A094705 -id:A094705 - cp's OEIS frontend

A061547 Number of 132 and 213-avoiding derangements of {1,2,...,n}.

1, 0, 1, 2, 6, 10, 26, 42, 106, 170, 426, 682, 1706, 2730, 6826, 10922, 27306, 43690, 109226, 174762, 436906, 699050, 1747626, 2796202, 6990506, 11184810, 27962026, 44739242, 111848106, 178956970, 447392426, 715827882, 1789569706, 2863311530, 7158278826

Offset: 0

Emeric Deutsch, May 16 2001

Or, number of permutations with no fixed points avoiding 213 and 132.
Number of derangements of {1,2,...,n} having ascending runs consisting of consecutive integers. Example: a(4)=6 because we have 234/1, 34/12, 34/2/1, 4/123, 4/3/12, 4/3/2/1, the ascending runs being as indicated. - Emeric Deutsch, Dec 08 2004
Let c be twice the sequence A002450 interlaced with itself (from the second term), i.e., c = 2*(0, 1, 1, 5, 5, 21, 21, 85, 85, 341, 341, ...). Let d be powers of 4 interlaced with the zero sequence: d = (1, 0, 4, 0, 16, 0, 64, 0, 256, 0, ...). Then a(n+1) = c(n) + d(n). - Creighton Dement, May 09 2005
Inverse binomial transform of A094705 (0, 1, 4, 15). - Paul Curtz, Jun 15 2008
Equals row sums of triangle A177993. - Gary W. Adamson, May 16 2010
a(n-1) is also the number of order preserving partial isometries (of an n-chain) of fix 1 (fix of alpha equals the number of fixed points of alpha). - Abdullahi Umar, Dec 28 2010
a(n+1) <= A218553(n) is also the Moore lower bound on the order of a (5,n)-cage. - Jason Kimberley, Oct 31 2011
For n > 0, a(n) is the location of the n-th new number to make a first appearance in A087230. E.g., the 17th number to make its first appearance in A087230 is 18 and this occurs at A087230(43690) and a(17)=43690. - K D Pegrume, Jan 26 2022
Position in A002487 of 2 adjacent terms of A000045. E.g., 3/5 at 10, 5/8 at 26, 8/13 at 42, ... - Ed Pegg Jr, Dec 27 2022

			a(4)=6 because the only 132 and 213-avoiding permutations of {1,2,3,4} without fixed points are: 2341, 3412, 3421, 4123, 4312 and 4321.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. Al-Kharousi, R. Kehinde and A. Umar, Combinatorial results for certain semigroups of partial isometries of a finite chain, The Australasian Journal of Combinatorics, Volume 58 (3) (2014), 363-375.
J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869 (contains the sequence of the odd-subscripted terms and that of the even-subscripted terms).
Emeric Deutsch, Derangements That Don't Rise Too Fast: 10902, Amer. Math. Monthly, Vol. 110, No. 7 (2003), pp. 639-640.
K. Dilcher and K. B. Stolarsky, Stern polynomials and double-limit continued fractions, Acta Arithmetica 140 (2009), 119-134
R. Kehinde and A. Umar, On the semigroup of partial isometries of a finite chain, arXiv:1101.0049 [math.GR], 2010.
T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, arXiv:math/0204005 [math.CO], 2002
Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

Cf. A000166, A020988, A020989, A068018.
Cf. A177993. - Gary W. Adamson, May 16 2010
Cf. A183158, A183159. - Abdullahi Umar, Dec 28 2010
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), this sequence (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 31 2011

[(3/8)*2^n +(1/24)*(-2)^n - 2/3: n in [1..35]]; // Vincenzo Librandi, Aug 13 2011

A061547:=n->ceil(abs((3/8)*2^n +(1/24)*(-2)^n - 2/3)); seq(A061547(n), n=1..30); # Wesley Ivan Hurt, Apr 03 2014

f[n_] := (9*2^(n-3) - (-2)^(n-3) - 2)/3; Array[f, 32] (* Robert G. Wilson v, Aug 13 2011 *)

a(n)=(3/8)*2^n+(1/24)*(-2)^n-2/3 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (3/8)*2^n + (1/24)*(-2)^n - 2/3 for n>=1.
a(n) = 4*a(n-2) + 2, a(0)=1, a(1)=0, a(2)=1.
G.f: (5*z^3-3*z^2-z+1)/((z-1)*(4*z^2-1)).
a(n) = A020989((n-2)/2) for n=2, 4, 6, ... and A020988((n-3)/2) for n=3, 5, 7, ... .
a(n+1)-2*a(n) = A078008 signed. Differences: doubled A000302. - Paul Curtz, Jun 15 2008
a(2i+1) = 2*Sum_{j=0..i-1} 4^j = string "2"^i read in base 4.
a(2i+2) = 4^i + 2*Sum_{j=0..i-1} 4^j = string "1"*"2"^i read in base 4.
a(n+2) = Sum_{k=0..n} A144464(n,k)^2 = Sum_{k=0..n} A152716(n,k). - Philippe Deléham and Michel Marcus, Feb 26 2014
a(2*n-1) = A176965(2*n), a(2*n) = A176965(2*n-1) for n>0. - Yosu Yurramendi, Dec 23 2016
a(2*n-1) = A020988(k-1), a(2*n)= A020989(n-1) for n>0. - Yosu Yurramendi, Jan 03 2017
a(n+2) = 2*A086893(n), n > 0. - Yosu Yurramendi, Mar 07 2017
E.g.f.: (15 - 8*cosh(x) + 5*cosh(2*x) - 8*sinh(x) + 4*sinh(2*x))/12. - Stefano Spezia, Apr 07 2022

a(0)=1 prepended by Alois P. Heinz, Jan 27 2022

A137241 Number triples (k,3-k,2-2k), concatenated for k=0, 1, 2, 3,...

Original entry on oeis.org

0, 3, 2, 1, 2, 0, 2, 1, -2, 3, 0, -4, 4, -1, -6, 5, -2, -8, 6, -3, -10, 7, -4, -12, 8, -5, -14, 9, -6, -16, 10, -7, -18, 11, -8, -20, 12, -9, -22, 13, -10, -24, 14, -11, -26, 15, -12, -28, 16, -13, -30, 17, -14, -32, 18, -15, -34, 19, -16, -36, 20, -17, -38, 21, -18, -40

Offset: 0

Paul Curtz, Mar 09 2008

The entries are the coefficients in a family of Jacobsthal recurrences: a(n)=k*a(n-1)+(3-k)*a(n-2)+(2-2k)*a(n-3).
Examples for k=0 are in A001045 and A113954. Examples for k=1 are A001045, A078008.
Examples for k=2 are A000975, A087288, A084639, A000012 and A001045.
Examples for k=3 are A045883, A059570. Examples for k=4 are A094705 and A015518.

			The triples (k,3-k,2-2k) are (0,3,2), (1,2,0), (2,1,-2), (3,0,-4),...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 2, 0, 0, -1).

CoefficientList[Series[x*(3 + 2*x + x^2 - 4*x^3 - 4*x^4)/((x - 1)^2*(1 + x + x^2)^2), {x, 0, 50}], x] (* G. C. Greubel, Sep 28 2017 *)
Table[{n,3-n,2-2n},{n,0,30}]//Flatten (* or *) LinearRecurrence[ {0,0,2,0,0,-1},{0,3,2,1,2,0},100] (* Harvey P. Dale, Jun 23 2019 *)

x='x+O('x^50); Vec(x*(3+2*x+x^2-4*x^3-4*x^4)/((x-1)^2*(1+x +x^2 )^2)) \\ G. C. Greubel, Sep 28 2017

From R. J. Mathar, Feb 25 2009: (Start)
a(n) = 2*a(n-3) - a(n-6).
G.f.: x*(3+2*x+x^2-4*x^3-4*x^4)/((x-1)^2*(1+x+x^2)^2). (End)

Edited by R. J. Mathar, Jun 28 2008

A249992 Expansion of 1/((1+x)(1+2x)(1-3x)).

Original entry on oeis.org

1, 0, 7, 6, 49, 84, 379, 882, 3157, 8448, 27391, 78078, 242425, 710892, 2165443, 6430794, 19423453, 58008216, 174548935, 522598230, 1569891841, 4705481220, 14124832267, 42357719586, 127106713189, 381253030704, 1143893309839, 3431411494062, 10294771353097

Offset: 0

Alex Ratushnyak, Dec 27 2014

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,7,6).

Cf. A249993.
Cf. A000392: expansion of x^3/((1-x)*(1-2*x)*(1-3*x)).
Cf. A091002: expansion of x^2/((1-x)*(1+2*x)*(1-3*x)).
Cf. A094705: expansion of   x/((1+x)*(1-2*x)*(1-3*x)).

[(3^(n+2) + (-1)^n*(2^(n+4) - 5))/20: n in [0..50]]; // G. C. Greubel, Jul 21 2022

seq((9/20)*3^n+(4/5)*(-2)^n-(1/4)*(-1)^n, n=0 .. 100); # Robert Israel, Dec 28 2014

LinearRecurrence[{0, 7, 6}, {1, 0, 7}, 29] (* Jean-François Alcover, Oct 05 2017 *)
CoefficientList[Series[1/((1+x)(1+2x)(1-3x)),{x,0,30}],x] (* Harvey P. Dale, May 26 2020 *)

Vec(1/((1+x)*(1+2*x)*(1-3*x)) + O(x^50)) \\ Michel Marcus, Dec 28 2014

[(3^(n+2) +(-1)^n*(2^(n+4) -5))/20 for n in (0..50)] # G. C. Greubel, Jul 21 2022

G.f.: 1/((1+x)*(1+2*x)*(1-3*x)).
a(n) = ( 3^(n+2) + (2^(n+4) - 5)*(-1)^n )/20. - Colin Barker, Dec 28 2014
a(n) = 7*a(n-2) + 6*a(n-3). - Colin Barker, Dec 28 2014
E.g.f.: (9/20)*exp(3*x) + (4/5)*exp(-2*x) - (1/4)*exp(-x). - Robert Israel, Dec 28 2014

A099621 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k+1) * 3^(n-k-1)*(4/3)^k.

Original entry on oeis.org

0, 1, 6, 31, 144, 637, 2730, 11467, 47508, 194953, 794574, 3222583, 13023192, 52491349, 211161138, 848231779, 3403688796, 13647040225, 54685016022, 219030629455, 876994213920, 3510591943981, 14050213040826, 56224387958011

Offset: 0

Paul Barry, Oct 25 2004

In general a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k+1) * u^(n-k-1)* (v/u)^(k-1) has g.f. x^2/((1-u*x)*(1-u*x-v*x^2)) and satisfies the recurrence a(n) = 2*u*a(n-1) - (u^2-v)*a(n-2) - u*v*a(n-3).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5,-12).

Cf. A094705, A099622.

List([0..30], n-> (4^(n+2)-5*3^(n+1)-(-1)^n)/20) # G. C. Greubel, Jun 06 2019

[(4^(n+2)-5*3^(n+1)-(-1)^n)/20: n in [0..30]]; // G. C. Greubel, Jun 06 2019

Table[Sum[Binomial[n-k,k+1]3^(n-k-1) (4/3)^k,{k,0,Floor[n/2]}],{n,0,25}] (* or *) LinearRecurrence[{6,-5,-12},{0,1,6},30] (* Harvey P. Dale, Dec 13 2012 *)
Table[(4^(n+2)-5*3^(n+1)-(-1)^n)/20, {n,0,30}] (* G. C. Greubel, Jun 06 2019 *)

vector(30, n, n--; (4^(n+2)-5*3^(n+1)-(-1)^n)/20) \\ G. C. Greubel, Jun 06 2019

[(4^(n+2)-5*3^(n+1)-(-1)^n)/20 for n in (0..30)] # G. C. Greubel, Jun 06 2019

G.f.: x^2/((1-3*x)*(1-3*x-4*x^2)).
a(n) = 6*a(n-1) - 5*a(n-2) - 12*a(n-3).
From G. C. Greubel, Jun 06 2019: (Start)
a(n) = (4^(n+2) - 5*3^(n+1) - (-1)^n)/20.
E.g.f.: (-exp(-x) - 15*exp(3*x) + 16*exp(4*x))/20. (End)

A004054 Expansion of (1-x)/((1+x)(1-2x)(1-3x)).

Original entry on oeis.org

1, 3, 11, 35, 111, 343, 1051, 3195, 9671, 29183, 87891, 264355, 794431, 2386023, 7163531, 21501515, 64526391, 193622863, 580955971, 1743042675, 5229477551, 15689131703, 47068793211, 141209175835, 423633119911, 1270910544543, 3812754003251, 11438306748995

Offset: 0

N. J. A. Sloane

Number of paths with n+2 steps on the cycle graph C_6 which start at the first node and end at the 3rd node and each step is -1, 0 or +1. - Herbert Kociemba, Sep 30 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Xavier Acloque, Polynexus Numbers and other mathematical wonders.
Index entries for linear recurrences with constant coefficients, signature (4,-1,-6).

Cf. A001045, A001047.

Cf. A000244, A078008.

[Ceiling(3^(n+2)/6+(-1)^(n+2)/6-0^n/6-2^(n+2)/6) : n in [0..30]]; // Vincenzo Librandi, Oct 08 2011

Table[1/6 ((-1)^(2+n)-2^(n+2)+3^(n+2)),{n,0,30}] (* Herbert Kociemba, Sep 30 2020 *)

Vec((1-x)/((1+x)*(1-2*x)*(1-3*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

From Paul Barry, Sep 13 2003: (Start)
The sequence 0, 0, 1, ... has a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*A001045(2*k).
a(n) = 3^n/6 + (-1)^n/6 - 0^n/6 - 2^n/6. (End)
The signed sequence 0, 1, -3, ... has g.f. x*(1+x)/((1-x)*(1+2*x)*(1+3*x)) and a(n) = 1/6 + (-2)^n/3 - (-3)^n/2. It is the third inverse binomial transform of A001045(2*n-1) - 0^n/2. - Paul Barry, Apr 21 2004
From Paul Barry, Jul 22 2004: (Start)
Convolution of A000244 and A078008.
a(n) = Sum_{k=0..n} A078008(k)*3^(n-k).
a(n) = (3*A000244(n) - A001045(n+2))/2. (End)
a(n) = (A001047(n+2) + (-1)^n)/6. - Vladimir Pletser, Dec 02 2023
a(n) = A094705(n+1)-A094705(n). - R. J. Mathar, Dec 02 2023

A249999 Expansion of 1/((1-x)^2(1-2x)(1-3x)).

Original entry on oeis.org

1, 7, 32, 122, 423, 1389, 4414, 13744, 42245, 128771, 390396, 1179366, 3554467, 10696153, 32153978, 96592988, 290041089, 870647535, 2612991160, 7841070610, 23527406111, 70590606917, 211788597942, 635399348232, 1906265153533, 5718929678299, 17157057470324, 51471709281854

Offset: 0

Alex Ratushnyak, Dec 28 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Anthony G. Shannon, Hakan Akkuş, Yeşim Aküzüm, Ömür Deveci, and Engin Özkan, A partial recurrence Fibonacci link, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 530-537. See Table 2, p. 534.
Index entries for linear recurrences with constant coefficients, signature (7,-17,17,-6).

Cf. A000392 (first differences), A094705, A243869, A249997.

[(2*n +9 -2^(n+5) +3^(n+3))/4: n in [0..50]]; // G. C. Greubel, Jul 21 2022

LinearRecurrence[{7,-17,17,-6}, {1,7,32,122}, 50] (* G. C. Greubel, Jul 21 2022 *)
CoefficientList[Series[1/((1-x)^2(1-2x)(1-3x)),{x,0,30}],x] (* Harvey P. Dale, Feb 11 2025 *)

[(2*n+9 -2^(n+5) +3^(n+3))/4 for n in (0..50)] # G. C. Greubel, Jul 21 2022

G.f.: 1/((1-x)^2 * (1-2*x) * (1-3*x)).
a(n) = 9/4 - 2^(n+3) + n/2 + 3^(n+3)/4. - R. J. Mathar, Jan 09 2015
E.g.f.: (1/4)*((9 + 2*x) - 32*exp(x) + 27*exp(2*x))*exp(x). - G. C. Greubel, Jul 21 2022

A099622 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)4^(n-k-1)(5/4)^k.

Original entry on oeis.org

0, 1, 8, 53, 316, 1785, 9744, 51997, 273092, 1417889, 7299160, 37334661, 190028748, 963565513, 4871514656, 24572321645, 123720601684, 622038982257, 3123938806632, 15674669614549, 78593250398300, 393845861293721

Offset: 0

Paul Barry, Oct 25 2004

In general a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k+1) * u^(n-k-1) * (v/u)^(k-1) has g.f. x^2/((1-u*x) * (1-u*x-v*x^2)) and satisfies the recurrence a(n) = 2*u*a(n-1) - (u^2 - v)*a(n-2) - u*v*a(n-3).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-11,-20).

Cf. A094705, A099582, A099621.

[(5^(n+2) -6*4^(n+1) -(-1)^n)/30: n in [0..40]]; // G. C. Greubel, Jul 22 2022

LinearRecurrence[{8,-11,-20},{0,1,8},30] (* Harvey P. Dale, Nov 05 2017 *)

[(5^(n+2) -6*4^(n+1) -(-1)^n)/30 for n in (0..40)] # G. C. Greubel, Jul 22 2022

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*4^(n-k-1)*(5/4)^k.
a(n) = 8*a(n-1) - 11*a(n-2) - 20*a(n-3).
G.f.: x^2/((1-4*x)*(1-4*x-5*x^2)) = x^2/((1+x)*(1-4*x)*(1-5*x)).
From G. C. Greubel, Jul 22 2022: (Start)
a(n) = (1/30)*(5^(n+2) - 6*4^(n+1) - (-1)^n).
E.g.f.: (1/30)*(25*exp(5*x) - 24*exp(4*x) - exp(-x)). (End)

A155118 Array T(n,k) read by antidiagonals: the k-th term of the n-th iterated differences of A140429.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 4, 6, 9, 5, 8, 12, 18, 27, 11, 16, 24, 36, 54, 81, 21, 32, 48, 72, 108, 162, 243, 43, 64, 96, 144, 216, 324, 486, 729, 85, 128, 192, 288, 432, 648, 972, 1458, 2187, 171, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 341, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683

Offset: 0

Paul Curtz, Jan 20 2009

Deleting column k=0 and reading by antidiagonals yields A036561.

Deleting column k=0 and reading the antidiagonals downwards yields A175840.

			The array starts in row n=0 with columns k>=0 as:
   0   1    3    9    27    81    243    729    2187  ... A140429;
   1   2    6   18    54   162    486   1458    4374  ... A025192;
   1   4   12   36   108   324    972   2916    8748  ... A003946;
   3   8   24   72   216   648   1944   5832   17496  ... A080923;
   5  16   48  144   432  1296   3888  11664   34992  ... A257970;
  11  32   96  288   864  2592   7776  23328   69984  ...
  21  64  192  576  1728  5184  15552  46656  139968  ...
Antidiagonal triangle begins as:
   0;
   1,   1;
   1,   2,   3;
   3,   4,   6,   9;
   5,   8,  12,  18,  27;
  11,  16,  24,  36,  54,  81;
  21,  32,  48,  72, 108, 162, 243;
  43,  64,  96, 144, 216, 324, 486, 729;
  85, 128, 192, 288, 432, 648, 972, 1458, 2187; - _G. C. Greubel_, Mar 25 2021

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A001045, A004054, A046936.

t:= func< n,k | k eq 0 select (2^(n-k) -(-1)^(n-k))/3 else 2^(n-k)*3^(k-1) >;
[t(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 25 2021

T:=proc(n,k)if(k>0)then return 2^n*3^(k-1):else return (2^n - (-1)^n)/3:fi:end:
for d from 0 to 8 do for m from 0 to d do print(T(d-m,m)):od:od: # Nathaniel Johnston, Apr 13 2011

t[n_, k_]:= If[k==0, (2^(n-k) -(-1)^(n-k))/3, 2^(n-k)*3^(k-1)];
Table[t[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 25 2021 *)

def A155118(n,k): return (2^(n-k) -(-1)^(n-k))/3 if k==0 else 2^(n-k)*3^(k-1)
flatten([[A155118(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 25 2021

For the square array:
T(n,k) = 2^n*3^(k-1), k>0.
T(n,k) = T(n-1,k+1) - T(n-1,k), n>0.
Rows:
T(0,k) = A140429(k) = A000244(k-1).
T(1,k) = A025192(k).
T(2,k) = A003946(k).
T(3,k) = A080923(k+1).
T(4,k) = A257970(k+3).
Columns:
T(n,0) = A001045(n) (Jacobsthal numbers J_{n}).
T(n,1) = A000079(n).
T(n,2) = A007283(n).
T(n,3) = A005010(n).
T(n,4) = A175806(n).
T(0,k) - T(k+1,0) = 4*A094705(k-2).
From G. C. Greubel, Mar 25 2021: (Start)
For the antidiagonal triangle:
t(n, k) = T(n-k, k).
t(n, k) = (2^(n-k) - (-1)^(n-k))/3 (J_{n-k}) if k = 0 else 2^(n-k)*3^(k-1).
Sum_{k=0..n} t(n, k) = 3^n - J_{n+1}, where J_{n} = A001045(n).
Sum_{k=0..n} t(n, k) = A004054(n-1) for n >= 1. (End)

a(22) - a(57) from Nathaniel Johnston, Apr 13 2011

cp's OEIS Frontend

A061547 Number of 132 and 213-avoiding derangements of {1,2,...,n}.

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A137241 Number triples (k,3-k,2-2k), concatenated for k=0, 1, 2, 3,...

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A249992 Expansion of 1/((1+x)*(1+2*x)*(1-3*x)).

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A099621 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k+1) * 3^(n-k-1)*(4/3)^k.

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A004054 Expansion of (1-x)/((1+x)*(1-2*x)*(1-3*x)).

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Magma

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A249999 Expansion of 1/((1-x)^2*(1-2*x)*(1-3*x)).

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A099622 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*4^(n-k-1)*(5/4)^k.

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A249992 Expansion of 1/((1+x)(1+2x)(1-3x)).

A004054 Expansion of (1-x)/((1+x)(1-2x)(1-3x)).

A249999 Expansion of 1/((1-x)^2(1-2x)(1-3x)).

A099622 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)4^(n-k-1)(5/4)^k.