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

1, 1, 6, 1, 12, 36, 1, 18, 108, 216, 1, 24, 216, 864, 1296, 1, 30, 360, 2160, 6480, 7776, 1, 36, 540, 4320, 19440, 46656, 46656, 1, 42, 756, 7560, 45360, 163296, 326592, 279936, 1, 48, 1008, 12096, 90720, 435456, 1306368, 2239488, 1679616

Offset: 0

Author

N. J. A. Sloane

Keywords

• tabl
• nonn
• easy

Comments

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

Examples

			Triangle begins:
1;
1, 6;
1, 12, 36;
1, 18, 108, 216;
1, 24, 216, 864, 1296;
...
		

Links

• Michael De Vlieger and Reinhard Zumkeller, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened, rows 0..125 from Reinhard Zumkeller)
• Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.

Crossrefs

Cf. A038255 (mirrored).

Programs

Haskell

import Data.List (inits)
a013613 n k = a013613_tabl !! n !! k
a013613_row n = a013613_tabl !! n
a013613_tabl = zipWith (zipWith (*))
               (tail $ inits a000400_list) a007318_tabl
-- Reinhard Zumkeller, Nov 21 2013

Formula

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

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

T(n,k) = A007318(n,k)*A000400(k), 0 <= k <= n. - Reinhard Zumkeller, Nov 21 2013