A006117 Sum of Gaussian binomial coefficients [ n,k ] for q=3.
1, 2, 6, 28, 212, 2664, 56632, 2052656, 127902864, 13721229088, 2544826627424, 815300788443072, 452436459318538048, 434188323928823259776, 722197777341507864283008, 2078153254879878944892861184, 10366904326991986000747424911616, 89478415088556766546699920236339712, 1338962661056423158371347974009398601216
Offset: 0
Examples
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 + ...
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)
Programs
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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
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Maple
f:=n-> 1+ add( mul((3^(n-i)-1)/(3^(i+1)-1), i=0..k-1), k=1..n);
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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 *) -
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
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