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A038221
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Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
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11
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48
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LINKS
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FORMULA
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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
T(n, k) = T(n, n-k).
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)
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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
_ * _ = []
(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)
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CROSSREFS
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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).
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KEYWORD
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AUTHOR
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STATUS
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approved
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