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A027471
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a(n) = (n-1)*3^(n-2), n > 0.
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57
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0, 1, 6, 27, 108, 405, 1458, 5103, 17496, 59049, 196830, 649539, 2125764, 6908733, 22320522, 71744535, 229582512, 731794257, 2324522934, 7360989291, 23245229340, 73222472421, 230127770466, 721764371007, 2259436291848
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OFFSET
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1,3
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COMMENTS
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If X_1,X_2,...,X_n are 3-blocks of a (3n+1)-set X then, for n >= 1, a(n+2) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). > - Milan Janjic, Nov 18 2007
Let S be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xSy if x is a subset of y. Then a(n+1) = the sum of the differences in size (i.e., |y|-|x|) for all (x, y) of S. - Ross La Haye, Nov 19 2007
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LINKS
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FORMULA
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G.f.: (x/(1-3*x))^2.
E.g.f.: (1 + (3*x-1)*exp(3*x))/9.
a(n) = 3^(n-2)*(n-1) (convolution of A000244, powers of 3, with itself). (End)
a(n) = Sum_{k=0..n} 3^(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2.
a(n) = (1/3)*Sum_{k=0..2n} T(n, k)*k, where T(n, k) is given by A027907.
a(n) = (1/3)*Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j+k).
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j-k).
a(n+1) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j+k+1). (End)
Sum_{n>=2} (-1)^n/a(n) = 3*log(4/3). - Amiram Eldar, Oct 28 2020
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MAPLE
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MATHEMATICA
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Table[(n-1)3^(n-2), {n, 30}] (* or *)
LinearRecurrence[{6, -9}, {0, 1}, 30] (* Harvey P. Dale, Apr 14 2016 *)
Range[0, 24]! CoefficientList[ Series[x*Exp[3 x], {x, 0, 24}], x] (* Robert G. Wilson v, Aug 03 2018 *)
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PROG
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(PARI) a(n)=if(n<1, 0, (n-1)*3^(n-2));
(Sage) [3^(n-2)*(n-1) for n in (1..30)] # G. C. Greubel, May 20 2021
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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