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

3,2

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

With offset = 2, 2*a(n) is the number of length n words on {a,b,c,d,e} that contain exactly one a and exactly one b. - Enrique Navarrete, Dec 07 2025

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).

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)

From Enrique Navarrete, Dec 07 2025: (Start)

a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3).

E.g.f.: (1/54)*(9*x^2 - 6*x + 2)*exp(3*x) - 1/27. (End)

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 *)

Prog

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

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).

Partial sums of A383136.

First differences of A367591.

Sequence in context: A079817 A169796 A359722 * 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

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

Status

approved