login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A006117 Sum of Gaussian binomial coefficients [ n,k ] for q=3.
(Formerly M1687)
9
1, 2, 6, 28, 212, 2664, 56632, 2052656, 127902864, 13721229088, 2544826627424, 815300788443072, 452436459318538048, 434188323928823259776, 722197777341507864283008, 2078153254879878944892861184, 10366904326991986000747424911616, 89478415088556766546699920236339712, 1338962661056423158371347974009398601216 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
REFERENCES
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.
LINKS
R. Chapman et al., 2-modular lattices from ternary codes, J. Th. des Nombres de Bordeaux, 14 (2002), 73-85.
S. Hitzemann, W. Hochstattler, On the combinatorics of Galois numbers, Discr. Math. 310 (2010) 3551-3557, Galois Numbers G_{n}^(2).
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
FORMULA
O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - 3^k*x). - Paul D. Hanna, Dec 06 2007
a(n) = 2*a(n-1)+(3^(n-1)-1)*a(n-2), n>1. [Hitzemann and Hochstattler] - R. J. Mathar, Aug 21 2013
a(n) ~ c * 3^(n^2/4), where c = EllipticTheta[3,0,1/3] / QPochhammer[1/3,1/3] = 3.019783845699... if n is even and c = EllipticTheta[2,0,1/3]/QPochhammer[1/3,1/3] = 3.018269046371... if n is odd. - Vaclav Kotesovec, Aug 21 2013
0 = a(n)*(2*a(n+1) + 2*a(n+2) - a(n+3)) + a(n+1)*(-6*a(n+1) + 3*a(n+2)) for all n in Z. - Michael Somos, Jan 25 2014
EXAMPLE
O.g.f.: A(x) = 1/(1-x) + x/((1-x)*(1-3x)) + x^2/((1-x)*(1-3x)*(1-9x)) + x^3/((1-x)*(1-3x)*(1-9x)*(1-27x)) + ...
Also generated by iterated binomial transforms in the following way:
[1,2,6,28,212,2664,56632,...] = BINOMIAL([1,1,3,15,129,1833,43347,..]);
[1,3,15,129,1833,43347,1705623,...] = BINOMIAL^2([1,1,7,67,1081,...]);
[1,7,67,1081,29185,1277887,...] = BINOMIAL^6([1,1,19,415,12961,...]);
[1,19,415,12961,684361,58352707,...] = BINOMIAL^18([1,1,55,3187,...]);
[1,55,3187,219673,22634209,...] = BINOMIAL^54([1,1,163,27055,4805569,...]);
etc.
G.f. = 1 + 2*x + 6*x^2 + 28*x^3 + 212*x^4 + 2664*x^5 + 56632*x^6 + 2052656*x^7 + ...
MAPLE
f:=n-> 1+ add( mul((3^(n-i)-1)/(3^(i+1)-1), i=0..k-1), k=1..n);
MATHEMATICA
Flatten[{1, RecurrenceTable[{a[n]==2*a[n-1]+(3^(n-1)-1)*a[n-2], a[0]==1, a[1]==2}, a, {n, 1, 15}]}] (* Vaclav Kotesovec, Aug 21 2013 *)
Table[Sum[QBinomial[n, k, 3], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2016 *)
PROG
(PARI) a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-3^j*x+x*O(x^n))), n) \\ Paul D. Hanna, Dec 06 2007
(Magma) [n le 2 select n else 2*Self(n-1)+(3^(n-2)-1)*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 13 2016
CROSSREFS
Sequence in context: A093657 A355064 A305627 * A118025 A226773 A370926
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 07:33 EDT 2024. Contains 371235 sequences. (Running on oeis4.)