A027472 Third convolution of the powers of 3 (A000244).

1, 9, 54, 270, 1215, 5103, 20412, 78732, 295245, 1082565, 3897234, 13817466, 48361131, 167403915, 573956280, 1951451352, 6586148313, 22082967873, 73609892910, 244074908070, 805447196631, 2646469360359, 8661172452084, 28242953648100, 91789599356325, 297398301914493, 960825283108362, 3095992578904722

Offset: 3

Author

Olivier Gérard

Keywords

• nonn
• easy

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

• G. C. Greubel, Table of n, a(n) for n = 3..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Crossrefs

Cf. A000244, A006043, A027465, A075513, A152818.

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), this sequence (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

Programs

Magma

[3^(n-3)*Binomial(n-1, 2): n in [3..40]]; // G. C. Greubel, May 12 2021

Mathematica

nn=41; Drop[Range[0,nn]!CoefficientList[Series[Exp[x]^3 x^2/2!,{x,0,nn}],x],2] (* Geoffrey Critzer, Oct 03 2013 *)
LinearRecurrence[{9,-27,27}, {1,9,54}, 40] (* G. C. Greubel, May 12 2021 *)
Abs[Take[CoefficientList[Series[1/(1+3x^2)^3,{x,0,60}],x],{1,-1,2}]] (* Harvey P. Dale, Mar 03 2022 *)

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

Sage

[3^(n-3)*binomial(n-1,2) for n in range(3, 40)] # Zerinvary Lajos, Mar 10 2009

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.

From Wolfdieter Lang: (Start)

a(n) = 3^(n-3)*binomial(n-1, 2).

G.f.: (x/(1-3*x))^3. (Third convolution of A000244, powers of 3.) (End)

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.: (x + 3*x^2/2)*exp(3*x)/. - 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.: x^2*exp(3*x)/2. - Geoffrey Critzer, Oct 03 2013

From Amiram Eldar, Jan 05 2022: (Start)

Sum_{n>=3} 1/a(n) = 6 - 12*log(3/2).

Sum_{n>=3} (-1)^(n+1)/a(n) = 24*log(4/3) - 6. (End)

Extensions

Corrected by T. D. Noe, Nov 07 2006

Better name from Wolfdieter Lang

Terms a(23) onward added by G. C. Greubel, May 12 2021