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A006117 Sum of Gaussian binomial coefficients [ n,k ] for q=3.
(Formerly M1687)
8
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

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.

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: A272662 A125812 A093657 * A118025 A226773 A119966

Adjacent sequences:  A006114 A006115 A006116 * A006118 A006119 A006120

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified July 25 15:34 EDT 2017. Contains 289795 sequences.