A203243 -id:A203243 - cp's OEIS frontend

A006100 Gaussian binomial coefficient [n, 2] for q = 3.

1, 13, 130, 1210, 11011, 99463, 896260, 8069620, 72636421, 653757313, 5883904390, 52955405230, 476599444231, 4289397389563, 38604583680520, 347441274648040, 3126971536402441

Offset: 2

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=2..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)
Index entries for linear recurrences with constant coefficients, signature (13,-39,27).

Cf. A203243.

a:=n->sum((9^(n-j)-3^(n-j))/6,j=0..n): seq(a(n), n=1..17); # Zerinvary Lajos, Jan 15 2007
A006100:=-1/(z-1)/(3*z-1)/(9*z-1); # Simon Plouffe in his 1992 dissertation with offset 0

f[k_] := 3^(k - 1); t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 32}]    (* A203243 *)
Table[a[n]/3, {n, 2, 32}]  (* A006100 *)

[gaussian_binomial(n,2,3) for n in range(2,19)] # Zerinvary Lajos, May 25 2009

G.f.: x^2/[(1-x)(1-3x)(1-9x)].
a(n) = (9^n - 4*3^n + 3)/48. - Mitch Harris, Mar 23 2008
a(n) = 4*a(n-1) -3*a(n-2) +9^(n-2), n>=4. - Vincenzo Librandi, Mar 20 2011
From Peter Bala, Jul 01 2025: (Start)
G.f. with an offset of 0: exp(Sum_{n >= 1} b(3*n)/b(n)*x^n/n) = 1 + 13*x + 130*x^2 + ..., where b(n) = 3^n - 1.
The following series telescope:
Sum_{n >= 2} 3^n/a(n) = 12; Sum_{n >= 2} 3^n/(a(n)*a(n+3)) = 129/16900;
Sum_{n >= 2} 9^n/(a(n)*a(n+3)) = 1227/16900;
Sum_{n >= 2} 3^n/(a(n)*a(n+3)*a(n+6)) = 156706257/18829431219368770. (End)

A157783 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (3^(i-1)-x) in row n, column k, 0 <= k <= n.

Original entry on oeis.org

1, 1, -1, 3, -4, 1, 27, -39, 13, -1, 729, -1080, 390, -40, 1, 59049, -88209, 32670, -3630, 121, -1, 14348907, -21493836, 8027019, -914760, 33033, -364, 1, 10460353203, -15683355351, 5873190687, -674887059, 24995817, -298389, 1093

Offset: 0

Roger L. Bagula, Mar 06 2009

Row sums except n=0 are zero.

Triangle T(n,k), 0 <= k <= n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,...] (for q=3)=[1,2,9,24,81,234,729,2160,6561,...] DELTA [ -1,0,-3,0,-9,0,-27,0,-81,0,-243,0,...] where DELTA is the operator defined in A084938; see A122006 and A000244. - Philippe Deléham, Mar 09 2009

			Triangle begins
  1;
  1, -1;
  3, -4, 1;
  27, -39, 13, -1;
  729, -1080, 390, -40, 1;
  59049, -88209, 32670, -3630, 121, -1;
  14348907, -21493836, 8027019, -914760, 33033, -364, 1;
  10460353203, -15683355351, 5873190687, -674887059, 24995817, -298389, 1093, -1;
  22876792454961, -34309958505840, 12860351387820, -1481851188720, 55340738838, -677572560, 2688780, -3280, 1;
Row n=3 is 27 - 39*x + 13*x^2 - x^3.

Cf. A157832, A135950, A022166, A047656 (column k=1), A003462 (subdiagonal k=n-1), A203243 (subdiagonal k=n-2).

A157783 := proc(n,k)
    product( 3^(i-1)-x,i=1..n) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Oct 15 2013

Clear[f, q, M, n, m];
q = 3;
f[k_, m_] := If[k == m, q^(n - k), If[m == 1 && k < n, q^(n - k), If[k == n && m == 1, -(n-1), If[k == n && m > 1, 1, 0]]]];
M[n_] := Table[f[k, m], {k, 1, n}, {m, 1, n}];
Table[M[n], {n, 1, 10}];
Join[{1}, Table[Expand[CharacteristicPolynomial[M[n], x]], {n, 1, 7}]];
a = Join[{{ 1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 1, 7}]];
Flatten[a]

A203197 (n-1)-st elementary symmetric function of the first n terms of (1,3,9,27,...)=A000244.

Original entry on oeis.org

1, 4, 39, 1080, 88209, 21493836, 15683355351, 34309958505840, 225130514549271201, 4431394012508602048404, 261672339357326993189906439, 46354644349343413982791427120040, 24634789450813795903041020740742981169

Offset: 1

Clark Kimberling, Dec 30 2011

nonn

Cf. A000244, A003462 (1st symm. func.), A203243 (2nd symm. func.).

f[k_] := 3^(k - 1); t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}]  (* A203197 *)
Table[1/2 (3 - 1/3^(n-1)) 3^Binomial[n, 2], {n, 1, 20}] (* Emanuele Munarini, Sep 14 2017 *)

a(n) = (1/2)*(3-1/3^(n-1))*3^(binomial(n,2)). - Emanuele Munarini, Sep 14 2017

cp's OEIS Frontend

A006100 Gaussian binomial coefficient [n, 2] for q = 3.

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A157783 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (3^(i-1)-x) in row n, column k, 0 <= k <= n.

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A203197 (n-1)-st elementary symmetric function of the first n terms of (1,3,9,27,...)=A000244.

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