A006117 Sum of Gaussian binomial coefficients [ n,k ] for q=3. (Formerly M1687)

1, 2, 6, 28, 212, 2664, 56632, 2052656, 127902864, 13721229088, 2544826627424, 815300788443072, 452436459318538048, 434188323928823259776, 722197777341507864283008, 2078153254879878944892861184, 10366904326991986000747424911616, 89478415088556766546699920236339712, 1338962661056423158371347974009398601216

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

Vincenzo Librandi, Table of n, a(n) for n = 0..90

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

Adjacent sequences: A006114 A006115 A006116 * A006118 A006119 A006120

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved