A038231 Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy

Comments

Triangle of coefficients in expansion of (4+x)^n. - N-E. Fahssi, Apr 13 2008

Examples

			Triangle begins:
      1;
      4,      1;
     16,      8,      1;
     64,     48,     12,     1;
    256,    256,     96,    16,     1;
   1024,   1280,    640,   160,    20,    1;
   4096,   6144,   3840,  1280,   240,   24,   1;
  16384,  28672,  21504,  8960,  2240,  336,  28,  1;
  65536, 131072, 114688, 57344, 17920, 3584, 448, 32, 1;
		

Links

• Indranil Ghosh, Rows 0..125 of triangle, flattened
• Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
• B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Crossrefs

Cf. A000302, A013611 (row-reversed), A000351 (row sums).

Programs

GAP

Flat(List([0..10], n-> List([0..n], k-> 4^(n-k)*Binomial(n, k) ))); # G. C. Greubel, Jul 20 2019

Magma

[4^(n-k)*Binomial(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jul 20 2019

Maple

for i from 0 to 10 do seq(binomial(i, j)*4^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007
# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> 4^(n-1)); # Peter Luschny, Oct 09 2022

Mathematica

Table[4^(n-k)*Binomial[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 20 2019 *)

PARI

T(n,k) = 4^(n-k)*binomial(n, k); \\ G. C. Greubel, Jul 20 2019

Sage

[[4^(n-k)*binomial(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 20 2019

Formula

G.f. for j-th column is (x^j)/(1-4*x)^(j+1).

Convolution triangle of A000302 (powers of 4).

Sum_{k=0..n} T(n,k)*(-1)^k*A000108(k) = A001700(n). - Philippe Deléham, Nov 27 2009

See A038207 and A027465 and replace 2 and 3 in analogous formulas with 4. - Tom Copeland, Oct 26 2012