
%A A013613 _N. J. A. Sloane_