A057083 -id:A057083 - cp's OEIS frontend

A106233 An inverse Catalan transform of A003462.

0, 1, 3, 5, 5, 0, -14, -41, -81, -121, -121, 0, 364, 1093, 2187, 3281, 3281, 0, -9842, -29525, -59049, -88573, -88573, 0, 265720, 797161, 1594323, 2391485, 2391485, 0, -7174454, -21523361, -43046721, -64570081, -64570081, 0, 193710244, 581130733, 1162261467

Offset: 0

Paul Barry, Apr 26 2005

The g.f. is obtained from that of A003462 through the mapping g(x)->g(x(1-x)). A003462 may be retrieved through the mapping g(x)->g(xc(x)), where c(x) is the g.f. of A000108. Binomial transform of x(1+x)/(1+x^2+x^4).

The sequence is identical to its sixth differences. See A140344. - Paul Curtz, Nov 09 2012

			From _Paul Curtz_, Nov 09 2012: (Start)
The sequence and its higher-order differences (periodic after 6 rows):
   0,  1,  3,  5,  5,   0, -14, ...
   1,  2,  2,  0, -5, -14, -27, ...
   1,  0, -2, -5, -9, -13, -13, ...
  -1, -2, -3, -4, -4,   0,  13, ...   = -A134581(n+1)
  -1, -1, -1,  0,  4,  13,  27, ...
   0,  0,  1,  4,  9,  14,  14, ...   = A140343(n+2)
   0,  1,  3,  5,  5,   0, -14, ...
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-7,6,-3).

Cf. A103368.

I:=[0,1,3,5]; [n le 4 select I[n] else 4*Self(n-1)-7*Self(n-2)+ 6*Self(n-3)-3*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 24 2018

LinearRecurrence[{4, -7, 6, -3}, {0, 1, 3, 5}, 35] (* Vincenzo Librandi, Dec 24 2018 *)

G.f.: x(1-x)/((1-x+x^2)*(1-3*x+3*x^2));
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*(3^(n-k)-1)/2.
a(n) = Sum_{k=0..n} A109466(n,k)*A003462(k). - Philippe Deléham, Oct 30 2008
a(n) = (1/2)*[A057083(n) - [1,1,0,0,-1,-1]6 ]. - _Ralf Stephan, Nov 15 2010
a(n) = 4*a(n-1) - 7*a(n-2) + 6*a(n-3) - 3*a(n-4) = A140343(n+2) - A140343(n+1). - Paul Curtz, Nov 09 2012
a(n) is the binomial transform of the sequence 0, 1, 1, -1, -1, 0, ... = A103368(n+5). - Paul Curtz, Nov 09 2012

A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - kx + kx^2).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, -1, 0, 1, 4, 6, 0, -1, 0, 1, 5, 12, 9, -4, 0, 0, 1, 6, 20, 32, 9, -8, 1, 0, 1, 7, 30, 75, 80, 0, -8, 1, 0, 1, 8, 42, 144, 275, 192, -27, 0, 0, 0, 1, 9, 56, 245, 684, 1000, 448, -81, 16, -1, 0, 1, 10, 72, 384, 1421, 3240, 3625, 1024, -162, 32, -1, 0

Offset: 0

Seiichi Manyama, Feb 28 2021

			Square array begins:
  1,  1,  1, 1,   1,    1, ...
  0,  1,  2, 3,   4,    5, ...
  0,  0,  2, 6,  12,   20, ...
  0, -1,  0, 9,  32,   75, ...
  0, -1, -4, 9,  80,  275, ...
  0,  0, -8, 0, 192, 1000, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Chebyshev polynomials.

Columns 0..10 give A000007, A010892, A099087, A057083, A001787(n+1), A030191, A030192, A030240, A057084, A057085(n+1), A057086.
Rows 0..1 give A000012, A001477.
Main diagonal gives (-1) * A109519(n+1).
Cf. A342120, A342133, A342134.

T:= (n, k)-> (<<0|1>, <-k|k>>^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Mar 01 2021

T[n_, k_] := (-1)^n * Sum[If[k == j == 0, 1, (-k)^j] * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)

T(n, k) = (-1)^n*sum(j=0, n\2, (-k)^(n-j)*binomial(n-j, j));

T(n, k) = (-1)^n*sum(j=0, n, (-k)^j*binomial(j, n-j));

T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)/2));

T(0,k) = 1, T(1,k) = k and T(n,k) = k*(T(n-1,k) - T(n-2,k)) for n > 1.
T(n,k) = (-1)^n * Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n-j,j) = (-1)^n * Sum_{j=0..n} (-k)^j * binomial(j,n-j).
T(n,k) = sqrt(k)^n * S(n, sqrt(k)) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind.

A368153 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3x - x^2.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 4, -2, 4, 5, 5, 4, -10, 5, 8, 10, -3, 4, -25, 6, 13, 16, 1, -29, 14, -49, 7, 21, 28, -8, -24, -78, 56, -84, 8, 34, 47, -12, -88, -26, -162, 168, -132, 9, 55, 80, -31, -140, -200, 100, -330, 408, -195, 10, 89, 135, -58, -301, -230, -296

Offset: 1

Clark Kimberling, Jan 20 2024

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
   1
   1   2
   2   1   3
   3   4  -2    4
   5   5   4  -10    5
   8  10  -3    4  -25    6
  13  16   1  -29   14  -49    7
  21  28  -8  -24  -78   56  -84   8
Row 4 represents the polynomial p(4,x) = 3 + 4*x - 2*x^2 + 4*x^3, so (T(4,k)) = (3,4,-2,4), k=0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), 1-28, Paper No. A14.

Cf. A000045 (column 1); A000027 (p(n,n-1)); A057083 (row sums), (p(n,1)); A182228 (alternating row sums), (p(n,-1)); A190970, (p(n,2)); A030195, (p(n,-2)); A052918, (p(n,-3)); A190972, (p(n,-4)); A057085, (p(n,-5)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152.

p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 3x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 - 3*x - x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 - 8*x), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

A094717 a(n) = n! * Sum_{i+2j+3k=n} 1/(i!(2j)!(3k)!).

Original entry on oeis.org

1, 1, 2, 5, 12, 36, 113, 351, 1080, 3281, 9882, 29646, 88817, 266085, 797526, 2391485, 7173360, 21520080, 64563521, 193700403, 581120892, 1743392201, 5230206126, 15690618378, 47071766561, 141215033961, 423644570442, 1270932914165, 3812797945332, 11438393835996

Offset: 0

Benoit Cloitre, May 23 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,10,-6,12,-9).

Cf. A024493, A049347, A057083, A057682, A094715, A099837.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-5*x+8*x^2-5*x^3+2*x^4-2*x^5)/((1-x)*(1-3*x)*(1+x+x^2)*(1-3*x+3*x^2)) )); // G. C. Greubel, Jul 14 2023

A094717_list := proc(n) local i; exp(z)*cosh(z)*(exp(z)+2*exp(-z/2)* cos(z*sqrt(3/4)))/3; series(%,z,n+2); seq(simplify(i!*coeff(%,z,i)),i=0..n) end: A094717_list(27); # Peter Luschny, Jul 11 2012

a[n_]:= n! Sum[Boole[i +2j +3k ==n]/(i! (2j)! (3k)!), {i,0,n}, {j,0,n}, {k,0,n}]; Table[a[n], {n,0,27}] (* Jean-François Alcover, Jul 06 2019 *)
LinearRecurrence[{6,-12,10,-6,12,-9}, {1,1,2,5,12,36}, 40] (* G. C. Greubel, Jul 14 2023 *)

a(n)=sum(i=0,n,sum(j=0,n,sum(k=0,n,if(n-i-2*j-3*k,0,n!/(i)!/(2*j)!/(3*k)!))))

def A094717_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-5*x+8*x^2-5*x^3+2*x^4-2*x^5)/((1-x)*(1-3*x)*(1+x+x^2)*(1-3*x+3*x^2)) ).list()
A094717_list(40) # G. C. Greubel, Jul 14 2023

Limit_{n->oo} a(n)/3^n = 1/6.
E.g.f.: exp(z)*cosh(z)*(exp(z) + 2*exp(-z/2)*cos(z*sqrt(3/4)))/3. - Peter Luschny, Jul 11 2012
G.f.: (1-5*x+8*x^2-5*x^3+2*x^4-2*x^5)/((1-x)*(1-3*x)*(1+x+x^2)*(1-3*x+3*x^2)). - Colin Barker, Dec 24 2012
From G. C. Greubel, Jul 14 2023: (Start)
a(n) = (1/6)*(1 + 3^n + 2*A049347(n) + A049347(n-1) + 2*A057083(n) - 3*A057083(n-1)).
a(n) = (1/6)*(1 + 3^n + A099837(n+3) + A057682(n+3)). (End)

A100239 G.f. A(x) satisfies: 3^n + 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (3+z)^n + (1+z)^n - z^n = Sum_{k=0..n} [x^k](A(x) + z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.

Original entry on oeis.org

1, 3, -3, 9, -36, 162, -783, 3969, -20817, 112023, -615033, 3431403, -19398690, 110880900, -639730305, 3720657807, -21790419444, 128398625658, -760668489729, 4528069760691, -27070491820644, 162464919528222, -978463778897637, 5911727071716891, -35821932198013809

Offset: 0

Paul D. Hanna, Nov 30 2004

sign

			From the table of powers of A(x), we see that
3^n+1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1 = [1,  3], -3,    9,  -36,  162, -783, 3969, -20817, 112023, ...
A^2 = [1,  6,   3],   0,   -9,   54, -297, 1620,  -8910,  49572, ...
A^3 = [1,  9,  18,    0],   0,    0,  -27,  243,  -1701,  10935, ...
A^4 = [1, 12,  42,   36,   -9],   0,    0,    0,    -81,    972, ...
A^5 = [1, 15,  75,  135,   45,  -27],   0,    0,      0,      0, ...
A^6 = [1, 18, 117,  324,  324,    0,  -54],   0,      0,      0, ...
A^7 = [1, 21, 168,  630, 1071,  567, -189,  -81],     0,      0, ...
A^8 = [1, 24, 228, 1080, 2610, 2808,  540, -648,    -81],     0, ...
the main diagonal of which is:
[x^n]A(x)^(n+1) = (n+1)*A057083(n) for n>=0.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A057083, A100226, A100239, A107264.

a[n_]:= a[n]= 3^n*Boole[n<2] + 3*(-1)^(n+1)*Sum[Binomial[k+1, n-k-1]*Binomial[n-2,k]*3^k/(k+1), {k,0,n-2}];
Table[a[n], {n,0,40}] (* G. C. Greubel, May 21 2022 *)

a(n)=if(n==0, 1, (3^n+1-sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j)^n + x*O(x^k), k)))/n)

a(n)=polcoeff((1+3*x+sqrt(1+6*x-3*x^2+x^2*O(x^n)))/2,n)

def A100239_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+3*x+sqrt(1+6*x-3*x^2))/2 ).list()
A100239_list(40) # G. C. Greubel, May 21 2022

G.f.: A(x) = (1+3*x+sqrt(1+6*x-3*x^2))/2.
Given g.f. A(x), then B(x) = A(x) - (1+2*x) series reversion is -B(-x). - Michael Somos, Sep 07 2005
Given g.f. A(x) and C(x) = g.f. of A025226, then B(x)=A(x)-1-2x satisfies B(x) = x - C(x*B(x)). - Michael Somos, Sep 07 2005
a(n) = 3^n*[n<2] + 3*(-1)^(n+1)*A107264(n-2). - G. C. Greubel, May 21 2022

A131292 a(2n+1)=3a(2n)-3a(2n-1)+2a(2n-2), a(2n+2)=3a(2n+1)-3a(2n), a(0)=a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 9, 9, 12, 9, 9, 0, -9, -27, -54, -81, -135, -162, -243, -243, -324, -243, -243, 0, 243, 729, 1458, 2187, 3645, 4374, 6561, 6561, 8748, 6561, 6561, 0, -6561, -19683, -39366, -59049, -98415, -118098, -177147, -177147, -236196, -177147, -177147, 0, 177147, 531441, 1062882, 1594323

Offset: 0

Paul Curtz, Sep 29 2007

Sequence is identical to its third differences in absolute value.

Index entries for linear recurrences with constant coefficients, signature (0, 3, 0, -3).

Cf. A131665 (0, 0, 1, 3, 6, 11).

Cf. A057083 (bisection). - R. J. Mathar, Jul 17 2009

A131292 := proc(n) option remember ; if n <= 2 then 1; elif n mod 2 = 1 then 3*(A131292(n-1)-A131292(n-2))+2*A131292(n-3) ; else 3*(A131292(n-1)-A131292(n-2)) ; fi ; end: seq(A131292(n),n=0..80) ; # R. J. Mathar, Oct 18 2007

Join[{1, 1},LinearRecurrence[{0, 3, 0, -3},{1, 2, 3, 5},51]] (* Ray Chandler, Sep 23 2015 *)

G.f.: (2*x^3-x^2-x+1)*(1+x)^2/(1-3*x^2+3*x^4). - R. J. Mathar, Jul 17 2009

More terms from R. J. Mathar, Oct 18 2007

A133474 Inverse binomial transform of (A113405 preceded by 0).

Original entry on oeis.org

0, 0, 0, 1, 6, 24, 81, 252, 756, 2241, 6642, 19764, 59049, 176904, 530712, 1592865, 4780782, 14346720, 43046721, 129146724, 387440172, 1162300833, 3486843450, 10460412252, 31381059609, 94143001680, 282429005040, 847287546561

Offset: 0

Paul Curtz, Nov 29 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,9).

Cf. A047790, A131571.

a:=[0,0,1];; for n in [4..30] do a[n]:=6*a[n-1]-12*a[n-2]+9*a[n-3]; od; a; # G. C. Greubel, Nov 21 2019

R:=PowerSeriesRing(Integers(), 30); [0,0,0] cat Coefficients(R!( x^3/((1-3*x)*(1-3*x+3*x^2)) )); // G. C. Greubel, Nov 21 2019

seq(coeff(series(x^3/((1-3*x)(1-3*x+3*x^2)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Nov 21 2019

LinearRecurrence[{6,-12,9}, {0,0,0,1}, 30] (* G. C. Greubel, Nov 21 2019 *)

my(x='x+O('x^30)); concat([0,0,0], Vec(x^3/((1-3*x)*(1-3*x+3*x^2)))) \\ G. C. Greubel, Nov 21 2019

def A133474_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^3/((1-3*x)*(1-3*x+3*x^2))).list()
A133474_list(30) # G. C. Greubel, Nov 21 2019

b(n) = a(n) with one 0; c(n)=1, 3, 6, 9, 9, 0, -27, ... = A057083; b(n+1) = 3*b(n) + c(n)?
From R. J. Mathar, Apr 02 2008: (Start)
O.g.f.: x^3/((1-3*x)*(1-3*x+3*x^2)).
a(n) = 6*a(n-1) - 12*a(n-2) + 9*a(n-3). (End)

A139687 Basis of degenerate cases of sequences identical to its p-th differences. Complement to A140344 which is based on natural Catalan's triangle. Triangle without first term (probable 1) on line.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 3, 6, 9, 9, 1, 4, 10, 19, 28, 28, 1, 4, 10, 20, 34, 48, 48, 1, 5, 15, 35, 69, 117, 165, 165, 1, 5, 15, 35, 70, 125, 200, 275, 275, 1, 6, 21, 56, 126, 251, 451, 726, 1001, 1001, 1, 6, 21, 56, 126, 252, 461, 780, 1209, 1638, 1638

Offset: 0

Paul Curtz, Jun 13 2008

Triangle from A140344:
(1;)
0, 1, 1;
0, 0, 1, 2, 2;
0, 0, 0, 1, 3, 5, 5; see A138112,
0, 0, 0, 0, 1, 4, 9, 14, 14; see A140343,
begins (without 0's) like a(n).

Cf. A135356, A140344.

First four rows of triangle from second row: 1, 1; 1, 2, 2; see A099087, 1, 3, 5, 5; 1, 3, 6, 9, 9; see A057083 which can be preceded with 3 leading 0's, are, as said, from natural Catalan's triangle A009766. Origin of a(n) explained later.

A168615 Inverse binomial transform of A169609, or of A144437 preceded by 1.

Original entry on oeis.org

1, 2, -2, 0, 6, -18, 36, -54, 54, 0, -162, 486, -972, 1458, -1458, 0, 4374, -13122, 26244, -39366, 39366, 0, -118098, 354294, -708588, 1062882, -1062882, 0, 3188646, -9565938, 19131876, -28697814, 28697814, 0, -86093442, 258280326, -516560652

Offset: 0

Paul Curtz, Dec 01 2009

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3,-3).

Cf. A169609, A144437, A000748, A057083, A057682.

[ n le 2 select n else n eq 3 select -2 else -3*Self(n-1)-3*Self(n-2): n in [1..37] ]; // Klaus Brockhaus, Dec 03 2009

Join[{1,2,-2}, LinearRecurrence[{-3, -3}, {0, 6}, 25]] (* G. C. Greubel, Jul 27 2016 *)
LinearRecurrence[{-3,-3},{1,2,-2},40] (* Harvey P. Dale, Jul 21 2024 *)

a(n) = -3*a(n-1) - 3*a(n-2) for n > 2; a(0) = 1, a(1) = 2, a(2) = -2.
a(n) = 2*A123877(n-1), n>0.
G.f.: 1+2*x*(1+2*x)/(1+3*x+3*x^2).
a(6*m + 3) = 0, m>=0. - G. C. Greubel, Jul 27 2016

Edited and extended by Klaus Brockhaus, Dec 03 2009

A103135 Expansion of (-3x^3-18x^2+14x-1)/(3x^4-5x^2+4x-1).

Original entry on oeis.org

1, -10, -27, -55, -82, -83, -3, 238, 721, 1445, 2166, 2153, -55, -6650, -19827, -39599, -59426, -59659, -987, 175550, 528857, 1058701, 1587558, 1583377, -17711, -4811626, -14395275, -28772839, -43168114, -43243139, -317811, 128625934, 386588449, 773494709, 1160083158, 1158736889

Offset: 0

Creighton Dement, Jan 24 2005

A floretion-generated sequence which emerges as a transformation of A000004. a(6n+6)= A103134(n).
It appears that Fib(6n+1) = a(6n+4) - a(6n+5). - Creighton Dement, Jan 31 2005
Floretion Algebra Multiplication Program. FAMP code: 4lesforcycseq[ - .25'i + .5'j - .25i' - .5j' + .5k' - .25'ii' + .75'jj' - .25'kk' + .5'ji' + .25'jk' + .25'kj' + .75e ] Note: vesforcycseq = A000004, 4lesforseq gives A000045, vesseq gives A057681.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,3).

Cf. A103134.

Vec((-3*x^3-18*x^2+14*x-1)/(3*x^4-5*x^2+4*x-1)+O(x^99)) \\ Charles R Greathouse IV, Feb 05 2013

a(n) = -9*A057083(n-1) - Fib(n-2). - Ralf Stephan, May 18 2007

a(n) = 4*a(n-1) - 5*a(n-2) + 3*a(n-4) for n>3. - Colin Barker, May 06 2019

Definition not clear to me. A000004 is the zero sequence! N. J. A. Sloane.

cp's OEIS Frontend

A106233 An inverse Catalan transform of A003462.

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A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - k*x + k*x^2).

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A368153 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3*x - x^2.

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Mathematica

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A094717 a(n) = n! * Sum_{i+2j+3k=n} 1/(i!*(2j)!*(3k)!).

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Magma

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SageMath

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A100239 G.f. A(x) satisfies: 3^n + 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (3+z)^n + (1+z)^n - z^n = Sum_{k=0..n} [x^k](A(x) + z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.

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PARI

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A131292 a(2n+1)=3a(2n)-3a(2n-1)+2a(2n-2), a(2n+2)=3a(2n+1)-3a(2n), a(0)=a(1)=a(2)=1.

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Extensions

A133474 Inverse binomial transform of (A113405 preceded by 0).

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A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - kx + kx^2).

A368153 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3x - x^2.

A094717 a(n) = n! * Sum_{i+2j+3k=n} 1/(i!(2j)!(3k)!).

A103135 Expansion of (-3x^3-18x^2+14x-1)/(3x^4-5x^2+4x-1).