A038221 Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.

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

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

Cf. A007318, A304236, A304249.

Cf. A000007, A000400, A027465, A030195, A057083.

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

Adjacent sequences: A038218 A038219 A038220 * A038222 A038223 A038224

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane

Status

approved