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A027472 Third convolution of the powers of 3 (A000244). 21
1, 9, 54, 270, 1215, 5103, 20412, 78732, 295245, 1082565, 3897234, 13817466, 48361131, 167403915, 573956280, 1951451352, 6586148313, 22082967873, 73609892910, 244074908070 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

Third column of A027465.

With offset = 2, a(n) is the number of length n words on alphabet {u,v,w,z} such that each word contains exactly 2 u's.. - Zerinvary Lajos, Dec 29 2007

LINKS

Table of n, a(n) for n=3..22.

Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

FORMULA

Numerators of sequence a[ 3, 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.

a(n) = 3^(n-3)*binomial(n-1, 2); G.f.: (x/(1-3*x))^3. (Third convolution of A000244, powers of 3) - Wolfdieter Lang.

a(n) = |A075513(n, 2)|/9, n>=3.

a(n) = A152818(n-3,2)/2 = A006043(n-3)/2. - Paul Curtz, Jan 07 2009

The sequence 0, 1, 9, 54, ... has e.g.f. exp(3x)(x+3x^2/2) - Paul Barry, Jul 23 2003

E.g.f.: E(0) where E(k)= 1 + 3*(2*k+3)*x/((2*k+1)^2 - 3*x*(k+2)*(2*k+1)^2/(3*x*(k+2) + 2*(k+1)^2/E(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 23 2012

With offset=2 e.g.f.: exp(3x)*x^2/2. - Geoffrey Critzer, Oct 03 2013

MAPLE

BB:=2^(n+3)*z^3/(2*z-3*z^2)^3: gser:=series(BB, z=0, 20): seq(coeff(gser, z, n), n=0..19); # Zerinvary Lajos, Mar 29 2007

seq(seq(binomial(i+1, j)*3^(i-1), j =i-1), i=1..20); # Zerinvary Lajos, Dec 29 2007

MATHEMATICA

nn=21; Drop[Range[0, nn]!CoefficientList[Series[Exp[x]^3 x^2/2!, {x, 0, nn}], x], 2] (* Geoffrey Critzer, Oct 03 2013 *)

PROG

(Sage) [lucas_number2(n, 3, 0)*binomial(n, 2)/9 for n in xrange(2, 22)] # Zerinvary Lajos, Mar 10 2009

(PARI) a(n)=([0, 1, 0; 0, 0, 1; 27, -27, 9]^(n-3)*[1; 9; 54])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016

CROSSREFS

Cf. A027465.

Sequence in context: A023008 A079817 A169796 * A022637 A001392 A188428

Adjacent sequences:  A027469 A027470 A027471 * A027473 A027474 A027475

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

EXTENSIONS

Corrected by T. D. Noe, Nov 07 2006

Better name from Wolfdieter Lang

STATUS

approved

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Last modified October 20 07:17 EDT 2018. Contains 316378 sequences. (Running on oeis4.)