A084057 -id:A084057 - cp's OEIS frontend

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

1, 1, 0, 2, 1, 0, 4, 3, 0, 0, 8, 8, 1, 0, 0, 16, 20, 5, 0, 0, 0, 32, 48, 18, 1, 0, 0, 0, 64, 112, 56, 7, 0, 0, 0, 0, 128, 256, 160, 32, 1, 0, 0, 0, 0, 256, 576, 432, 120, 9, 0, 0, 0, 0, 0, 512, 1280, 1120, 400, 50, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 03 2011

Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.
Skewed version of triangle in A200139.
Triangle without zeros: A207537.
For the version with negative odd numbered columns, which is Riordan ((1-x)/(1-2*x), -x^2/(1-2*x)) see comments on A028297 and A039991. - Wolfdieter Lang, Aug 06 2014
This is an example of a stretched Riordan array in the terminology of Section 2 of Corsani et al. - Peter Bala, Jul 14 2015

			The triangle T(n,k) begins:
  n\k      0     1     2     3     4    5   6  7 8 9 10 11 ...
  0:       1
  1:       1     0
  2:       2     1     0
  3:       4     3     0     0
  4:       8     8     1     0     0
  5:      16    20     5     0     0    0
  6:      32    48    18     1     0    0   0
  7:      64   112    56     7     0    0   0  0
  8:     128   256   160    32     1    0   0  0 0
  9:     256   576   432   120     9    0   0  0 0 0
  10:    512  1280  1120   400    50    1   0  0 0 0  0
  11:   1024  2816  2816  1232   220   11   0  0 0 0  0  0
  ...  reformatted and extended. - _Wolfdieter Lang_, Aug 06 2014

C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.

Columns include A011782, A001792, A001793, A001794, A006974, A006975, A006976.
Diagonals sums are in A052980.
Cf. A028297, A081265, A124182, A131577, A039991 (zero-columns deleted, unsigned and zeros appended).
Cf. A098158, A200139, A207537.
Cf. A028297 (signed version, zeros deleted). Cf. A034839.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 - #)/(1 - 2 #)&, #^2/(1 - 2 #)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 for k<0 or for n

Sum_{k=0..n} T(n,k)^2 = A002002(n) for n>0.

Sum_{k=0..n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n), A087455(n), A146559(n), A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 respectively.

G.f.: (1-x)/(1-2*x-y*x^2). - Philippe Deléham, Mar 03 2012

From Peter Bala, Jul 14 2015: (Start)

Factorizes as A034839 * A007318 = (1/(1 - x), x^2/(1 - x)^2) * (1/(1 - x), x/(1 - x)) as a product of Riordan arrays.

T(n,k) = Sum_{i = k..floor(n/2)} binomial(n,2*i) *binomial(i,k). (End)

Name changed, keyword:easy added, crossrefs A028297 and A039991 added, and g.f. corrected by Wolfdieter Lang, Aug 06 2014

A108404 Expansion of (1-4x)/(1-8x+11x^2).

Original entry on oeis.org

1, 4, 21, 124, 761, 4724, 29421, 183404, 1143601, 7131364, 44471301, 277325404, 1729418921, 10784771924, 67254567261, 419404046924, 2615432135521, 16310012568004, 101710347053301, 634272638178364, 3955367287840601

Offset: 0

Philippe Deléham, Jul 04 2005

Binomial transform of A098648. Second binomial transform of A001077. Third binomial transform of A084057. 4th binomial transform of (1, 0, 5, 0, 25, 0, 125, 0, 625, 0, 3125, ...).

Seiichi Manyama, Table of n, a(n) for n = 0..1258
Index entries for linear recurrences with constant coefficients, signature (8,-11).

Cf. A001077, A084057, A098648.

CoefficientList[Series[(1-4x)/(1-8x+11x^2),{x,0,30}],x] (* or *) LinearRecurrence[{8,-11},{1,4},30] (* Harvey P. Dale, Jan 03 2012 *)

E.g.f.: exp(4x)cosh(sqrt(5)x).
a(n) = 8a(n-1) - 11a(n-2), a(0) = 1, a(1) = 4.
a(n) = ((4+sqrt(5))^n + (4-sqrt(5))^n)/2.
a(n+1)/a(n) converges to 4 + sqrt(5) = 6.2360679774997896964... = 4+A002163.
a(n) = A091870(n+1)-4*A091870(n). - R. J. Mathar, Nov 10 2013

A162516 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = ((x+d)^n + (x-d)^n)/2, where d=sqrt(x+4).

Original entry on oeis.org

1, 1, 0, 1, 1, 4, 1, 3, 12, 0, 1, 6, 25, 8, 16, 1, 10, 45, 40, 80, 0, 1, 15, 75, 121, 252, 48, 64, 1, 21, 119, 287, 644, 336, 448, 0, 1, 28, 182, 588, 1457, 1360, 1888, 256, 256, 1, 36, 270, 1092, 3033, 4176, 6240, 2304, 2304, 0, 1, 45, 390, 1890, 5925, 10801, 17780, 11680, 12160, 1280, 1024

Offset: 0

Clark Kimberling, Jul 05 2009

			First six rows:
  1;
  1,  0;
  1,  1,  4;
  1,  3, 12,  0;
  1,  6, 25,  8, 16;
  1, 10, 48, 40, 80, 0;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A162514, A162515, A162517.
Cf. A000217, A026150, A084057, A125818, A199572.
For fixed k, the sequences P(n,k), for n=1,2,3,4,5, are A084057, A084059, A146963, A081342, A081343, respectively.

m:=12;
p:= func< n,x | ((x+Sqrt(x+4))^n + (x-Sqrt(x+4))^n)/2 >;
R:=PowerSeriesRing(Rationals(), m+1);
T:= func< n,k | Coefficient(R!( p(n,x) ), n-k) >;
[T(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Jul 09 2023

P[n_, x_]:= P[n, x]= ((x+Sqrt[x+4])^n + (x-Sqrt[x+4])^n)/2;
T[n_, k_]:= Coefficient[Series[P[n, x], {x,0,n-k+1}], x, n-k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 08 2020; Jul 09 2023 *)

def p(n,x): return ((x+sqrt(x+4))^n + (x-sqrt(x+4))^n)/2
def T(n,k):
    P. = PowerSeriesRing(QQ)
    return P( p(n,x) ).list()[n-k]
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 09 2023

P(n,x) = 2*x*P(n-1,x) - (x^2 -x -4)*P(n-2,x).
From G. C. Greubel, Jul 09 2023: (Start)
T(n, k) = [x^(n-k)] ( ((x+sqrt(x+4))^n + (x-sqrt(x+4))^n)/2 ).
T(n, 1) = A000217(n-1), n >= 1.
T(n, n) = A199572(n).
Sum_{k=0..n} T(n, k) = A084057(n).
Sum_{k=0..n} 2^k*T(n, k) = A125818(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A026150(n).
Sum_{k=0..n} (-2)^k*T(n, k) = A133343(n). (End)

A052899 Expansion of g.f.: (1-2x) / ((x-1)(4x^2+2x-1)).

Original entry on oeis.org

1, 1, 5, 13, 45, 141, 461, 1485, 4813, 15565, 50381, 163021, 527565, 1707213, 5524685, 17878221, 57855181, 187223245, 605867213, 1960627405, 6344723661, 20531956941, 66442808525, 215013444813, 695798123725, 2251650026701, 7286492548301, 23579585203405, 76305140600013

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

From L. Edson Jeffery, Apr 19 2011: (Start)
Let A be the unit-primitive matrix (see [Jeffery])
A = A_(10,4) =
  (0 0 0 0 1)
  (0 0 0 2 0)
  (0 0 2 0 1)
  (0 2 0 2 0)
  (2 0 2 0 1).
Then a(n) = (1/5)*trace(A^n). (End)
a(n-1)+1 is the number of paths to reach a position outside a 4 X 4 chessboard after n steps, starting in one of the corners, when performing a walk with unit steps on the square lattice. - Ruediger Jehn, Oct 10 2024

Harvey P. Dale, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 875
L. E. Jeffery, Unit-primitive matrices
Index entries for linear recurrences with constant coefficients, signature (3,2,-4).

Cf. A084057.

[(1/5)*(2^(n+1)*Lucas(n)+1): n in [0..50]]; // Vincenzo Librandi, Apr 20 2011

spec := [S,{S=Sequence(Prod(Union(Sequence(Union(Z,Z)),Z,Z),Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[(1-2x)/((x-1)(4x^2+2x-1)),{x,0,40}],x] (* or *) LinearRecurrence[{3,2,-4},{1,1,5},40] (* Harvey P. Dale, Jul 10 2017 *)

makelist(coeff(taylor((1-2*x)/(1-3*x-2*x^2+4*x^3),x,0,n),x,n),n,0,25); /* Bruno Berselli, May 30 2011 */

from sage.combinat.sloane_functions import recur_gen2b
it = recur_gen2b(1,1,2,4, lambda n:-1)
[next(it) for i in range(1,28)] # Zerinvary Lajos, Jul 09 2008

Recurrence: {a(1)=1, a(0)=1, -4*a(n) - 2*a(n+1) + a(n+2) + 1 = 0}.
a(n) = Sum((-1/25)*(-1-8*_alpha+4*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-3*_Z-2*_Z^2+4*_Z^3)).
a(n)/a(n-1) tends to (1 + sqrt(5)) = 3.236067... - Gary W. Adamson, Mar 01 2008
a(n) = (1/5) * Sum_{k=1..5} ((x_k)^4-3*(x_k)^2+1), x_k=2*cos((2*k-1)*Pi/10). Also, a(n)/a(n-1) -> spectral radius of matrix A_(10,4) above. - L. Edson Jeffery, Apr 19 2011
a(n) = (2*A087131(n)+1)/5. - Bruno Berselli, Apr 20 2011
a(n) = (2/5)*((1+sqrt(5))^n + (1-sqrt(5))^n + 1/2). - Ruediger Jehn, Sep 29 2024
E.g.f.: exp(x)*(1 + 4*cosh(sqrt(5)*x))/5. - Stefano Spezia, Oct 02 2024

More terms from James Sellers, Jun 08 2000

A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 1, 0, 0, 1, 10, 5, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (1/(1-x), x^2/(1-x)^2).
A skewed version of triangular array A085478.
Mirror image of triangle in A098158.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n),A087455(n), A146559(n), A000012(n), A011782(n), A001333(n),A026150(n), A046717(n), A084057(n), A002533(n), A083098(n),A084058(n), A003665(n), A002535(n), A133294(n), A090042(n),A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
From Gus Wiseman, Jul 08 2025: (Start)
After the first row this is also the number of subsets of {1..n-1} with k maximal runs (sequences of consecutive elements increasing by 1) for k = 0..n. For example, row n = 5 counts the following subsets:
  {}  {1}        {1,3}    .  .  .
      {2}        {1,4}
      {3}        {2,4}
      {4}        {1,2,4}
      {1,2}      {1,3,4}
      {2,3}
      {3,4}
      {1,2,3}
      {2,3,4}
      {1,2,3,4}
Requiring n-1 gives A202064.
For anti-runs instead of runs we have A384893.
(End)

			Triangle begins :
1
1, 0
1, 1, 0
1, 3, 0, 0
1, 6, 1, 0, 0
1, 10, 5, 0, 0, 0
1, 15, 15, 1, 0, 0, 0
1, 21, 35, 7, 0, 0, 0, 0
1, 28, 70, 28, 1, 0, 0, 0, 0

Column k = 1 is A000217.
Column k = 2 is A000332.
Row sums are A011782 (or A000079 shifted right).
Removing all zeros gives A034839 (requiring n-1 A034867).
Last nonzero term in each row appears to be A093178, requiring n-1 A124625.
Reversing rows gives A098158, without zeros A109446.
Without the k = 0 column we get A210039.
Row maxima appear to be A214282.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
A268193 counts integer partitions by number of maximal runs, for anti-runs A384881.
Cf. A000045, A000071, A007318, A010027, A053538, A084938, A202064, A208342, A210034, A384175, A384893.

Table[Length[Select[Subsets[Range[n-1]],Length[Split[#,#2==#1+1&]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Jul 08 2025 *)

T(n,k) = binomial(n,2k).
G.f.: (1-x)/((1-x)^2-y*x^2).
T(n,k)= Sum_{j, j>=0} T(n-1-j,k-1)*j with T(n,0)=1 and T(n,k)= 0 if k<0 or if nT(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k) for n>1, T(0,0) = T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Nov 10 2013

A084097 Square array whose rows have e.g.f. exp(x)cosh(sqrt(k)x), k>=0, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 7, 8, 1, 1, 1, 5, 10, 17, 16, 1, 1, 1, 6, 13, 28, 41, 32, 1, 1, 1, 7, 16, 41, 76, 99, 64, 1, 1, 1, 8, 19, 56, 121, 208, 239, 128, 1, 1, 1, 9, 22, 73, 176, 365, 568, 577, 256, 1, 1, 1, 10, 25, 92, 241, 576, 1093, 1552, 1393, 512, 1

Offset: 0

Paul Barry, May 11 2003

Rows are the binomial transforms of expansions of cosh(sqrt(k)*x), k >= 0.

			Array, A(n,k), begins:
.n\k.........0..1...2...3....4.....5......6......7.......8........9.......10
.0: A000012..1..1...1...1....1.....1......1......1.......1........1........1
.1: A000079..1..1...2...4....8....16.....32.....64.....128......256......512
.2: A001333..1..1...3...7...17....41.....99....239.....577.....1393.....3363
.3: A026150..1..1...4..10...28....76....208....568....1552.....4240....11584
.4: A046717..1..1...5..13...41...121....365...1093....3281.....9841....29525
.5: A084057..1..1...6..16...56...176....576...1856....6016....19456....62976
.6: A002533..1..1...7..19...73...241....847...2899...10033....34561...119287
.7: A083098..1..1...8..22...92...316...1184...4264...15632....56848...207488
.8: A084058..1..1...9..25..113...401...1593...5993...23137....88225...338409
.9: A003665..1..1..10..28..136...496...2080...8128...32896...130816...524800
10: A002535..1..1..11..31..161...601...2651..10711...45281...186961...781451
11: A133294..1..1..12..34..188...716...3312..13784...60688...259216..1125312
12: A090042..1..1..13..37..217...841...4069..17389...79537...350353..1575613
13: A125816..1..1..14..40..248...976...4928..21568..102272...463360..2153984
14: A133343..1..1..15..43..281..1121...5895..26363..129361...601441..2884575
15: A133345..1..1..16..46..316..1276...6976..31816..161296...768016..3794176
16: A120612..1..1..17..49..353..1441...8177..37969..198593...966721..4912337
17: A133356..1..1..18..52..392..1616...9504..44864..241792..1201408..6271488
18: A125818..1..1..19..55..433..1801..10963..52543..291457..1476145..7907059
25: A083578
- _Robert G. Wilson v_, Jan 02 2013
Antidiagonal triangle, T(n,k), begins:
  1;
  1,  1;
  1,  1,  1;
  1,  1,  2,  1;
  1,  1,  3,  4,  1;
  1,  1,  4,  7,  8,   1;
  1,  1,  5, 10, 17,  16,   1;
  1,  1,  6, 13, 28,  41,  32,    1;
  1,  1,  7, 16, 41,  76,  99,   64,    1;
  1,  1,  8, 19, 56, 121, 208,  239,  128,    1;
  1,  1,  9, 22, 73, 176, 365,  568,  577,  256,   1;
  1,  1, 10, 25, 92, 241, 576, 1093, 1552, 1393, 512,  1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf  A140895, A221131.
Rows: A000012, A000079, A001333, A026150, A046717, A084057, A002533, A083098, A084058, A003665,
Rows: A002535, A133294, A090042, A125816, A133343, A133345, A120612, A133356, A125818, A083578.
Columns: A000012, A000012, A000027, A016777, A028884, A134593.
Rows include A011782, A001333, A026150, A046717, A002533.

function A084097(n,k)
  if k eq 0 then return 1;
  else return k*2^(k-1)*(&+[ Binomial(k-j,j)*((n-k-1)/4)^j/(k-j): j in [0..Floor(k/2)]]);
  end if; return A084097; end function;
[A084097(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 15 2022

T[j_, k_] := Expand[((1 + Sqrt[j])^k + (1 - Sqrt[j])^k)/2]; T[1, 0] = 1; Table[ T[j - k, k], {j, 0, 11}, {k, 0, j}] // Flatten (* Robert G. Wilson v, Jan 02 2013 *)

def A084097(n,k):
    if (k==0): return 1
    else: return k*2^(k-1)*sum( binomial(k-j,j)*((n-k-1)/4)^j/(k-j) for j in range( (k+2)//2 ) )
flatten([[A084097(n,k) for k in range(n+1)] for n in range(15)]) # G. C. Greubel, Oct 15 2022

From Robert G. Wilson v, Jan 02 2013: (Start)
A(n, k) = (1/2)*( (1 + sqrt(n))^k + (1 - sqrt(n))^k ) (array).
T(n, k) = A(n-k, k). (End)
T(n, k) = Sum_{j=0..floor(k/2)} binomial(k-j, j)*((n-k-1)/4)^j/(k-j), with T(n, 0) = 1 (antidiagonal triangle T(n,k)). - G. C. Greubel, Oct 15 2022

Edited by N. J. A. Sloane, Jul 14 2010

Previous Showing 21-30 of 36 results. Next

Binomial transform of A001077. Second binomial transform of A084057. Third binomial transform of 1/(1-5*x^2). Let A=[1,1,1,1;3,1,-1,-3;3,-1,-1,3;1,-1,1,-1], the 4 X 4 Krawtchouk matrix. Then a(n)=trace((16(A*A`)^(-1))^n)/4.

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0,...] where DELTA is the operator defined in A084938. Falling diagonal sums in A052980.

Binomial transform is A055271. May be seen as a ibasefor-transform of the zero-sequence A000004 with respect to the floretion given in the program code.
The sequence is the INVERT transform of (1, 5, 10, 20, 40, 80, 160, ...) and can be obtained by extracting the upper left terms of matrix powers of [(1,5); (1,2)]. These results are a case (a=5, b=2) of the conjecture:  The INVERT transform of a sequence starting (1, a, a*b, a*b^2, a*b^3, ...) is equivalent to extracting the upper left terms of powers of the  2x2 matrix [(1,a); (1,b)]. - Gary W. Adamson, Jul 31 2016
From Klaus Purath, Mar 09 2023: (Start)
For any terms (a(n), a(n+1)) = (x, y), -3*x^2 - 3*x*y + y^2 = 15*(-3)^n = A082111(2)*(-3)^n. This is valid in general for all recursive sequences (t) with constant coefficients (3,3) and t(0) = 1: -3*x^2 - 3*x*y + y^2 = A082111(t(1)-4)*(-3)^n.
By analogy to this, for three consecutive terms (x, y, z) of any sequence (t) of the form (3,3) with t(0) = 1: y^2 - x*z = A082111(t(1)-4)*(-3)^n. (End)

Binomial transform of A123011. Inverse binomial transform of A164550.
INVERT transform of the sequence (1, 5, 5*3, 5*3^2, 5*3^3, 5*3^4, ...); i.e., of (1, 5, 15, 45, 135, 405, ...). The sequence can also be obtained by extracting the upper left terms in matrix powers of [(1,5); (1,3)]. - Gary W. Adamson, Jul 31 2016
The sequence is A090017 (1, 4, 18, 80, 356, ...) convolved with (1, 2, 0, 0, 0, ...). Also, the upper left terms extracted from matrix powers of [(1,5); (1,3)]. - Gary W. Adamson, Aug 20 2016

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A098648 Expansion of (1-3*x)/(1 - 6*x + 4*x^2).

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A124182 A skewed version of triangular array A081277.

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A108306 Expansion of (3*x+1)/(1-3*x-3*x^2).

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A164549 a(n) = 4*a(n-1) + 2*a(n-2) for n > 1; a(0) = 1, a(1) = 6.

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A201701 Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).

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A108404 Expansion of (1-4x)/(1-8x+11x^2).

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A162516 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = ((x+d)^n + (x-d)^n)/2, where d=sqrt(x+4).

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A052899 Expansion of g.f.: (1-2*x) / ((x-1)*(4*x^2+2*x-1)).

A098648 Expansion of (1-3x)/(1 - 6x + 4*x^2).

A108306 Expansion of (3x+1)/(1-3x-3*x^2).

A164549 a(n) = 4a(n-1) + 2a(n-2) for n > 1; a(0) = 1, a(1) = 6.

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

A052899 Expansion of g.f.: (1-2x) / ((x-1)(4x^2+2x-1)).

A084097 Square array whose rows have e.g.f. exp(x)cosh(sqrt(k)x), k>=0, read by ascending antidiagonals.