OFFSET
0,2
COMMENTS
Triangle of coefficients in expansion of (3 + 3x)^n = 3^n (1 +x)^n, where n is a nonnegative integer. (Coefficients in expansion of (1 +x)^n are given in A007318: Pascal's triangle). - Zagros Lalo, Jul 23 2018
REFERENCES
Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48
LINKS
Indranil Ghosh, Rows 0..100 of triangle, flattened
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.: 1/(1 - 3*x - 3*x*y). - Ilya Gutkovskiy, Apr 21 2017
T(0,0) = 1; T(n,k) = 3 T(n-1,k) + 3 T(n-1,k-1) for k = 0...n; T(n,k)=0 for n or k < 0. - Zagros Lalo, Jul 23 2018
From G. C. Greubel, Oct 17 2022: (Start)
T(n, k) = T(n, n-k).
T(n, n) = A000244(n).
T(n, n-1) = 3*A027471(n).
T(n, n-2) = 9*A027472(n+1).
T(n, n-3) = 27*A036216(n-3).
T(n, n-4) = 81*A036217(n-4).
T(n, n-5) = 243*A036219(n-5).
Sum_{k=0..n} T(n, k) = A000400(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A030195(n+1), n >= 0.
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A057083(n).
T(n, k) = 3^k * A027465(n, k). (End)
EXAMPLE
Triangle begins as:
1;
3, 3;
9, 18, 9;
27, 81, 81, 27;
81, 324, 486, 324, 81;
243, 1215, 2430, 2430, 1215, 243;
729, 4374, 10935, 14580, 10935, 4374, 729;
2187, 15309, 45927, 76545, 76545, 45927, 15309, 2187;
6561, 52488, 183708, 367416, 459270, 367416, 183708, 52488, 6561;
MATHEMATICA
(* programs from Zagros Lalo, Jul 23 2018 *)
t[0, 0]=1; t[n_, k_]:= t[n, k]= If[n<0 || k<0, 0, 3 t[n-1, k] + 3 t[n-1, k-1]]; Table[t[n, k], {n, 0, 10}, {k, 0, n}]//Flatten
Table[CoefficientList[Expand[3^n *(1+x)^n], x], {n, 0, 10}]//Flatten
Table[3^n Binomial[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* End *)
PROG
(Haskell)
a038221 n = a038221_list !! n
a038221_list = concat $ iterate ([3, 3] *) [1]
instance Num a => Num [a] where
fromInteger k = [fromInteger k]
(p:ps) + (q:qs) = p + q : ps + qs
ps + qs = ps ++ qs
(p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
_ * _ = []
-- Reinhard Zumkeller, Apr 02 2011
(GAP) Flat(List([0..8], i->List([0..i], j->Binomial(i, j)*3^(i-j)*3^j))); # Muniru A Asiru, Jul 23 2018
(Magma) [3^n*Binomial(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 17 2022
(SageMath)
def A038221(n, k): return 3^n*binomial(n, k)
flatten([[A038221(n, k) for k in range(n+1)] for n in range(10)]) # G. C. Greubel, Oct 17 2022
CROSSREFS
KEYWORD
AUTHOR
STATUS
approved