OFFSET
0,2
COMMENTS
Triangle of coefficients in expansion of (4+x)^n. - N-E. Fahssi, Apr 13 2008
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.
FORMULA
G.f. for j-th column is (x^j)/(1-4*x)^(j+1).
Convolution triangle of A000302 (powers of 4).
See A038207 and A027465 and replace 2 and 3 in analogous formulas with 4. - Tom Copeland, Oct 26 2012
EXAMPLE
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;
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 *)
PROG
(PARI) T(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
(Sage) [[4^(n-k)*binomial(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 20 2019
(GAP) Flat(List([0..10], n-> List([0..n], k-> 4^(n-k)*Binomial(n, k) ))); # G. C. Greubel, Jul 20 2019
CROSSREFS
KEYWORD
AUTHOR
STATUS
approved