A013611 Triangle of coefficients in expansion of (1+4x)^n.

1, 1, 4, 1, 8, 16, 1, 12, 48, 64, 1, 16, 96, 256, 256, 1, 20, 160, 640, 1280, 1024, 1, 24, 240, 1280, 3840, 6144, 4096, 1, 28, 336, 2240, 8960, 21504, 28672, 16384, 1, 32, 448, 3584, 17920, 57344, 114688, 131072, 65536, 1, 36, 576, 5376, 32256, 129024, 344064, 589824, 589824, 262144

Offset

0,3

Comments

T(n,k) equals the number of n-length words on {0,1,2,3,4} having n-k zeros. - Milan Janjic, Jul 24 2015

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1034 (rows 0..44 flattened, missing terms added by Sean A. Irvine, Apr 21 2019)

J. Goldman, J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 1-rook coefficients for k rooks on the 4xn board, all heights 4.

Formula

G.f.: 1 / (1 - x(1+4y)).

T(n,k) = 4^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*3^(n-i). Row sums are 5^n = A000351. - Mircea Merca, Apr 28 2012

Example

Triangle begins
  1;
  1,    4;
  1,    8,   16;
  1,   12,   48,   64;
  1,   16,   96,  256,  256;
  1,   20,  160,  640, 1280, 1024;
  1,   24,  240, 1280, 3840, 6144, 4096;

Maple

T:= n-> (p-> seq(coeff(p, x, k), k=0..n))((1+4*x)^n):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 24 2015

Mathematica

Flatten[Table[CoefficientList[Series[(1+4x)^n, {x, 0, 10}], x], {n, 0, 15}]] (* Harvey P. Dale, Oct 10 2011 *)

Crossrefs

Cf. A000351 (5^n).

Sequence in context: A324780 A280108 A299583 * A297194 A077910 A366399

Adjacent sequences: A013608 A013609 A013610 * A013612 A013613 A013614

Keyword

tabl,nonn,easy

Author

N. J. A. Sloane

Status

approved