A015109 -id:A015109 - cp's OEIS frontend

A015323 Gaussian binomial coefficient [ n,6 ] for q = -2.

1, 43, 3655, 208335, 14208447, 882215391, 57344000415, 3642010817055, 233988483199263, 14946527496991519, 957498220445101855, 61250446192484546335, 3920970870875818419999, 250911985465716094666527

Offset: 6

Olivier Gérard, Dec 11 1999

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.

Vincenzo Librandi, Table of n, a(n) for n = 6..200
Index entries for linear recurrences with constant coefficients, signature (43,1806,-26488,-211904,924672,1409024,-2097152).

Diagonal k=6 of the triangular array A015109. See there for further references and programs. - M. F. Hasler, Nov 04 2012

Table[QBinomial[n, 6, -2], {n, 6, 20}] (* Vincenzo Librandi, Oct 29 2012 *)

[gaussian_binomial(n,6,-2) for n in range(6,20)] # Zerinvary Lajos, May 27 2009

A015323(n) = T[n,6] where T is the triangular array A015109. - M. F. Hasler, Nov 04 2012

G.f.: x^6 / ( (x-1)*(8*x+1)*(64*x-1)*(2*x+1)*(32*x+1)*(4*x-1)*(16*x-1) ). - R. J. Mathar, Aug 04 2016

A015338 Gaussian binomial coefficient [ n,7 ] for q = -2.

Original entry on oeis.org

1, -85, 14535, -1652145, 225683007, -28005209505, 3642010817055, -462535373765985, 59438516325245343, -7593183562134412385, 972884994173649887135, -124468028808034701006945

Offset: 7

Olivier Gérard, Dec 11 1999

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.

G. C. Greubel, Table of n, a(n) for n = 7..450[Terms 7 through 200 were computed by Vincenzo Librandi; terms 201 to 450 by G. C. Greubel, Nov 06 2016]

Diagonal k=7 of the triangular array A015109. See there for further references and programs. - M. F. Hasler, Nov 04 2012

/* By definition: */ r:=7; q:=-2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Bruno Berselli, Oct 30 2012

Table[QBinomial[n, 7, -2], {n, 7, 20}] (* Vincenzo Librandi, Oct 29 2012 *)

[gaussian_binomial(n,7,-2) for n in range(7,19)] # Zerinvary Lajos, May 27 2009

A015405 Gaussian binomial coefficient [ n,11 ] for q=-2.

Original entry on oeis.org

1, -1365, 3727815, -6785865905, 14824402656063, -29439916001972385, 61250446192484546335, -124468028808034701006945, 255910660218571393553843871, -523082886040328458081329117025

Offset: 11

Olivier Gérard

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.

Vincenzo Librandi, Table of n, a(n) for n = 11..200
Index entries related to Gaussian binomial coefficients.

Diagonal k=11 of the triangular array A015109. See there for further references and programs. - M. F. Hasler, Nov 04 2012

r:=11; q:=-2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 05 2012

Table[QBinomial[n, 11, -2], {n, 11, 20}] (* Vincenzo Librandi, Nov 05 2012 *)

[gaussian_binomial(n,11,-2) for n in range(11,21)] # Zerinvary Lajos, May 28 2009

a(n) = Product_{i=1..11} ((-2)^(n-i+1)-1)/((-2)^i-1). - Vincenzo Librandi, Nov 05 2012

A015423 Gaussian binomial coefficient [ n,12 ] for q=-2.

Original entry on oeis.org

1, 2731, 14913991, 54301841231, 237244744338239, 942314556807454559, 3920970870875818419999, 15935828658299317547308959, 65529064844612576067331339935, 267883966717492783113707839256735

Offset: 12

Olivier Gérard

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.

Vincenzo Librandi, Table of n, a(n) for n = 12..200

Diagonal k=12 of the triangular array A015109. See there for further references and programs. - M. F. Hasler, Nov 04 2012

r:=12; q:=-2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 06 2012

Table[QBinomial[n, 12, -2], {n, 12, 20}] (* Vincenzo Librandi, Nov 06 2012 *)

[gaussian_binomial(n,12,-2) for n in range(12,22)] # Zerinvary Lajos, May 28 2009

a(n) = Product_{i=1..12} ((-2)^(n-i+1)-1)/((-2)^i-1) (by definition). - Vincenzo Librandi, Nov 06 2012

A172349 Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=4.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 9, 45, 9, 1, 1, 29, 261, 261, 29, 1, 1, 65, 1885, 3393, 1885, 65, 1, 1, 181, 11765, 68237, 68237, 11765, 181, 1, 1, 441, 79821, 1037673, 3343613, 1037673, 79821, 441, 1, 1, 1165, 513765, 18598293, 134321005, 134321005

Offset: 0

Roger L. Bagula and Gary W. Adamson, Feb 01 2010

Start from the generalized Fibonacci sequence A006131 and its partial products c(n) = 1, 1, 1, 5, 45, 1305, 84825, 15353325, 6770816325, 7888001018625... Then t(n,k) = c(n)/(c(k)*c(n-k)).

Row sums are 1, 2, 3, 12, 65, 582, 7295, 160368, 5579485, 306868458, 26280601275,...

			1;
1, 1;
1, 1, 1;
1, 5, 5, 1;
1, 9, 45, 9, 1;
1, 29, 261, 261, 29, 1;
1, 65, 1885, 3393, 1885, 65, 1;
1, 181, 11765, 68237, 68237, 11765, 181, 1;
1, 441, 79821, 1037673, 3343613, 1037673, 79821, 441, 1;
1, 1165, 513765, 18598293, 134321005, 134321005, 18598293, 513765, 1165, 1;
1, 2929, 3412285, 300963537, 6052711133, 13566421505, 6052711133, 300963537, 3412285, 2929, 1;

Cf. A010048 (m=1), A015109 (m=2), A172347 (m=3), A172350 (m=5).

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]

A172350 Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=5.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 6, 6, 1, 1, 11, 66, 11, 1, 1, 41, 451, 451, 41, 1, 1, 96, 3936, 7216, 3936, 96, 1, 1, 301, 28896, 197456, 197456, 28896, 301, 1, 1, 781, 235081, 3761296, 14019376, 3761296, 235081, 781, 1, 1, 2286, 1785366, 89565861, 781665696

Offset: 0

Roger L. Bagula and Gary W. Adamson, Feb 01 2010

Start from the generalized Fibonacci sequence A015440 and its partial products c(n) = 1, 1, 1, 6, 66, 2706, 259776, 78192576, 61068401856, 139602366642816... Then t(n,k) = c(n)/(c(k)*c(n-k)).

Row sums are 1, 2, 3, 14, 90, 986, 15282, 453308, 22013694, 1746038420, 222562828116,...

			1;
1, 1;
1, 1, 1;
1, 6, 6, 1;
1, 11, 66, 11, 1;
1, 41, 451, 451, 41, 1;
1, 96, 3936, 7216, 3936, 96, 1;
1, 301, 28896, 197456, 197456, 28896, 301, 1;
1, 781, 235081, 3761296, 14019376, 3761296, 235081, 781, 1;
1, 2286, 1785366, 89565861, 781665696, 781665696, 89565861, 1785366, 2286, 1;
1, 6191, 14152626, 1842200151, 50409295041, 118031520096, 50409295041, 1842200151, 14152626, 6191, 1;

Cf. A010048 (m=1), A015109 (m=2), A172349 (m=4), A172351 (m=6).

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]

A172351 Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 7, 7, 1, 1, 13, 91, 13, 1, 1, 55, 715, 715, 55, 1, 1, 133, 7315, 13585, 7315, 133, 1, 1, 463, 61579, 483835, 483835, 61579, 463, 1, 1, 1261, 583843, 11093017, 46931995, 11093017, 583843, 1261, 1, 1, 4039, 5093179, 336877411

Offset: 0

Roger L. Bagula and Gary W. Adamson, Feb 01 2010

Start from the generalized Fibonacci sequence A015441 and its partial products c(n) = 1, 1, 1, 7, 91, 5005, 665665, 308202895, 388643850595, 1569732512553205... Then t(n,k) = c(n)/(c(k)*c(n-k)).

Row sums are 1, 2, 3, 16, 119, 1542, 28483, 1091756, 70288239, 7576979362, 1345651717403,..

			1;
1, 1;
1, 1, 1;
1, 7, 7, 1;
1, 13, 91, 13, 1;
1, 55, 715, 715, 55, 1;
1, 133, 7315, 13585, 7315, 133, 1;
1, 463, 61579, 483835, 483835, 61579, 463, 1;
1, 1261, 583843, 11093017, 46931995, 11093017, 583843, 1261, 1;
1, 4039, 5093179, 336877411, 3446515051, 3446515051, 336877411, 5093179, 4039, 1;

Cf. A010048 (m=1), A015109 (m=2), A172350 (m=5), A172352 (m=7).

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]

A216206 a(n) = Product_{i=1..n} ((-2)^i-1).

Original entry on oeis.org

1, -3, -9, 81, 1215, -40095, -2525985, 325852065, 83092276575, -42626337882975, -43606743654283425, 89350217747626737825, 365889141676531491393375, -2997729737755822508985921375, -49111806293653640164716349886625, 1609344780436736134557590069434814625

Offset: 0

R. J. Mathar, Mar 12 2013

Signed partial products of A062510. This implies that all terms from a(1) on are multiples of 3.

Cf. A005329, A015013, A015109, A027871, A062510, A330862.

A216206 := proc(n)
        mul( (-2)^i-1, i=1..n) ;
end proc:

Table[(-1)^n QPochhammer[-2, -2, n], {n, 0, 15}] (* Bruno Berselli, Mar 13 2013 *)
Table[Product[(-2)^k-1,{k,n}],{n,0,20}] (* Harvey P. Dale, Oct 21 2024 *)

A015109(n,k) = a(n)/(a(k)*a(n-k)).
a(n) = (-3)^n*A015013(n) for n>0, a(0)=1. - Bruno Berselli and Alonso del Arte, Mar 13 2013
a(n) ~ (-1)^(floor(n/2)+1) * c * 2^(n*(n+1)/2), where c = Product_{k>=1} (1 - 1/(-2)^k) = 1.21072413030105918013... (A330862). - Amiram Eldar, Aug 10 2025

A172347 Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=3.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 7, 28, 7, 1, 1, 19, 133, 133, 19, 1, 1, 40, 760, 1330, 760, 40, 1, 1, 97, 3880, 18430, 18430, 3880, 97, 1, 1, 217, 21049, 210490, 571330, 210490, 21049, 217, 1, 1, 508, 110236, 2673223, 15275560, 15275560, 2673223, 110236, 508

Offset: 0

Roger L. Bagula and Gary W. Adamson, Feb 01 2010

Start from the generalized Fibonacci sequence A006130 and its partial products c(n) = 1, 1, 1, 4, 28, 532, 21280, 2064160, 447922720, 227544741760... Then t(n,k) = c(n)/(c(k)*c(n-k)).

Row sums are 1, 2, 3, 10, 44, 306, 2932, 44816, 1034844, 36119056, 1882089488,...

			1;
1, 1;
1, 1, 1;
1, 4, 4, 1;
1, 7, 28, 7, 1;
1, 19, 133, 133, 19, 1;
1, 40, 760, 1330, 760, 40, 1;
1, 97, 3880, 18430, 18430, 3880, 97, 1;
1, 217, 21049, 210490, 571330, 210490, 21049, 217, 1;
1, 508, 110236, 2673223, 15275560, 15275560, 2673223, 110236, 508, 1;
1, 1159, 588772, 31940881, 442609351, 931809160, 442609351, 31940881, 588772, 1159, 1;

Cf. A010048 (m=1), A015109 (m=2), A172349 (m=4).

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]

A172352 Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 8, 1, 1, 15, 120, 15, 1, 1, 71, 1065, 1065, 71, 1, 1, 176, 12496, 23430, 12496, 176, 1, 1, 673, 118448, 1051226, 1051226, 118448, 673, 1, 1, 1905, 1282065, 28205430, 133505702, 28205430, 1282065, 1905, 1, 1, 6616, 12603480

Offset: 0

Roger L. Bagula and Gary W. Adamson, Feb 01 2010

Start from the generalized Fibonacci sequence A015442 and its partial products c(n) = 1, 1, 1, 8, 120, 8520, 1499520, 1009176960, 1922482108800... Then t(n,k) = c(n)/(c(k)*c(n-k)).

Row sums are 1, 2, 3, 18, 152, 2274, 48776, 2340696, 192484504, 27026705688, 6379354108992,...

			1;
1, 1;
1, 1, 1;
1, 8, 8, 1;
1, 15, 120, 15, 1;
1, 71, 1065, 1065, 71, 1;
1, 176, 12496, 23430, 12496, 176, 1;
1, 673, 118448, 1051226, 1051226, 118448, 673, 1;
1, 1905, 1282065, 28205430, 133505702, 28205430, 1282065, 1905, 1;
1, 6616, 12603480, 1060267755, 12440474992, 12440474992, 1060267755, 12603480, 6616, 1;

CF. A010048 (m=1), A015109 (m=2), A172351 (m=6).

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]

cp's OEIS Frontend

A015323 Gaussian binomial coefficient [ n,6 ] for q = -2.

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A015338 Gaussian binomial coefficient [ n,7 ] for q = -2.

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A015405 Gaussian binomial coefficient [ n,11 ] for q=-2.

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A015423 Gaussian binomial coefficient [ n,12 ] for q=-2.

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A172349 Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=4.

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A172350 Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=5.

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A172351 Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=6.

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A216206 a(n) = Product_{i=1..n} ((-2)^i-1).

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