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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 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

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