1,3

Arithmetic derivative of 3^n: a(n) = A003415(A000244(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Feb 26 2002

Binomial transform of A053220(n+1) is a(n+2). Binomial transform of A001787 is a(n+1). Binomial transform of A045883(n-1). - Michael Somos, Jul 10 2003

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 R. Janjic (agnus(AT)blic.net), 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 (rlahaye(AT)new.rr.com), Nov 19 2007

With a different offset, number of n-permutations of 4 objects u,v,w,z, with repetition allowed, containing exactly one u. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 27 2007

Milan Janjic, Two Enumerative Functions

F. Ellermann, Illustration of binomial transforms

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 715

G.f.: (x/(1-3*x))^2. E.g.f.: (1+(3x-1)exp(3x))/9. a(n) = 3^(n-2)*(n-1); (convolution of A000244, powers of 3, with itself) - from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de).

a(n)=6a(n-1)-9a(n-2); n>2; a(1)=0, a(2)=1. - Barry E. Williams, Jan 13 2000

A027471(n)=A036290(n)/3 - Paul Barry (pbarry(AT)wit.ie), Feb 06 2004

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)=sum{k=0..2n, T(n, k)*k}/3, where T(n, k) is given by A027907; a(n)=sum{k=0..n, sum{j=0..n, C(n, j)C(j, k)(j+k)}}/3; 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)}}. - Paul Barry (pbarry(AT)wit.ie), Feb 15 2005

Numerators of sequence a[ 2, n ] in (b^2)[ i, j ] where b[ i, j ] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i.

(PARI) a(n)=if(n<1, 0, (n-1)*3^(n-2))

Second column of A027465.

Cf. A006234.

Partial sums of A081038.

Adjacent sequences: A027468 A027469 A027470 this_sequence A027472 A027473 A027474

Sequence in context: A005325 A099623 A119852 this_sequence A037695 A094829 A055145

nonn

Olivier Gerard (ogerard(AT)ext.jussieu.fr)

Edited by Michael Somos, Jul 10, 2003