A226804 -id:A226804 - cp's OEIS frontend

A006102 Gaussian binomial coefficient [ n,4 ] for q=3.

1, 121, 11011, 925771, 75913222, 6174066262, 500777836042, 40581331447162, 3287582741506063, 266307564861468823, 21571273555248777493, 1747282899667791058573, 141530177899268957392924, 11463951511551877750726204, 928580264181940191843785764, 75215006575885931519565302404

Offset: 4

N. J. A. Sloane

nonn

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

T. D. Noe, Table of n, a(n) for n=4..100
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

Partial sums of A226804. - Christian Krause, Dec 26 2022

r:=4; q:=3; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 02 2016

A006102:=-1/((z-1)*(81*z-1)*(3*z-1)*(9*z-1)*(27*z-1)); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

Table[QBinomial[n, 4, 3], {n, 4, 24}] (* Vincenzo Librandi, Aug 02 2016 *)

[gaussian_binomial(n,4,3) for n in range(4,20)] # Zerinvary Lajos, May 25 2009

A227524 Expansion of 1/((1-3x)(1-9x)(1-27x)).

Original entry on oeis.org

1, 39, 1170, 32670, 891891, 24169509, 653373540, 17648258940, 476567558181, 12867905191779, 347438670325110, 9380891170278810, 253284485241566871, 6838684914320250849, 184644527001833063880, 4985402537886183692280, 134605871302457221445961

Offset: 0

Vincenzo Librandi, Jul 17 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (39,-351,729).

Cf. A016142, A226804.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-9*x)*(1-27*x))));

I:=[1, 39, 1170]; [n le 3 select I[n] else 39*Self(n-1)-351*Self(n-2)+729*Self(n-3): n in [1..25]];

CoefficientList[Series[1 / ((1 - 3 x) (1 - 9 x) (1 - 27 x)), {x, 0, 30}], x]
LinearRecurrence[{39,-351,729},{1,39,1170},20] (* Harvey P. Dale, Jul 04 2022 *)

G.f.: 1/((1-3*x)*(1-9*x)*(1-27*x)).
a(n) = 3^n*(3^(n+1)-1)*(3^(n+2)-1)/16.
a(0)=1, a(1)=39, a(2)=1170; for n>2, a(n) = 39*a(n-1)-351*a(n-2)+729*a(n-3).
a(n)-27*a(n-1) = A016142(n), with a(-1)=0; a(n) = A226804(n)-81*A226804(n-1), with A226804(-1)=0. [Bruno Berselli, Jul 17 2013]

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A006102 Gaussian binomial coefficient [ n,4 ] for q=3.

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A227524 Expansion of 1/((1-3x)(1-9x)(1-27x)).

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