A006105 -id:A006105 - cp's OEIS frontend

A022168 Triangle of Gaussian binomial coefficients [ n,k ] for q = 4.

1, 1, 1, 1, 5, 1, 1, 21, 21, 1, 1, 85, 357, 85, 1, 1, 341, 5797, 5797, 341, 1, 1, 1365, 93093, 376805, 93093, 1365, 1, 1, 5461, 1490853, 24208613, 24208613, 1490853, 5461, 1, 1, 21845, 23859109, 1550842085, 6221613541

Offset: 0

N. J. A. Sloane

The coefficients of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^n*A157784(n,k). - R. J. Mathar, Mar 12 2013

			Triangle begins:
  1;
  1,    1;
  1,    5,       1;
  1,   21,      21,        1;
  1,   85,     357,       85,        1;
  1,  341,    5797,     5797,      341,       1;
  1, 1365,   93093,   376805,    93093,    1365,    1;
  1, 5461, 1490853, 24208613, 24208613, 1490853, 5461, 1;

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.

T. D. Noe, Rows n=0..50 of triangle, flattened
R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.
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)
Index entries for sequences related to Gaussian binomial coefficients

Cf. A006118 (row sums), A002450 (k=1), A006105 (k=2), A006106 (k=3).

A022168 := proc(n,m)
        A027637(n)/A027637(n-m)/A027637(m) ;
end proc: # R. J. Mathar, Nov 14 2011

gaussianBinom[n_, k_, q_] := Product[q^i - 1, {i, n}]/Product[q^j - 1, {j, n - k}]/Product[q^l - 1, {l, k}]; Column[Table[gaussianBinom[n, k, 4], {n, 0, 8}, {k, 0, n}], Center] (* Alonso del Arte, Nov 14 2011 *)
Table[QBinomial[n,k,4], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 4; T[n_, 0]:= 1; T[n_,n_]:= 1; T[n_,k_]:= T[n,k] = If[k < 0 || n < k, 0, T[n-1, k -1] +q^k*T[n-1,k]]; Table[T[n,k], {n,0,10}, {k,0,n}] // Flatten  (* G. C. Greubel, May 27 2018 *)

{q=4; T(n,k) = if(k==0,1, if (k==n, 1, if (k<0 || nG. C. Greubel, May 27 2018

T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017

G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 4^j - 1. - Seiichi Manyama, May 09 2025

A166915 a(n) = 20a(n-1) - 64a(n-2) - 45 for n>1; a(0) = 399, a(1) = 5695.

Original entry on oeis.org

399, 5695, 88319, 1401855, 22384639, 357974015, 5726863359, 91626930175, 1466019348479, 23456263438335, 375300030463999, 6004799749226495, 96076793034833919, 1537228676746182655, 24595658780694282239

Offset: 0

Klaus Brockhaus, Oct 27 2009

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n+3) = 15*a(n).

lim_{n -> infinity} a(n)/a(n-1) = 16.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (21, -84, 64).

Cf. A075153, A166912, A166913, A166914, A166916, A166917, A006105, A167031, A167032, A167033.

LinearRecurrence[{21, -84, 64}, {399, 5695, 88319}, 50] (* G. C. Greubel, May 28 2016 *)

m=15; v=concat([399, 5695], vector(m-2)); for(n=3, m, v[n]=20*v[n-1]-64*v[n-2]-45); v

a(n) = (1024*16^n + 176*4^n - 3)/3.
G.f.: (399 - 2684*x + 2240*x^2)/((1-x)*(1-4*x)*(1-16*x)).
From G. C. Greubel, May 28 2016: (Start)
a(n) = 21*a(n-1) - 84*a(n-2) + 64*a(n-3).
E.g.f.: (1/3)*(1024*exp(16*x) + 176*exp(4*x) - 3*exp(x)). (End)

A166916 a(n) = 20a(n-1) - 64a(n-2) - 15 for n > 1; a(0) = 357, a(1) = 5525.

Original entry on oeis.org

357, 5525, 87637, 1399125, 22373717, 357930325, 5726688597, 91626231125, 1466016552277, 23456252253525, 375299985724757, 6004799570269525, 96076792319006037, 1537228673882871125, 24595658769241036117

Offset: 0

Klaus Brockhaus, Oct 27 2009

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n+4) = 30*a(n).

lim_{n -> infinity} a(n)/a(n-1) = 16.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (21, -84, 64).

Cf. A075153, A166912, A166913, A166914, A166915, A166917, A006105, A167031, A167032, A167033.

LinearRecurrence[{21,-84,64},{357,5525,87637},20] (* Harvey P. Dale, Sep 24 2012 *)

m=15; v=concat([357, 5525], vector(m-2)); for(n=3, m, v[n]=20*v[n-1]-64*v[n-2]-15); v

a(n) = (1024*16^n + 48*4^n - 1)/3.
G.f.: (357 - 1972*x + 1600*x^2)/((1-x)*(1-4*x)*(1-16*x)).
a(0)=357, a(1)=5525, a(2)=87637, a(n)=21*a(n-1)-84*a(n-2)+64*a(n-3). - Harvey P. Dale, Sep 24 2012
E.g.f.: (1/3)*(1024*exp(16*x) + 48*exp(4*x) - exp(x)). - G. C. Greubel, May 28 2016

A166984 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 1, a(1) = 20.

Original entry on oeis.org

1, 20, 336, 5440, 87296, 1397760, 22368256, 357908480, 5726601216, 91625881600, 1466015154176, 23456246661120, 375299963355136, 6004799480791040, 96076791961092096, 1537228672451215360, 24595658763514413056, 393530540233410478080, 6296488643803287126016

Offset: 0

Klaus Brockhaus, Oct 26 2009

Partial sums of A166965.

First differences of A006105. - Klaus Purath, Oct 15 2020

Seiichi Manyama, Table of n, a(n) for n = 0..830 (terms 0..200 from Vincenzo Librandi)
E. Saltürk and I. Siap, Generalized Gaussian Numbers Related to Linear Codes over Galois Rings, European Journal of Pure and Applied Mathematics, Vol. 5, No. 2, 2012, 250-259; ISSN 1307-5543. - From _N. J. A. Sloane_, Oct 23 2012
Index entries for linear recurrences with constant coefficients, signature (20,-64).

Cf. A000302, A002450, A002452, A002453, A006105, A079598, A083584.

Cf. A115490, A166914, A166917, A166927, A166965, A307695.

[n le 2 select 19*n-18 else 20*Self(n-1)-64*Self(n-2): n in [1..17] ];

LinearRecurrence[{20,-64},{1,20},30] (* Harvey P. Dale, Jul 04 2012 *)

a(n) = (4*16^n - 4^n)/3 \\ Charles R Greathouse IV, Jun 21 2022

A166984=BinaryRecurrenceSequence(20,-64,1,20)
[A166984(n) for n in range(31)] # G. C. Greubel, Oct 02 2024

a(n) = (4*16^n - 4^n)/3.
G.f.: 1/((1-4*x)*(1-16*x)).
Limit_{n -> oo} a(n)/a(n-1) = 16.
a(n) = A115490(n+1)/3.
Sum_{n>=0} a(n) x^(2*n+4)/(2*n+4)! = ( sinh(x) )^4/4!. - Robert A. Russell, Apr 03 2013
From Klaus Purath, Oct 15 2020: (Start)
a(n) = A002450(n+1)*(A002450(n+2) - A002450(n))/5.
a(n) = (A083584(n+1)^2 - A083584(n)^2)/80. (End)
a(n) = (A079598(n) - A000302(n))/24. - César Aguilera, Jun 21 2022
a(n) = 16*a(n-1) + 4^n with a(0) = 1. - Nadia Lafreniere, Aug 08 2022
E.g.f.: (4/3)*exp(10*x)*sinh(6*x + log(2)). - G. C. Greubel, Oct 02 2024

A166965 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 1, a(1) = 19.

Original entry on oeis.org

1, 19, 316, 5104, 81856, 1310464, 20970496, 335540224, 5368692736, 85899280384, 1374389272576, 21990231506944, 351843716694016, 5629499517435904, 90071992480301056, 1441151880490123264, 23058430091063197696

Offset: 0

Klaus Brockhaus, Oct 25 2009

nonn

lim_{n -> infinity} a(n)/a(n-1) = 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (20, -64).

Cf. A166927, A006105 (Gaussian binomial coefficient [ n, 2 ] for q=4).

[ n le 2 select 18*n-17 else 20*Self(n-1)-64*Self(n-2): n in [1..17] ];

LinearRecurrence[{20,-64},{1,19},20] (* Harvey P. Dale, Aug 24 2014 *)

a(n) = (5*16^n - 4^n)/4.
G.f.: (1-x)/((1-4*x)*(1-16*x)).
E.g.f.: (1/4)*(5*exp(16*x) - exp(4*x)). - G. C. Greubel, May 29 2016

A167031 a(n) = 20a(n-1) - 64a(n-2) + 1 for n > 1; a(0) = 1, a(1) = 20.

Original entry on oeis.org

1, 20, 337, 5461, 87653, 1403557, 22461349, 359399333, 5750460325, 92007649189, 1472123522981, 23553980911525, 376863712759717, 6029819476856741, 96477111920512933, 1543633791891427237, 24698140674915717029

Offset: 0

Klaus Brockhaus, Oct 27 2009

nonn

lim_{n -> infinity} a(n)/a(n-1) = 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (21,-84,64).

Cf. A166915, A166916, A006105, A167032, A167033.

[ n le 2 select 19*n-18 else 20*Self(n-1)-64*Self(n-2)+1: n in [1..17] ];

LinearRecurrence[{21, -84, 64}, {1, 20, 337}, 50] (* G. C. Greubel, May 30 2016 *)

a(n) = (241*16^n - 65*4^n + 4)/180.
G.f.: (1-x+x^2)/((1-x)*(1-4*x)*(1-16*x)).
From G. C. Greubel, May 30 2016: (Start)
a(n) = 21*a(n-1) - 84*a(n-2) + 64*a(n-2).
E.g.f.: (1/180)*(241*exp(16*x) - 65*exp(4*x) + 4*exp(x)). (End)

A167032 a(n) = 20a(n-1) - 64a(n-2) + 2 for n > 1; a(0) = 1, a(1) = 21.

Original entry on oeis.org

1, 21, 358, 5818, 93450, 1496650, 23952202, 383258442, 6132227914, 98116017994, 1569857773386, 25117730316106, 401883708825418, 6430139436277578, 102882231360724810, 1646115703292731210, 26337851258768236362

Offset: 0

Klaus Brockhaus, Oct 27 2009

lim_{n -> infinity} a(n)/a(n-1) = 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (21,-84,64).

Cf. A006105, A166915, A166916, A167031, A167033.

[ n le 2 select 20*n-19 else 20*Self(n-1)-64*Self(n-2)+2: n in [1..17] ];

A167032:=n->(257*16^n - 85*4^n + 8)/180: seq(A167032(n), n=0..25); # Wesley Ivan Hurt, May 30 2016

LinearRecurrence[{21, -84, 64}, {1,21,358}, 50] (* G. C. Greubel, May 30 2016 *)
RecurrenceTable[{a[0]==1,a[1]==21,a[n]==20a[n-1]-64a[n-2]+2},a,{n,20}] (* Harvey P. Dale, Oct 27 2019 *)

a(n) = (257*16^n - 85*4^n + 8)/180.
G.f.: (1+x^2)/((1-x)*(1-4*x)*(1-16*x)).
From G. C. Greubel, May 30 2016: (Start)
a(n) = 21*a(n-1) - 84*a(n-2) + 64*a(n-3) for n>2.
E.g.f.: (1/180)*(257*exp(16*x) - 85*exp(4*x) + 8*exp(x)). (End)

A167033 a(n) = 20a(n-1) - 64a(n-2) + 3 for n > 1; a(0) = 1, a(1) = 22.

Original entry on oeis.org

1, 22, 379, 6175, 99247, 1589743, 25443055, 407117551, 6513995503, 104224386799, 1667592023791, 26681479720687, 426903704891119, 6830459395698415, 109287350800936687, 1748597614694035183, 27977561842620755695

Offset: 0

Klaus Brockhaus, Oct 27 2009

nonn

lim_{n -> infinity} a(n)/a(n-1) = 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (21,-84,64).

Cf. A166915, A166916, A006105, A167031, A167032.

[ n le 2 select 21*n-20 else 20*Self(n-1)-64*Self(n-2)+3: n in [1..17] ];

LinearRecurrence[{21, -84, 64}, {1, 22, 379}, 50] (* G. C. Greubel, May 30 2016 *)

a(n) = (91*16^n - 35*4^n + 4)/60.
G.f.: (1+x+x^2)/((1-x)*(1-4*x)*(1-16*x)).
From G. C. Greubel, May 30 2016: (Start)
a(n) = 21*a(n-1) - 84*a(n-2) + 64*a(n-3).
E.g.f.: (1/60)*(91*exp(16*x) - 35*exp(4*x) + 4*exp(x)). (End)

A203244 Second elementary symmetric function of the first n terms of (1,4,16,64,256,...).

Original entry on oeis.org

4, 84, 1428, 23188, 372372, 5963412, 95436436, 1527070356, 24433475220, 390937001620, 6254997618324, 100079984262804, 1601279837683348, 25620477760847508, 409927645605215892, 6558842335410077332, 104941477389467729556

Offset: 2

Clark Kimberling, Dec 31 2011

Index entries for linear recurrences with constant coefficients, signature (21,-84,64).

Cf. A006105.

f[k_] := 4^(k - 1); t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 32}]    (* A203244 *)
Table[a[n]/4, {n, 2, 32}]  (* A006105 *)
LinearRecurrence[{21,-84,64},{4,84,1428},20] (* Harvey P. Dale, Aug 12 2015 *)

Vec(-4*x^2/((x-1)*(4*x-1)*(16*x-1)) + O(x^100)) \\ Colin Barker, Aug 15 2014

a(n) = 4*A006105(n).
From Colin Barker, Aug 15 2014: (Start)
a(n) = (4-5*4^n+16^n)/45.
a(n) = 21*a(n-1)-84*a(n-2)+64*a(n-3).
G.f.: -4*x^2 / ((x-1)*(4*x-1)*(16*x-1)). (End)

Typo in formula fixed by Colin Barker, Aug 15 2014

cp's OEIS Frontend

A022168 Triangle of Gaussian binomial coefficients [ n,k ] for q = 4.

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A166915 a(n) = 20*a(n-1) - 64*a(n-2) - 45 for n>1; a(0) = 399, a(1) = 5695.

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A166916 a(n) = 20*a(n-1) - 64*a(n-2) - 15 for n > 1; a(0) = 357, a(1) = 5525.

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A166984 a(n) = 20*a(n-1) - 64*a(n-2) for n > 1; a(0) = 1, a(1) = 20.

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A166965 a(n) = 20*a(n-1) - 64*a(n-2) for n > 1; a(0) = 1, a(1) = 19.

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A167031 a(n) = 20*a(n-1) - 64*a(n-2) + 1 for n > 1; a(0) = 1, a(1) = 20.

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A167032 a(n) = 20*a(n-1) - 64*a(n-2) + 2 for n > 1; a(0) = 1, a(1) = 21.

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A166915 a(n) = 20a(n-1) - 64a(n-2) - 45 for n>1; a(0) = 399, a(1) = 5695.

A166916 a(n) = 20a(n-1) - 64a(n-2) - 15 for n > 1; a(0) = 357, a(1) = 5525.

A166984 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 1, a(1) = 20.

A166965 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 1, a(1) = 19.

A167031 a(n) = 20a(n-1) - 64a(n-2) + 1 for n > 1; a(0) = 1, a(1) = 20.

A167032 a(n) = 20a(n-1) - 64a(n-2) + 2 for n > 1; a(0) = 1, a(1) = 21.

A167033 a(n) = 20a(n-1) - 64a(n-2) + 3 for n > 1; a(0) = 1, a(1) = 22.