A015121
Triangle of q-binomial coefficients for q=-9.
Original entry on oeis.org
1, 1, 1, 1, -8, 1, 1, 73, 73, 1, 1, -656, 5986, -656, 1, 1, 5905, 484210, 484210, 5905, 1, 1, -53144, 39226915, -352504880, 39226915, -53144, 1, 1, 478297, 3177326971, 257015284435, 257015284435, 3177326971, 478297, 1, 1, -4304672, 257363962948
Offset: 0
Cf. analog triangles for other q:
A015109 (q=-2),
A015110 (q=-3),
A015112 (q=-4),
A015113 (q=-5),
A015116 (q=-6),
A015117 (q=-7),
A015118 (q=-8),
A015123 (q=-10),
A015124 (q=-11),
A015125 (q=-12),
A015129 (q=-13),
A015132 (q=-14),
A015133 (q=-15). -
M. F. Hasler, Nov 04 2012
Cf. analog triangles for positive q=2,...,24:
A022166 (q=2),
A022167 (q=3),
A022168,
A022169,
A022170,
A022171,
A022172,
A022173,
A022174 (q=10),
A022175,
A022176,
A022177,
A022178,
A022179,
A022180,
A022181,
A022182,
A022183,
A022184 (q=20),
A022185,
A022186,
A022187,
A022188. -
M. F. Hasler, Nov 05 2012
-
Table[QBinomial[n, k, -9], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)
-
T015121(n, k, q=-9)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) - M. F. Hasler, Nov 04 2012
A015123
Triangle of q-binomial coefficients for q=-10.
Original entry on oeis.org
1, 1, 1, 1, -9, 1, 1, 91, 91, 1, 1, -909, 9191, -909, 1, 1, 9091, 918191, 918191, 9091, 1, 1, -90909, 91828191, -917272809, 91828191, -90909, 1, 1, 909091, 9182728191, 917364637191, 917364637191, 9182728191, 909091, 1, 1, -9090909, 918273728191
Offset: 0
Cf. analog triangles for other q:
A015109 (q=-2),
A015110 (q=-3),
A015112 (q=-4),
A015113 (q=-5),
A015116 (q=-6),
A015117 (q=-7),
A015118 (q=-8),
A015121 (q=-9),
A015124 (q=-11),
A015125 (q=-12),
A015129 (q=-13),
A015132 (q=-14),
A015133 (q=-15). -
M. F. Hasler, Nov 04 2012
Cf. analog triangles for positive q=2,...,24:
A022166 (q=2),
A022167 (q=3),
A022168,
A022169,
A022170,
A022171,
A022172,
A022173,
A022174 (q=10),
A022175,
A022176,
A022177,
A022178,
A022179,
A022180,
A022181,
A022182,
A022183,
A022184 (q=20),
A022185,
A022186,
A022187,
A022188. -
M. F. Hasler, Nov 05 2012
-
Table[QBinomial[n, k, -10], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)
-
T015123(n, k, q=-10)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) - M. F. Hasler, Nov 04 2012
A015124
Triangle of q-binomial coefficients for q=-11.
Original entry on oeis.org
1, 1, 1, 1, -10, 1, 1, 111, 111, 1, 1, -1220, 13542, -1220, 1, 1, 13421, 1637362, 1637362, 13421, 1, 1, -147630, 198134223, -2177691460, 198134223, -147630, 1, 1, 1623931, 23974093353, 2898705467483, 2898705467483, 23974093353, 1623931, 1, 1
Offset: 0
Cf. analog triangles for other negative q=-2,...,-15:
A015109 (q=-2),
A015110 (q=-3),
A015112 (q=-4),
A015113 (q=-5),
A015116 (q=-6),
A015117 (q=-7),
A015118 (q=-8),
A015121 (q=-9),
A015123 (q=-10),
A015125 (q=-12),
A015129 (q=-13),
A015132 (q=-14),
A015133 (q=-15). -
M. F. Hasler, Nov 04 2012
Cf. analog triangles for positive q=2,...,24:
A022166 (q=2),
A022167 (q=3),
A022168,
A022169,
A022170,
A022171,
A022172,
A022173,
A022174 (q=10),
A022175,
A022176,
A022177,
A022178,
A022179,
A022180,
A022181,
A022182,
A022183,
A022184 (q=20),
A022185,
A022186,
A022187,
A022188. - M. F. Hasler, Nov 05 2012
-
T015124(n, k, q=-11)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) - M. F. Hasler, Nov 04 2012
A015125
Triangle of q-binomial coefficients for q=-12.
Original entry on oeis.org
1, 1, 1, 1, -11, 1, 1, 133, 133, 1, 1, -1595, 19285, -1595, 1, 1, 19141, 2775445, 2775445, 19141, 1, 1, -229691, 399683221, -4793193515, 399683221, -229691, 1, 1, 2756293, 57554154133, 8283038077141, 8283038077141, 57554154133, 2756293, 1, 1
Offset: 0
Cf. analog triangles for other negative q=-2,...,-15:
A015109 (q=-2),
A015110 (q=-3),
A015112 (q=-4),
A015113 (q=-5),
A015116 (q=-6),
A015117 (q=-7),
A015118 (q=-8),
A015121 (q=-9),
A015123 (q=-10),
A015124 (q=-11),
A015129 (q=-13),
A015132 (q=-14),
A015133 (q=-15). -
M. F. Hasler, Nov 04 2012
Cf. analog triangles for positive q=2,...,24:
A022166 (q=2),
A022167 (q=3),
A022168,
A022169,
A022170,
A022171,
A022172,
A022173,
A022174 (q=10),
A022175,
A022176,
A022177,
A022178,
A022179,
A022180,
A022181,
A022182,
A022183,
A022184 (q=20),
A022185,
A022186,
A022187,
A022188. -
M. F. Hasler, Nov 05 2012
-
T015125(n, k, q=-12)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) - M. F. Hasler, Nov 04 2012
A015132
Triangle of (Gaussian) q-binomial coefficients for q=-14.
Original entry on oeis.org
1, 1, 1, 1, -13, 1, 1, 183, 183, 1, 1, -2561, 36051, -2561, 1, 1, 35855, 7063435, 7063435, 35855, 1, 1, -501969, 1384469115, -19375002205, 1384469115, -501969, 1, 1, 7027567, 271355444571, 53166390519635, 53166390519635, 271355444571
Offset: 0
Cf. analog triangles for other negative q=-2,...,-15:
A015109 (q=-2),
A015110 (q=-3),
A015112 (q=-4),
A015113 (q=-5),
A015116 (q=-6),
A015117 (q=-7),
A015118 (q=-8),
A015121 (q=-9),
A015123 (q=-10),
A015124 (q=-11),
A015125 (q=-12),
A015129 (q=-13),
A015133 (q=-15). -
M. F. Hasler, Nov 04 2012
Cf. analog triangles for positive q=2,...,24:
A022166 (q=2),
A022167 (q=3),
A022168,
A022169,
A022170,
A022171,
A022172,
A022173,
A022174 (q=10),
A022175,
A022176,
A022177,
A022178,
A022179,
A022180,
A022181,
A022182,
A022183,
A022184 (q=20),
A022185,
A022186,
A022187,
A022188. -
M. F. Hasler, Nov 05 2012
-
T015132(n, k, q=-14)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) - M. F. Hasler, Nov 04 2012
A015195
Sum of Gaussian binomial coefficients for q=9.
Original entry on oeis.org
1, 2, 12, 184, 9104, 1225248, 540023488, 652225844096, 2584219514040576, 28081351726592246272, 1001235747932175990213632, 97915621602690773814148184064, 31420034518763282871588038742544384, 27654326463468067495668136467306727743488
Offset: 0
- 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.
- M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
-
Total/@Table[QBinomial[n, m, 9], {n, 0, 20}, {m, 0, n}] (* Vincenzo Librandi, Nov 01 2012 *)
Flatten[{1,RecurrenceTable[{a[n]==2*a[n-1]+(9^(n-1)-1)*a[n-2],a[0]==1,a[1]==2},a,{n,1,15}]}] (* Vaclav Kotesovec, Aug 21 2013 *)
A173585
Triangle T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, q) = Product_{j=1..n} t(2*j, q), t(n, q) = (1/4)*( (2 + sqrt(q))^n + (2 - sqrt(q))^n - 2 ), and q = 3, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 16, 1, 1, 225, 225, 1, 1, 3136, 44100, 3136, 1, 1, 43681, 8561476, 8561476, 43681, 1, 1, 608400, 1660970025, 23150231104, 1660970025, 608400, 1, 1, 8473921, 322220846025, 62555239000969, 62555239000969, 322220846025, 8473921, 1
Offset: 0
Triangle begins as:
1;
1, 1;
1, 16, 1;
1, 225, 225, 1;
1, 3136, 44100, 3136, 1;
1, 43681, 8561476, 8561476, 43681, 1;
1, 608400, 1660970025, 23150231104, 1660970025, 608400, 1;
-
b:= func< n, k | n eq 0 select 1 else k eq 0 select Factorial(n) else (&*[1 - Evaluate(ChebyshevT(j), k+1)^2 : j in [1..n]]) >;
T:= func< n,k,m | b(n,m)/(b(k,m)*b(n-k,m)) >;
[T(n,k,1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 06 2021
-
(* First program *)
f[n_, q_]:= (1/4)*((2+Sqrt[q])^n + (2-Sqrt[q])^n -2);
c[n_, q_]:= Product[f[k, q], {k, 2, n, 2}]//Simplify;
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n - k, q]);
Table[T[n, k, 3], {n, 0, 10, 2}, {k, 0, n, 2}]//Flatten (* modified by G. C. Greubel, Jul 06 2021 *)
(* Second program *)
t[n_, q_]:= (1/4)*(Round[(2+Sqrt[q])^n + (2-Sqrt[q])^n] -2);
c[n_, q_]:= Product[t[2*j, q], {j,n}];
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
Table[T[n, k, 3], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 06 2021 *)
-
@CachedFunction
def f(n,q): return (1/4)*( round((2 + sqrt(q))^n + (2 - sqrt(q))^n) - 2 )
def c(n,q): return product( f(2*j, q) for j in (1..n))
def T(n,k,q): return c(n, q)/(c(k, q)*c(n-k, q))
flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 06 2021
A347491
Irregular triangle read by rows: T(n, k) is the q-multinomial coefficient defined by the k-th partition of n in Abramowitz-Stegun order, evaluated at q = 9.
Original entry on oeis.org
1, 1, 10, 1, 91, 910, 1, 820, 7462, 74620, 746200, 1, 7381, 605242, 6052420, 55077022, 550770220, 5507702200, 1, 66430, 49031983, 441826660, 490319830, 40206226060, 365876657146, 402062260600, 3658766571460, 36587665714600, 365876657146000, 1, 597871
Offset: 1
The number of subspace chains 0 < V_1 < V_2 < (F_9)^3 is 910 = T(3, (1, 1, 1)). There are 91 = A022173(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 10 = A022173(2, 1) extensions to a two-dimensional subspace V_2.
Triangle begins:
k: 1 2 3 4 5
-----------------------
n=1: 1
n=2: 1 10
n=3: 1 91 910
n=4: 1 820 7462 74620 746200
- R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.
A347975
Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_9)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).
Original entry on oeis.org
1, 1, 1, 1, 6, 1, 1, 21, 21, 1, 1, 64, 374, 64, 1, 1, 163, 5900, 5900, 163, 1, 1, 380, 82587, 644680, 82587, 380, 1, 1, 809, 1018388, 66136870, 66136870, 1018388, 809, 1, 1, 1619, 11174165, 6057912073, 52901629980, 6057912073, 11174165, 1619, 1, 1, 3049, 110404788
Offset: 0
Triangle begins:
k: 0 1 2 3 4 5
--------------------------
n=0: 1
n=1: 1 1
n=2: 1 6 1
n=3: 1 21 21 1
n=4: 1 64 374 64 1
n=5: 1 163 5900 5900 163 1
There are 10 = A022173(2, 1) one-dimensional subspaces in (F_9)^2. Among them, <(1, 1)> and <(1, 2)> are invariant by coordinate swap and the rest are grouped in orbits of size two. Hence, T(2, 1) = 6.
A173583
Triangle T(n, k, q) = q-binomial(n, k, q^2), for q = 5, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 26, 1, 1, 651, 651, 1, 1, 16276, 407526, 16276, 1, 1, 406901, 254720026, 254720026, 406901, 1, 1, 10172526, 159200423151, 3980255126276, 159200423151, 10172526, 1, 1, 254313151, 99500274641901, 62191645548485651, 62191645548485651, 99500274641901, 254313151, 1
Offset: 0
Triangle begins as:
1;
1, 1;
1, 26, 1;
1, 651, 651, 1;
1, 16276, 407526, 16276, 1;
1, 406901, 254720026, 254720026, 406901, 1;
1, 10172526, 159200423151, 3980255126276, 159200423151, 10172526, 1;
-
q:=5;; [q^(k*(n-k))*GaussianBinomial(n, k, q): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 22 2021
-
(* First program *)
c[n_, q_]:= Product[(1 -q^(2*j))/(1-q), {j,1,n}];
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
Table[T[n, k, 5], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
Table[QBinomial[n,k,5^2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 22 2021 *)
T[n_, k_, p_]:= T[n, k, p] = If[k==0 || k==n, 1, T[n-1, k-1, p] + p^k*T[n-1, k, q]]; Table[T[n, k, 25], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 22 2021 *)
-
flatten([[q_binomial(n, k, 5^2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 22 2021
Comments