A030191 -id:A030191 - cp's OEIS frontend

A084326 a(0)=0, a(1)=1; for n>1, a(n) = 6a(n-1)-4a(n-2).

0, 1, 6, 32, 168, 880, 4608, 24128, 126336, 661504, 3463680, 18136064, 94961664, 497225728, 2603507712, 13632143360, 71378829312, 373744402432, 1956951097344, 10246728974336, 53652569456640, 280928500842496, 1470960727228416, 7702050360000512, 40328459251089408

Offset: 0

Benoit Cloitre, Jun 21 2003

Binomial transform of A001076. - Paul Barry, Aug 25 2003

The ratio a(n+1)/(a(n+1)-4*a(n)) converges to 2 + sqrt(5). - Karl V. Keller, Jr., May 17 2015

			a(5) = 6 * a(4) - 4 * a(3) = 6*168 - 4*32 = 880.

S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000
Pamela Fleischmann, Jonas Höfer, Annika Huch, and Dirk Nowotka, alpha-beta-Factorization and the Binary Case of Simon's Congruence, arXiv:2306.14192 [math.CO], 2023.
Yuhan Jiang, The doubly asymmetric simple exclusion process, the colored Boolean process, and the restricted random growth model, arXiv:2312.09427 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (6,-4).

Cf. A030191.

[n le 2 select (n-1) else 6*Self(n-1)-4*Self(n-2): n in [1..25]]; // Vincenzo Librandi, May 15 2015

Join[{a = 0, b = 1}, Table[c = 6 * b - 4 * a; a = b; b = c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
LinearRecurrence[{6, -4}, {0, 1}, 30] (* Vincenzo Librandi, May 15 2015 *)

a(n)=(1/2)*sum(k=0,n,binomial(n,k)*fibonacci(3*k))

a(n)={2^(n-1)*fibonacci(2*n)} \\ Andrew Howroyd, Oct 27 2020

[lucas_number1(n,6,4) for n in range(0, 22)] # Zerinvary Lajos, Apr 22 2009

a(n) = (1/2)*sum(k = 0, n, binomial(n, k)*F(3*k)) where F(k) denotes the k-th Fibonacci number.
a(n) = sqrt(5)((3+sqrt(5))^n - (3-sqrt(5))^n)/10. - Paul Barry, Aug 25 2003
a(n) = Sum(C(n, 2k+1)5^k 3^(n-2k-1), k = 0, .., Floor[(n-1)/2]). a(n) = 2^(n-1)F(2n). - Mario Catalani (mario.catalani(AT)unito.it), Jul 22 2004
a(n) is the rightmost term in M^n * [1 0] where M is the 2X2 matrix [5 1 / 1 1]. The characteristic polynomial of M = x^2 - 6x + 4. a(n)/a(n-1) tends to (3 + sqrt(5)), a root of the polynomial and an eigenvalue of M. - Gary W. Adamson, Dec 16 2004
a(n) = sum{k = 0..n, sum{j = 0..n, C(n, j)C(j, k)F(j+k)/2}}. - Paul Barry, Feb 14 2005
G.f.: x/(1 - 6x + 4x^2). - R. J. Mathar, Sep 09 2008
If p[i] = (4^i-1)/3, and if A is the Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i <= j), A[i,j] = -1, (i = j+1), and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = det A. - Milan Janjic, May 08 2010
a(n) = 5a(n - 1) + a(n - 2) + a(n - 3) + ... + a(1) + 1. - Gary W. Adamson, Feb 18 2011
a(n) = 2^(n-1)*A001906(n). - R. J. Mathar, Apr 03 2011

A093129 Binomial transform of Fibonacci(2n-1) (A001519).

Original entry on oeis.org

1, 2, 5, 15, 50, 175, 625, 2250, 8125, 29375, 106250, 384375, 1390625, 5031250, 18203125, 65859375, 238281250, 862109375, 3119140625, 11285156250, 40830078125, 147724609375, 534472656250, 1933740234375, 6996337890625

Offset: 0

Paul Barry, Mar 23 2004

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

Cf. A000032, A000045, A001519, A020876, A030191, A093123.

a:=[1,2];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; a; # G. C. Greubel, Dec 27 2019

I:=[1,2]; [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // G. C. Greubel, Dec 27 2019

a:= n-> (<<0|1>, <-5|5>>^n. <<1,2>>)[1,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 29 2015

LinearRecurrence[{5, -5}, {1, 2}, 25] (* Jean-François Alcover, May 11 2019 *)
Table[If[EvenQ[n], 5^(n/2)*Fibonacci[n-1], 5^((n-1)/2)*LucasL[n-1]], {n,0,30}] (* G. C. Greubel, Dec 27 2019 *)

my(x='x+O('x^30)); Vec((1-3*x)/(1-5*x+5*x^2)) \\ G. C. Greubel, Dec 27 2019

[lucas_number2(n,5,5) for n in range(-1,25)] # Zerinvary Lajos, Jul 08 2008

G.f.: (1-3*x)/(1-5*x+5*x^2).
a(n) = (5-sqrt(5))*((5+sqrt(5))/2)^n/10 + (5+sqrt(5))*((5-sqrt(5))/2)^n/10.
a(n) = A093123(n)/2^n.
a(n) = A020876(n-1). - R. J. Mathar, Sep 05 2008
a(n) = A030191(n) - 3*A030191(n-1). - R. J. Mathar, Jun 29 2012
a(2*n) = 5^n*Fibonacci(2*n-1), a(2*n+1) = 5^n*Lucas(2*n). - G. C. Greubel, Dec 27 2019
E.g.f.: (1/10)*exp((1/2)*(5-sqrt(5))*x)*(5 + sqrt(5) + (5 - sqrt(5))*exp(sqrt(5)*x)). - Stefano Spezia, Dec 28 2019

A057085 a(n) = 9a(n-1) - 9a(n-2) for n>1, with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 9, 72, 567, 4455, 34992, 274833, 2158569, 16953624, 133155495, 1045816839, 8213952096, 64513217313, 506693386953, 3979621526760, 31256353258263, 245490585583527, 1928108090927376, 15143557548094641, 118939045114505385, 934159388097696696

Offset: 0

Wolfdieter Lang, Aug 11 2000

Scaled Chebyshev U-polynomials evaluated at 3/2.

Colin Barker, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=9, q=-9.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(38) and (45),lhs, m=9.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (9,-9).

Cf. A000045, A000244, A001906, A004187, A030191, A033890, A049310, A109466.

[3^(n-1)*Fibonacci(2*n): n in [0..30]]; // G. C. Greubel, May 02 2022

f[n_]:= Fibonacci[2n]*3^(n-1); Table[f@n, {n, 0, 20}] (* or *)
a[0]=0; a[1]=1; a[n_]:= a[n]= 9(a[n-1] -a[n-2]); Table[a[n], {n, 0, 20}] (* or *)
CoefficientList[Series[x/(1-9x +9x^2), {x, 0, 20}], x] (* Robert G. Wilson v Sep 21 2006 *)

a(n)=(1/3)*sum(k=0,n,binomial(n,k)*fibonacci(4*k)) \\ Benoit Cloitre

concat(0, Vec(x/(1-9*x+9*x^2) + O(x^30))) \\ Colin Barker, Jun 14 2015

[lucas_number1(n,9,9) for n in range(0, 21)] # Zerinvary Lajos, Apr 23 2009

a(n) = A001906(n)*3^(n-1).
a(n) = S(n, 3)*3^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
a(n) = A001906(n)*A000244(n)/3. - Robert G. Wilson v, Sep 21 2006
a(2k) = A004187(k)*9^k/3, a(2k-1) = A033890(k)*9^k.
G.f.: x/(1-9*x+9*x^2).
a(n) = (1/3)*Sum_{k=0..n} binomial(n, k)*Fibonacci(4*k). - Benoit Cloitre, Jun 21 2003
a(n+1) = Sum_{k=0..n} A109466(n,k)*9^k. - Philippe Deléham, Oct 28 2008
E.g.f.: 2*exp(9*x/2)*sinh(3*sqrt(5)*x/2)/(3*sqrt(5)). - Stefano Spezia, Sep 01 2025

Edited by N. J. A. Sloane, Sep 16 2005

A057086 Scaled Chebyshev U-polynomials evaluated at sqrt(10)/2.

Original entry on oeis.org

1, 10, 90, 800, 7100, 63000, 559000, 4960000, 44010000, 390500000, 3464900000, 30744000000, 272791000000, 2420470000000, 21476790000000, 190563200000000, 1690864100000000, 15003009000000000, 133121449000000000, 1181184400000000000, 10480629510000000000

Offset: 0

Wolfdieter Lang, Aug 11 2000

This is the m=10 member of the m-family of sequences S(n,sqrt(m))*(sqrt(m))^n; for S(n,x) see Formula. The m=4..9 instances are A001787, A030191, A030192, A030240, A057084-5 and the m=1..3 signed sequences are A010892, A009545, A057083.

The characteristic roots are rp(m) := (m + sqrt(m*(m-4)))/2 and rm(m) := (m-sqrt(m*(m-4)))/2 and a(n,m)= (rp(m)^(n+1) - rm(m)^(n+1))/(rp(m) - rm(m)) is the Binet form of these m-sequences.

Colin Barker, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=10, q=-10.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(38) and (45),lhs, m=10.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (10,-10).

Cf. A001090, A001787, A030191, A030192, A030240, A049310.

Cf. A009545, A010892, A057080, A057083, A057084, A057085.

[(10)^n*Evaluate(DicksonSecond(n, 1/10), 1): n in [0..30]]; // G. C. Greubel, May 02 2022

Join[{a=1,b=10},Table[c=10*b-10*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 20 2011 *)

Vec(1/(1-10*x+10*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015

[lucas_number1(n,10,10) for n in range(1, 20)] # Zerinvary Lajos, Apr 26 2009

a(n) = 10*(a(n-1) - a(n-2)), a(-1)=0, a(0)=1.
a(n) = S(n, sqrt(10))*(sqrt(10))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
a(2*k) = A057080(k)*10^k, a(2*k+1) = A001090(k)*10^(k+1).
G.f.: 1/(1-10*x+10*x^2).
a(n) = Sum_{k=0..n} A109466(n,k)*10^k. - Philippe Deléham, Oct 28 2008

A099453 Expansion of 1/(1 - 7x + 11x^2).

Original entry on oeis.org

1, 7, 38, 189, 905, 4256, 19837, 92043, 426094, 1970185, 9104261, 42057792, 194257673, 897167999, 4143341590, 19134543141, 88365044497, 408075336928, 1884511869029, 8702754376995, 40189650079646, 185597252410577, 857094615997933, 3958092535469184, 18278606972307025

Offset: 0

Paul Barry, Oct 16 2004

Associated to the knot 8_12 by the modified Chebyshev transform A(x)-> (1/(1+x^2)^2)*A(x/(1+x^2)). See A099454 and A099455.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Dror Bar-Natan, The Rolfsen Knot Table.
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
J. Pan, Multiple Binomial Transforms and Families of Integer Sequences , J. Int. Seq. 13 (2010), 10.4.2, F^(3).
Index entries for linear recurrences with constant coefficients, signature (7,-11).

Cf. A030191, A099454, A099455.

a:=[1,7];; for n in [3..30] do a[n]:=7*a[n-1]-11*a[n-2]; od; a; # G. C. Greubel, May 21 2019

I:=[1,7]; [n le 2 select I[n] else 7*Self(n-1) -11*Self(n-2): n in [1..30]]; // G. C. Greubel, May 21 2019

LinearRecurrence[{7,-11}, {1,7}, 30] (* G. C. Greubel, May 21 2019 *)

Vec(1/(1-7*x+11*x^2) + O(x^30)) \\ Michel Marcus, Sep 09 2017

[lucas_number1(n,7,11) for n in range(1, 30)] # Zerinvary Lajos, Apr 23 2009

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-11)^k*7^(n-2k).
a(n) = ((7+sqrt(5))^n - (7-sqrt(5))^n)/(2^n*sqrt(5)), n > 0. Binomial transform of A030191 (Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2); 3rd binomial transform of Fibonacci(n). - Creighton Dement, Apr 19 2005
a(n) = 7*a(n-1) - 11*a(n-2), n >= 2. - Vincenzo Librandi, Mar 18 2011
E.g.f.: exp(7*x/2)*(5*cosh(sqrt(5)*x/2) + 7*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, May 13 2024

A216219 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=5 or if k-n>=5, T(4,0) = T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 0, 5, 10, 10, 5, 0, 0, 5, 15, 20, 15, 5, 0, 0, 0, 20, 35, 35, 20, 0, 0, 0, 0, 20, 55, 70, 55, 20, 0, 0, 0, 0, 0, 75, 125, 125, 75, 0, 0, 0, 0, 0, 0, 75, 200, 250, 200

Offset: 0

Philippe Deléham, Mar 13 2013

			Square array begins:
1, 1,  1,  1,   1,   0,   0,   0,    0,    0, 0, ...
1, 2,  3,  4,   5,   5,   0,   0,    0,    0, 0, ...
1, 3,  6, 10,  15,  20,  20,   0,    0,    0, 0, ...
1, 4, 10, 20,  35,  55,  75,  75,    0,    0, 0, ...
1, 5, 15, 35,  70, 125, 200, 275,  275,    0, 0, ...
0, 5, 20, 55, 125, 250, 450, 725, 1000, 1000, 0, ...
0, 0, 20, 75, 200, 450, 900, ...

Cf. A030191, A039717, A068913, A081567, A147748, A216210, A216212, A216218

T(n,n) = A147748(n).
T(n+1,n) = T(n,n+1) = A081567(n).
T(n+2,n) = T(n,n+2) = A039717(n+1).
T(n+3,n) = T(n+4,n) = T(n,n+3) = T(n,n+4) = A030191(n).
Sum_{k, 0<=k<=n} T(n-k,k) = A068913(4,n) = A216212(n).

A093131 Binomial transform of Fibonacci(2n).

Original entry on oeis.org

0, 1, 5, 20, 75, 275, 1000, 3625, 13125, 47500, 171875, 621875, 2250000, 8140625, 29453125, 106562500, 385546875, 1394921875, 5046875000, 18259765625, 66064453125, 239023437500, 864794921875, 3128857421875, 11320312500000, 40957275390625, 148184814453125

Offset: 0

Paul Barry, Mar 23 2004

Second binomial transform of Fibonacci(n). - Paul Barry, Apr 22 2005

Michael De Vlieger, Table of n, a(n) for n = 0..1791
G. Dresden and Y. Li, Periodic Weighted Sums of Binomial Coefficients, arXiv:2210.04322 [math.NT], 2022.
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
M. Griffiths, Families of Sequences From a Class of Multinomial Sums, Journal of Integer Sequences, 15 (2012), #12.1.8.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
J. Pan, Multiple Binomial Transforms and Families of Integer Sequences, J. Int. Seq. 13 (2010), 10.4.2, F^(2) and absolute values of F^(-2).
J. Pan, Some Properties of the Multiple Binomial Transform and the Hankel Transform of Shifted Sequences, J. Int. Seq. 14 (2011) # 11.3.4, remark 14.
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Index entries for linear recurrences with constant coefficients, signature (5,-5).

Cf. A000032, A000045, A020876, A030191.

a:=[0,1];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; a; # G. C. Greubel, Dec 27 2019

I:=[0,1]; [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // G. C. Greubel, Dec 27 2019

seq(coeff(series(x/(1-5*x+5*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 27 2019

CoefficientList[Series[x/(1-5x+5x^2), {x,0,30}], x] (* Michael De Vlieger, Dec 22 2019 *)
Table[If[EvenQ[n], 5^(n/2)*Fibonacci[n], 5^((n-1)/2)*LucasL[n]], {n,0,30}] (* G. C. Greubel, Dec 27 2019 *)
LinearRecurrence[{5,-5},{0,1},30] (* Harvey P. Dale, Mar 21 2023 *)

my(x='x+O('x^30)); concat([0], Vec(x/(1-5*x+5*x^2))) \\ G. C. Greubel, Dec 27 2019

def A093131_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x/(1-5*x+5*x^2) ).list()
A093131_list(30) # G. C. Greubel, Dec 27 2019

G.f.: x/(1 - 5*x + 5*x^2).
a(n) = 5*a(n-1) - 5*a(n-2).
a(n) = (((5 + sqrt(5))/2)^n - ((5 - sqrt(5))/2)^n)/sqrt(5).
a(n) = A093130(n)/2^n.
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*Fibonacci(j-k). - Paul Barry, Feb 15 2005
a(n) = Sum_{k=0..n} C(n, k)*2^k*Fibonacci(n-k) = Sum_{k=0..n} C(n, k)*2^(n-k) * Fibonacci(k). - Paul Barry, Apr 22 2005
a(n) = A030191(n-1), n > 0. - R. J. Mathar, Sep 05 2008
E.g.f.: 2*exp(5*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Ilya Gutkovskiy, Aug 11 2017
From Kai Wang, Dec 22 2019: (Start)
a(n) = Sum_{i=0..n-1; j=0..n-1; i+2*j=n-1} 5^i*((i+j)!/(i!*j!)).
a(n*k)/a(k) = Sum_{i=0..n-1; j=0..n-1; i+2*j=n-1} (-1)^(j*(k-1))*b(k)^i*((i+j)!/(i!*j!)).
a((2*m+1)*k)/a(k) = Sum_{i=0..m-1} (-1)^(i*k)*A020876((2*m-2*i)*k) + 5^(m*k).
a(2*m*k)/a(k) = Sum_{i=0..m-1} (-1)^(i*k)*A020876((2*m-2*i-1)*k}.
a(m+r)*a(n+s) - a(m+s)*a(n+r) = -5^(n+s)*a(m-n)*a(r-s).
a(m+r)*a(n+s) + a(m+s)*a(n+r) = (2*A020876(m+n+r+s) - 5^(n+s)*A020876(m-n)*A020876(r-s))/5.
A020876(m+r)*A020876(n+s) - A020876(m+s)*A020876(n+r) = 5^(n+s+1)*a(m-n)*a(r-s).
A020876(m+r)*A020876(n+s) - 5*a(m+s)*a(n+r) = 5^(n+s)*A020876(m-n)*A020876(r-s).
A020876(m+r)*A020876(n+s) + 5*a(m+s)*a(n+r) = 2*A020876(m+n+r+s) + 5^(n+s+1)*a(m-n)*a(r-s).
a(n)^2 - a(n+1)*a(n-1) = 5^(n-1).
a(n)^2 - a(n+r)*a(n-r) = 5^(n-r)*a(r)^2.
a(m)*a(n+1) - a(m+1)*a(n) = 5^n*a(m-n).
a(m-n) = (a(m)*A020876(n) - A020876(m)*a(n))/(2*5^n).
a(m+n) = (a(m)*A020876(n) + A020876(m)*a(n))/2.
A020876(n)^2 - A020876(n+r)*A020876(n-r) = -5^(n-r+1)*a(r)^2.
A020876(m)*A020876(n+1) - A020876(m+1)*A020876(n) = -5^(n+1)*a(m-n).
A020876(m+n) - 5^(n)*A020876(m-n) = 5*a(m)*a(n).
A020876(m-n) = (A020876(m)*A020876(n) - 5*a(m)*a(n))/(2*5^n).
A020876(m+n) = (A020876(m)*A020876(n) + 5*a(m)*a(n))/2.  (End)
a(2*n) = 5^n*Fibonacci(2*n), a(2*n+1) = 5^n*Lucas(2*n+1). - G. C. Greubel, Dec 27 2019
a(n) = Sum_{k=0..n} (-1)^(k+1)*binomial(2*n, n+k)*(k|5), where (k|5) is the Legendre symbol. - Greg Dresden, Oct 14 2022

A260585 Number of ways to place 2n rooks on an n X n board, with 2 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 2 rooks below the main diagonal.

Original entry on oeis.org

1, 11, 72, 367, 1630, 6680, 26082, 98870, 368045, 1354850, 4953503, 18035279, 65499031, 237511321, 860471110, 3115667369, 11277816388, 40814611818, 147692103728, 534404499040, 1933597628291, 6996040095316, 25312367524557, 91581960107817, 331348634005165

Offset: 2

Jeffrey Davis, Jul 29 2015

a(n) is the number of minimal multiplex juggling patterns of period n using exactly 2 balls when we can catch/throw up to 2 balls at a time. (Minimal in the sense that each of the n throws is between 0 and n-1.)

Colin Barker, Table of n, a(n) for n = 2..1000
Esther M. Banaian, Generalized Eulerian Numbers and Multiplex Juggling Sequences, (2016). All College Thesis Program. Paper 24.
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (12,-59,155,-236,209,-100,20).

Column k=2 of A269742.

Cf. A260575, A260582, A260583, A260584, A260727.

CoefficientList[Series[-(5*x^4 - 3*x^3 - x^2 - x + 1)/(20*x^7 - 100*x^6 + 209*x^5 - 236*x^4 + 155*x^3 - 59*x^2 + 12*x - 1), {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 16 2015 *)

Vec(-(5*x^6 - 3*x^5 - x^4 - x^3 + x^2)/(20*x^7 - 100*x^6 + 209*x^5 - 236*x^4 + 155*x^3 - 59*x^2 + 12*x - 1) + O(x^40)) \\ Michel Marcus, Aug 17 2015

G.f.: -x^2*(5*x^4-3*x^3-x^2-x+1)/((1-5*x+5*x^2)*(2*x-1)^2*(x-1)^3).
a(n) = 12*a(n-1) - 59*a(n-2) + 155*a(n-3) - 236*a(n-4) + 209*a(n-5) - 100*a(n-6) + 20*a(n-7). - Wesley Ivan Hurt, Jan 01 2024
a(n) = (n+2)*(n-1)/2-2^n*(1+3*n/2)+2*A030191(n)-5*A030191(n-1). - R. J. Mathar, Aug 26 2025

A129267 Triangle with T(n,k) = T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k) and T(0,0)=1 .

Original entry on oeis.org

1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -1, -3, -2, 1, 1, 0, -2, -5, -3, 1, 1, 1, 2, -2, -7, -4, 1, 1, 1, 5, 7, -1, -9, -5, 1, 1, 0, 3, 12, 15, 1, -11, -6, 1, 1, -1, -3, 3, 21, 26, 4, -13, -7, 1, 1, -1, -7, -15, -3, 31, 40, 8, -15, -8, 1, 1

Offset: 0

Philippe Deléham, Jun 08 2007

Triangle T(n,k), 0<=k<=n, read by rows given by [1,-1,1,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] where DELTA is the operator defined in A084938 . Riordan array (1/(1-x+x^2),(x*(1-x))/(1-x+x^2)); inverse array is (1/(1+x),(x/(1+x))*c(x/(1+x))) where c(x)is g.f. of A000108 .

Row sums are ( with the addition of a first row {0}): 0, 1, 2, 2, 0, -4, -8, -8, 0, 16, 32,... (see A009545). - Roger L. Bagula, Nov 15 2009

			Triangle begins:
   1;
   1,  1;
   0,  1,   1;
  -1, -1,   1,  1;
  -1, -3,  -2,  1,  1;
   0, -2,  -5, -3,  1,   1;
   1,  2,  -2, -7, -4,   1,   1;
   1,  5,   7, -1, -9,  -5,   1,   1;
   0,  3,  12, 15,  1, -11,  -6,   1,  1;
  -1, -3,   3, 21, 26,   4, -13,  -7,  1, 1;
  -1, -7, -15, -3, 31,  40,   8, -15, -8, 1, 1;

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

Cf. A063967, A167925, A199324, A202503, A202551.

T:= proc(n, k) option remember;
      if k<0 or  k>n  then 0
    elif n=0 and k=0 then 1
    else T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 14 2020

m = {{a, 1}, {-1, 1}}; v[0]:= {0, 1}; v[n_]:= v[n] = m.v[n-1]; Table[CoefficientList[v[n][[1]], a], {n, 0, 10}]//Flatten (* Roger L. Bagula, Nov 15 2009 *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n==0 && k==0, 1, T[n-1, k-1] + T[n-1, k] - T[n-2, k-1] - T[n-2, k] ]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 14 2020 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (n==0 and k==0): return 1
    else: return T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 14 2020

Sum{k=0..n} T(n,k)*x^k = { (-1)^n*A057093(n), (-1)^n*A057092(n), (-1)^n*A057091(n), (-1)^n*A057090(n), (-1)^n*A057089(n), (-1)^n*A057088(n), (-1)^n*A057087(n), (-1)^n*A030195(n+1), (-1)^n*A002605(n), A039834(n+1), A000007(n), A010892(n), A099087(n), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n), A057086(n) } for x=-11, -10, ..., 8, 9, respectively .
Sum{k=0..n} T(n,k)*A000045(k) = A100334(n).
Sum{k=0..floor(n/2)} T(n-k,k) = A050935(n+2).
T(n,k)= Sum{j>=0} A109466(n,j)*binomial(j,k).
T(n,k) = (-1)^(n-k)*A199324(n,k) = (-1)^k*A202551(n,k) = A202503(n,n-k). - Philippe Deléham, Mar 26 2013
G.f.: 1/(1-x*y+x^2*y-x+x^2). - R. J. Mathar, Aug 11 2015

Riordan array definition corrected by Ralf Stephan, Jan 02 2014

A147748 Row sums of Riordan array ((1-3x+x^2)/(1-4x+3x^2), x(1-2x)/(1-4x+3x^2)).

Original entry on oeis.org

1, 2, 6, 20, 70, 250, 900, 3250, 11750, 42500, 153750, 556250, 2012500, 7281250, 26343750, 95312500, 344843750, 1247656250, 4514062500, 16332031250, 59089843750, 213789062500, 773496093750, 2798535156250, 10125195312500

Offset: 0

Paul Barry, Nov 11 2008

Row sums of A147747. Binomial transform of A061646.
Counts all paths of length (2*n), n>=0, starting at the initial node on the path graph P_9, see the Maple program. - Johannes W. Meijer, May 29 2010
From L. Edson Jeffery, Apr 19 2011: (Start)
For the 5 X 5 unit-primitive matrix (see [Jeffery])
A_(10,1) = [0,1,0,0,0; 1,0,1,0,0; 0,1,0,1,0; 0,0,1,0,1; 0,0,0,2,0],
a(n) = (Trace([A_(10,1)]^(2*n)))/5. (See also A189315.) (End)

S. Felsner and D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
L. Edson Jeffery, Unit-primitive matrices.
Index entries for linear recurrences with constant coefficients, signature (5,-5).

Cf. A000984, A030191, A061646, A081567, A147746, A147747, A178381, A189315.

with(GraphTheory): G:=PathGraph(9): A:= AdjacencyMatrix(G): nmax:=24; n2:=nmax*2: for n from 0 to n2 do B(n):=A^n; a(n):= add(B(n)[1,k], k=1..9); od: seq(a(2*n), n=0..nmax); # Johannes W. Meijer, May 29 2010

(1 - 3x + x^2)/(1 - 5x + 5x^2) + O[x]^25 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 05 2016 *)

G.f.: (1-3*x+x^2)/(1-5*x+5*x^2).
a(n) = 5*a(n-1) - 5*a(n-2) for n > 2, a(0)=1, a(1)=2, a(2)=6. - Philippe Deléham, Nov 13 2008
For n >= 1: a(n) = (2/5)*((5-sqrt(5))/2)^n + (2/5)*((5+sqrt(5))/2)^n. - Richard Choulet, Nov 14 2008
G.f.: 1/(1-2x/(1-x/(1-x/(1-x)))) (hence sequence approximates A000984 in first few terms). - Paul Barry, Aug 05 2009
a(n) = (1/5)*Sum_{k=1..5} (x_k)^(2*n), x_k=2*cos((2*k-1)*Pi/10). - L. Edson Jeffery, Apr 19 2011
From R. J. Mathar, Apr 20 2011: (Start)
a(n) = A030191(n) - 3*A030191(n-1) + A030191(n-2).
a(n) = 2*A081567(n-1), n > 0. (End)
a(n) = Sum_{k=0..n} A147746(n,k)*2^k. - Philippe Deléham, Oct 30 2011
E.g.f.: (1 + 4*exp(5*x/2)*cosh(sqrt(5)*x/2))/5. - Stefano Spezia, Jul 09 2024

cp's OEIS Frontend

A084326 a(0)=0, a(1)=1; for n>1, a(n) = 6*a(n-1)-4*a(n-2).

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A093129 Binomial transform of Fibonacci(2n-1) (A001519).

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A057085 a(n) = 9*a(n-1) - 9*a(n-2) for n>1, with a(0)=0, a(1)=1.

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A057086 Scaled Chebyshev U-polynomials evaluated at sqrt(10)/2.

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A099453 Expansion of 1/(1 - 7*x + 11*x^2).

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A216219 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=5 or if k-n>=5, T(4,0) = T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A084326 a(0)=0, a(1)=1; for n>1, a(n) = 6a(n-1)-4a(n-2).

A057085 a(n) = 9a(n-1) - 9a(n-2) for n>1, with a(0)=0, a(1)=1.

A099453 Expansion of 1/(1 - 7x + 11x^2).