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A038221 Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j. 11
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
Columns k: A000244 (k=0), 3*A027471 (k=1), 3^2*A027472 (k=2), 3^3*A036216 (k=3), 3^4*A036217 (k=4), 3^5*A036219 (k=5), 3^6*A036220 (k=6), 3^7*A036221 (k=7), 3^8*A036222 (k=8), 3^9*A036223 (k=9), 3^10*A172362 (k=10).
Sequence in context: A241357 A217450 A089892 * A099465 A099094 A222169
KEYWORD
nonn,tabl,easy
AUTHOR
STATUS
approved

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Last modified April 22 20:16 EDT 2024. Contains 371906 sequences. (Running on oeis4.)