A155116 -id:A155116 - cp's OEIS frontend

A155020 a(n) = 2a(n-1) + 2a(n-2) for n > 2, a(0)=1, a(1)=1, a(2)=3.

1, 1, 3, 8, 22, 60, 164, 448, 1224, 3344, 9136, 24960, 68192, 186304, 508992, 1390592, 3799168, 10379520, 28357376, 77473792, 211662336, 578272256, 1579869184, 4316282880, 11792304128, 32217174016, 88018956288, 240472260608, 656982433792, 1794909388800, 4903783645184, 13397386067968

Offset: 0

Philippe Deléham, Jan 19 2009

Equals 1 followed by A028859. - Klaus Brockhaus, Jul 18 2009
a(n) is the number of ways to arrange 1- and 2-cent postage stamps (totaling n cents) in a row so that the first stamp is correctly placed and any subsequent stamp may (or not) be placed upside down.
Number of compositions of n into parts k >= 1 where there are F(k+1) = A000045(k+1) sorts of part k. - Joerg Arndt, Sep 30 2012
a(n) is the top-left entry of the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 1, 1; 1, 1, 0] or of the 3 X 3 matrix [1, 1, 1; 1, 0, 1; 1, 1, 1].
From Tom Copeland, Nov 08 2014: (Start)
(Setting a(0)=0.)
This array is one of a family of Catalan arrays related by compositions of the special fractional linear (Möbius) transformations P(x,t) = x/(1-t*x); its inverse Pinv(x,t) = P(x,-t); and an o.g.f. of the Catalan numbers A000108, C(x) = (1-sqrt(1-4x))/2; and its inverse Cinv(x) = x*(1-x). (Cf. A091867.)
O.g.f.: G(x) = -P(P(Cinv(-x),1),1) = -P(Cinv(-x),2) = x(1+x)/(1-2x(1+x)) = (x+x^2)/(1-2(x+x^2)) = x + 3*x^2 + 8*x^3 + ... = A155020(x) with a(0)=0.
Ginv(x) = -C(P(P(-x,-1),-1)) = -C(P(-x,-2)) = (-1+sqrt(1+4*x/(1+2*x)))/2 = x*A064613(-x).
G(x) = x*(1+x) + 2*(x*(1+x))^2 + 2^2*(x*(1+x))^3 - ..., and so this array contains the row sums of A030528 * Diag(1, 2^1, 2^2, 2^3, ...). (End)
INVERT transform of Fibonacci(n+1). - Alois P. Heinz, Feb 11 2021

			a(2) = 3 because we have {1,1}, {1,_1} and {2}.
a(3) = 8 because we can order the stamps in eight ways: {1,1,1}  {1,1,_1}  {1,_1,1}  {1,_1,_1}  {2,1}   {2,_1}  {1,2}   {1,_2}, where _1 and _2 are upside down stamps.
a(4) = 22 = 2*3 + 2*8 because we can append 2 or _2 to the a(2) examples and 1 or _1 to the a(3) examples. - _Jon Perry_, Nov 10 2014

Indranil Ghosh, Table of n, a(n) for n = 0..2287
Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See p. 14.
I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239v1 [math.CO] 17 Sep 2015.
Juan B. Gil and Jessica A. Tomasko, Fibonacci colored compositions and applications, arXiv:2108.06462 [math.CO], 2021.
Jeffrey Shallit, Proof of Irvine's conjecture via mechanized guessing, arXiv preprint arXiv:2310.14252 [math.CO], October 22 2023.
Index entries for linear recurrences with constant coefficients, signature (2,2).

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: this sequence (m=2), A155116 (m=3), A155117 (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), A155144 (m=9), A155157 (m=10).
Cf. A028859 (essentially the same sequence). - Klaus Brockhaus, Jul 18 2009
Cf. A000045, A000108, A030528, A064613, A091867, A154929.
Row sums of A155112.

I:=[1,1,3,8]; [n le 4 select I[n] else 2*Self(n-1)+2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Nov 10 2014

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*combinat[fibonacci](1+i), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 11 2021

CoefficientList[Series[(1 -x -x^2)/(1 -2x -2x^2), {x,0,20}], x]
With[{m=2}, LinearRecurrence[{m, m}, {1, m-1, m^2-1}, 30]] (* G. C. Greubel, Mar 25 2021 *)

makelist(sum(binomial(n-k,k)*2^(n-k-1),k,0,floor(n/2)),n,1,12); /* Emanuele Munarini, Feb 04 2014 */

Vec( (1-x-x^2)/(1-2*x-2*x^2) + O(x^66) )  /* Joerg Arndt, Sep 30 2012 */

[1]+[(-1)*(sqrt(2)*i)^(n-2)*chebyshev_U(n, -sqrt(2)*i/2) for n in (1..30)] # G. C. Greubel, Mar 25 2021

G.f.: (1 - x - x^2)/(1 - 2*x - 2*x^2).
G.f.: 1/( 1 - Sum_{k>=1} (x+x^2)^k ) - 1/( 1 - Sum_{k>=1} F(k+1)*x^k ) where F(k) = A000045(k). - Joerg Arndt, Sep 30 2012
a(n+1) = Sum_{k=0..n} A154929(n,k) = A028859(n).
a(n) = Sum_{k=0..floor(n/2)} ( binomial(n-k,k)*2^(n-k-1) ) for n > 0. - Emanuele Munarini, Feb 04 2014
a(n) = (1/2)*[n=0] - (sqrt(2)*i)^(n-2)*ChebyshevU(n, -sqrt(2)*i/2). - G. C. Greubel, Mar 25 2021
E.g.f.: (3 + exp(x)*(3*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)))/6. - Stefano Spezia, Mar 02 2024

A155117 a(n) = 4a(n-1) + 4a(n-2), n>2, a(0)=1, a(1)=3, a(2)=15.

Original entry on oeis.org

1, 3, 15, 72, 348, 1680, 8112, 39168, 189120, 913152, 4409088, 21288960, 102792192, 496324608, 2396467200, 11571167232, 55870537728, 269766819840, 1302549430272, 6289265000448, 30367257722880, 146626090893312

Offset: 0

Philippe Deléham, Jan 20 2009

nonn

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

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: A155020 (m=2), A155116 (m=3), this sequence (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), A155144 (m=9), A155157 (m=10).

Cf. A000129.

m:=4; [1] cat [n le 2 select (m-1)*(m*n-(m-1)) else m*(Self(n-1) + Self(n-2)): n in [1..30]]; // G. C. Greubel, Mar 25 2021

1,seq(simplify(-3*(2*I)^(n-2)*ChebyshevU(n, -I)), n = 1..30); # G. C. Greubel, Mar 25 2021

With[{m=4}, LinearRecurrence[{m, m}, {1, m-1, m^2-1}, 30]] (* G. C. Greubel, Mar 25 2021 *)

m=4; [1]+[-(m-1)*(sqrt(m)*i)^(n-2)*chebyshev_U(n, -sqrt(m)*i/2) for n in (1..30)] # G. C. Greubel, Mar 25 2021

G.f.: (1-x-x^2)/(1-4*x-4*x^2) . a(n)=3*A086347(n), n>=1 .
From G. C. Greubel, Mar 25 2021: (Start)
a(n) = (1/4)*[n=0] - 3*(2*i)^(n-2)*ChebyshevU(n, -i).
a(n) = (1/4)*[n=0] + 3*2^(n-2)*P_{n+1}, where P_{n} = A000129(n) (Pell numbers). (End)

A155119 a(n) = 5a(n-1) + 5a(n-2), n > 2, a(0)=1, a(1)=4, a(2)=24.

Original entry on oeis.org

1, 4, 24, 140, 820, 4800, 28100, 164500, 963000, 5637500, 33002500, 193200000, 1131012500, 6621062500, 38760375000, 226907187500, 1328337812500, 7776225000000, 45522814062500, 266495195312500, 1560090046875000, 9132926210937500

Offset: 0

Philippe Deléham, Jan 20 2009

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

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: A155020 (m=2), A155116 (m=3), A155117 (m=4), this sequence (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), A155144 (m=9), A155157 (m=10).

m:=5; [1] cat [n le 2 select (m-1)*(m*n-(m-1)) else m*(Self(n-1) + Self(n-2)): n in [1..30]]; // G. C. Greubel, Mar 25 2021

With[{m=5}, LinearRecurrence[{m, m}, {1, m-1, m^2-1}, 30]] (* G. C. Greubel, Mar 25 2021 *)

m=5; [1]+[-(m-1)*(sqrt(m)*i)^(n-2)*chebyshev_U(n, -sqrt(m)*i/2) for n in (1..30)] # G. C. Greubel, Mar 25 2021

G.f.: (1 - x - x^2) / (1 - 5*x - 5*x^2).
a(n) = (1/5)*[n=0] - 4*(sqrt(5)*i)^(n-2)*ChebyshevU(n, -sqrt(5)*i/2). - G. C. Greubel, Mar 25 2021
E.g.f.: (3 + 4*exp(5*x/2)*(3*cosh(3*sqrt(5)*x/2) + sqrt(5)*sinh(3*sqrt(5)*x/2)))/15. - Stefano Spezia, May 31 2023

a(20) corrected and a(21) from Sean A. Irvine, May 19 2019

A155127 a(n) = 6a(n-1) + 6a(n-2), n>2, a(0)=1, a(1)=5, a(2)=35.

Original entry on oeis.org

1, 5, 35, 240, 1650, 11340, 77940, 535680, 3681720, 25304400, 173916720, 1195326720, 8215460640, 56464724160, 388081108800, 2667274997760, 18332136639360, 125996469822720, 865971638772480, 5951808651571200

Offset: 0

Philippe Deléham, Jan 20 2009

nonn

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

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: A155020 (m=2), A155116 (m=3), A155117 (m=4), A155119 (m=5), this sequence (m=6), A155130 (m=7), A155132 (m=8), A155144 (m=9), A155157 (m=10).

m:=6; [1] cat [n le 2 select (m-1)*(m*n-(m-1)) else m*(Self(n-1) + Self(n-2)): n in [1..30]]; // G. C. Greubel, Mar 25 2021

m:=6; 1,seq(simplify((1-m)*(sqrt(m)*I)^(n-2)*ChebyshevU(n, -I*sqrt(m)/2)), n = 1..30); # G. C. Greubel, Mar 25 2021

LinearRecurrence[{6,6},{1,5,35},20] (* Harvey P. Dale, Apr 14 2015 *)

m=6; [1]+[-(m-1)*(sqrt(m)*i)^(n-2)*chebyshev_U(n, -sqrt(m)*i/2) for n in (1..30)] # G. C. Greubel, Mar 25 2021

G.f.: (1-x-x^2)/(1-6*x-6*x^2) .

a(n) = (1/6)*[n=0] - 5*(sqrt(6)*i)^(n-2)*ChebyshevU(n, -sqrt(6)*i/2). - G. C. Greubel, Mar 25 2021

A155130 a(n) = 7a(n-1) + 7a(n-2), n>2, a(0)=1, a(1)=6, a(2)=48.

Original entry on oeis.org

1, 6, 48, 378, 2982, 23520, 185514, 1463238, 11541264, 91031514, 718009446, 5663286720, 44669073162, 352326519174, 2778969146352, 21919069658682, 172886271635238, 1363637389057440, 10755665624848746, 84835121097343302

Offset: 0

Philippe Deléham, Jan 20 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,7).

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: A155020 (m=2), A155116 (m=3), A155117 (m=4), A155119 (m=5), A155127 (m=6), this sequence (m=7), A155132 (m=8), A155144 (m=9), A155157 (m=10).

m:=7; [1] cat [n le 2 select (m-1)*(m*n-(m-1)) else m*(Self(n-1) + Self(n-2)): n in [1..30]]; // G. C. Greubel, Mar 25 2021

m:= 7; 1,seq(simplify((1-m)*(sqrt(m)*I)^(n-2)*ChebyshevU(n, -I*sqrt(m)/2)), n = 1..30); # G. C. Greubel, Mar 25 2021

LinearRecurrence[{7,7},{1,6,48},30] (* Harvey P. Dale, Mar 11 2018 *)

m=7; [1]+[-(m-1)*(sqrt(m)*i)^(n-2)*chebyshev_U(n, -sqrt(m)*i/2) for n in (1..30)] # G. C. Greubel, Mar 25 2021

G.f.: (1-x-x^2)/(1-7*x-7*x^2) .

a(n) = (1/7)*[n=0] - 6*(sqrt(7)*i)^(n-2)*ChebyshevU(n, -sqrt(7)*i/2). - G. C. Greubel, Mar 25 2021

A155132 a(n) = 8a(n-1) + 8a(n-2), n > 2, a(0)=1, a(1)=7, a(2)=63.

Original entry on oeis.org

1, 7, 63, 560, 4984, 44352, 394688, 3512320, 31256064, 278147072, 2475225088, 22026977280, 196017618944, 1744356769792, 15522995109888, 138138815037440, 1229294481178624, 10939466369728512, 97350086807257088

Offset: 0

Philippe Deléham, Jan 20 2009

nonn

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

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: A155020 (m=2), A155116 (m=3), A155117 (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), this sequence (m=8), A155144 (m=9), A155157 (m=10).

[1] cat [n le 2 select 7*(8*n-7) else 8*(Self(n-1) + Self(n-2)): n in [1..30]]; // G. C. Greubel, Mar 24 2021

With[{m=8}, LinearRecurrence[{m, m}, {1, m-1, m^2-1}, 30]] (* G. C. Greubel, Mar 24 2021 *)

[1]+[-7*(2*sqrt(2)*i)^(n-2)*chebyshev_U(n, -sqrt(2)*i) for n in (1..30)] # G. C. Greubel, Mar 24 2021

G.f.: (1-x-x^2)/(1-8*x-8*x^2) .

a(n) = (1/8)*[n=0] - 7*(2*sqrt(2)*i)^(n-2)*ChebyshevU(n, -sqrt(2)*I). - G. C. Greubel, Mar 24 2021

A155144 a(n) = 9a(n-1) + 9a(n-2), n>2; a(0)=1, a(1)=8, a(2)=80.

Original entry on oeis.org

1, 8, 80, 792, 7848, 77760, 770472, 7634088, 75641040, 749476152, 7426054728, 73579777920, 729052493832, 7223690445768, 71574686456400, 709185392119512, 7026840707183208, 69624234893724480, 689859680408169192

Offset: 0

Philippe Deléham, Jan 21 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..998
Index entries for linear recurrences with constant coefficients, signature (9,9).

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: A155020 (m=2), A155116 (m=3), A155117 (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), this sequence (m=9), A155157 (m=10).

[1] cat [n le 2 select 8*(9*n-8) else 9*(Self(n-1) + Self(n-2)): n in [1..30]]; // G. C. Greubel, Mar 24 2021

m:=30; S:=series( (1-x-x^2)/(1-9*x-9*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 24 2021

Join[{1},LinearRecurrence[{9,9},{8,80},20]] (* or *) CoefficientList[ Series[ (1-x-x^2)/(1-9x-9x^2),{x,0,20}],x] (* Harvey P. Dale, Jun 19 2011 *)

[1]+[-8*(3*i)^(n-2)*chebyshev_U(n, -3*i/2) for n in (1..30)] # G. C. Greubel, Mar 24 2021

G.f.: (1-x-x^2)/(1-9*x-9*x^2).
From G. C. Greubel, Mar 24 2021: (Start)
a(n) = (1/9)*[n=0] - 8*3^(n-2)*ChebyshevU(n, -3*i/2).
a(n) = (1/9)*[n=0] + 8*3^(n-2)*Fibonacci(n+1, 3). (End)

A155157 a(n) = 10a(n-1) + 10a(n-2), with a(0)=1, a(1)=9, a(2)=99.

Original entry on oeis.org

1, 9, 99, 1080, 11790, 128700, 1404900, 15336000, 167409000, 1827450000, 19948590000, 217760400000, 2377089900000, 25948503000000, 283255929000000, 3092044320000000, 33753002490000000, 368450468100000000

Offset: 0

Philippe Deléham, Jan 21 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..950
Index entries for linear recurrences with constant coefficients, signature (10,10).

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: A155020 (m=2), A155116 (m=3), A155117 (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), A155144 (m=9), this sequence (m=10).

Cf. A057093.

[1]cat[n le 2 select 9*(10*n-9) else 10*(Self(n-1) + Self(n-2)): n in [1..30]]; // G. C. Greubel, Mar 20 2021

1,seq( simplify(9*(-I*sqrt(10))^n*ChebyshevU(n, I*sqrt(10)/2)/10), n=1..30); # G. C. Greubel, Mar 20 2021

LinearRecurrence[{10,10},{1,9,99},20] (* Harvey P. Dale, Jan 27 2016 *)

[1]+[(9/10)*(-i*sqrt(10))^n*chebyshev_U(n, i*sqrt(10)/2) for n in (1..30)] # G. C. Greubel, Mar 20 2021

G.f.: (1-x-x^2)/(1-10*x-10*x^2).
From G. C. Greubel, Mar 20 2021: (Start)
a(n) = ([n=0] + 9*A057093(n))/10.
a(n) = (1/10)*([n=0] + 9*(-i*sqrt(10))^n*ChebyshevU(n, i*sqrt(10)/2)). (End)

A180147 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 4x - 3x^2 + 6x^3).

Original entry on oeis.org

1, 7, 31, 139, 607, 2659, 11623, 50827, 222223, 971635, 4248247, 18574555, 81213151, 355086787, 1552539271, 6788138539, 29679651247, 129767784979, 567381262423, 2480750497147, 10846539065983, 47424120180835

Offset: 0

Johannes W. Meijer, Aug 13 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a rook on the eight side and corner squares but on the central square the rook goes berserk and turns into a berserker, see A180140.
On a 3 X 3 chessboard there are 2^9 = 512 ways to go berserk on the central square (we assume here that a berserker might behave like a rook). The berserker is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program. For the central squares the 512 berserkers lead to 42 berserker sequences, see the cross-references for some examples.
The sequence above corresponds to six A[5] vectors with decimal values between 191 and 506. These vectors lead for the corner squares to A180145 and for the side squares to A180146.

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

Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).

Cf. Berserker sequences central square [numerical values A[5]]: A000007 [0], A000012 [16], 2*A001835 [17, n>=1 and a(0)=1], A155116 [3], A077829 [7], A000302 [15], 6*A179606 [111, with leading 1 added], 2*A033887 [95, n>=1 and a(0)=1], A180147 [191, this sequence], 2*A180141 [495, n>=1 and a(0)=1], 4*A107979 [383, with leading 1 added].

with(LinearAlgebra): nmax:=22; m:=5; A[5]:=[0,1,0,1,1,1,1,1,1]: A:= Matrix([[0,1,1,1,0,0,1,0,0], [1,0,1,0,1,0,0,1,0], [1,1,0,0,0,1,0,0,1], [1,0,0,0,1,1,1,0,0], A[5], [0,0,1,1,1,0,0,0,1], [1,0,0,1,0,0,0,1,1], [0,1,0,0,1,0,1,0,1], [0,0,1,0,0,1,1,1,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

CoefficientList[Series[(1+3x)/(1-4x-3x^2+6x^3),{x,0,40}],x] (* or *) LinearRecurrence[{4,3,-6},{1,7,31},40] (* Harvey P. Dale, Oct 10 2011 *)

G.f.: (1+3*x)/(1 - 4*x - 3*x^2 + 6*x^3).
a(n) = 4*a(n-1) + 3*a(n-2) - 6*a(n-3) with a(0)=1, a(1)=7 and a(2)=31.
a(n) = -1/2 + (7+6*A)*A^(-n-1)/22 + (7+6*B)*B^(-n-1)/22 with A=(-3+sqrt(33))/12 and B=(-3-sqrt(33))/12.
a(n) = A180146(n) + 3*A180146(n-1) with A180146(-1) = 0.

A155112 Triangle T(n,k), 0<=k<=n, read by rows given by [0,2,-1/2,-1/2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 10, 6, 1, 0, 8, 22, 21, 8, 1, 0, 13, 45, 59, 36, 10, 1, 0, 21, 88, 147, 124, 55, 12, 1, 0, 34, 167, 339, 366, 225, 78, 14, 1, 0, 55, 310, 741, 976, 770, 370, 105, 16, 1, 0, 89, 566, 1557, 2422, 2337, 1443, 567, 136, 18, 1, 0, 144, 1020, 3174, 5696, 6505, 4920, 2485, 824, 171, 20, 1

Offset: 0

Philippe Deléham, Jan 20 2009

A Fibonacci convolution triangle; Riordan array (1, x*(1+x)/(1-x-x^2)).

			Triangle begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  4,  1;
  0,  5, 10,  6,  1;
  0,  8, 22, 21,  8,  1;
  0, 13, 45, 59, 36, 10, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000007, A155020, A155116, A155117, A155119, A155127, A155130, A155132, A155144, A155157.
Cf. A000012, A155020, A154964, A154968, A154996, A154997, A154999, A155000, A155001, A155017.
Cf. A000045, A154929.
T(2n,n) gives A262910.
Cf. A291385.

T:= func< n,k | n eq 0 select 1 else (&+[ Binomial(n-j,j)*Binomial(n-j,k)*k/(n-j): j in [0..Floor(n/2)]]) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 26 2021

# Uses function PMatrix from A357368.
PMatrix(10, n -> combinat:-fibonacci(n+1)); # Peter Luschny, Oct 19 2022

T[n_, k_]:= If[n==0, 1, Sum[Binomial[n-j, j]*Binomial[n-j, k]*k/(n-j), {j, 0, Floor[n/2]}]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 26 2021 *)

def T(n,k): return 1 if n==0 else sum( binomial(n-j,j)*binomial(n-j,k)*k/(n-j) for j in (0..n//2) )
flatten([[T(n,k) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Mar 26 2021

Recurrence: T(n+2,k+1) = T(n+1,k+1) + T(n+1,k) + T(n,k+1) + T(n,k).
Explicit formula: T(n,k) = Sum_{i=0..floor(n/2)} binomial(n-i, i)*binomial(n-i, k)*k/(n-i), for n > 0.
G.f.: (1-x-x^2)/(1-(1+y)*x-(1+y)*x^2). - Philippe Deléham, Feb 21 2012
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A155020(n), A154964(n), A154968(n), A154996(n), A154997(n), A154999(n), A155000(n), A155001(n), A155017(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A155020(n), A155116(n), A155117(n), A155119(n), A155127(n), A155130(n), A155132(n), A155144(n), A155157(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Feb 21 2012
Sum_{k=0..n} T(n, k)*(m-1)^k = (1/m)*[n=0] - (m-1)*(i*sqrt(m))^(n-2)*ChebyshevU(n, -i*sqrt(m)/2). - G. C. Greubel, Mar 26 2021
Sum_{k=0..n} k * T(n,k) = A291385(n-1) for n>=1. - Alois P. Heinz, Sep 29 2022

Typos in two terms corrected by Alois P. Heinz, Aug 08 2015

cp's OEIS Frontend

A155020 a(n) = 2*a(n-1) + 2*a(n-2) for n > 2, a(0)=1, a(1)=1, a(2)=3.

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A155117 a(n) = 4*a(n-1) + 4*a(n-2), n>2, a(0)=1, a(1)=3, a(2)=15.

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A155119 a(n) = 5*a(n-1) + 5*a(n-2), n > 2, a(0)=1, a(1)=4, a(2)=24.

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A155127 a(n) = 6*a(n-1) + 6*a(n-2), n>2, a(0)=1, a(1)=5, a(2)=35.

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A155130 a(n) = 7*a(n-1) + 7*a(n-2), n>2, a(0)=1, a(1)=6, a(2)=48.

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A155132 a(n) = 8*a(n-1) + 8*a(n-2), n > 2, a(0)=1, a(1)=7, a(2)=63.

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A155144 a(n) = 9*a(n-1) + 9*a(n-2), n>2; a(0)=1, a(1)=8, a(2)=80.

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A155020 a(n) = 2a(n-1) + 2a(n-2) for n > 2, a(0)=1, a(1)=1, a(2)=3.

A155117 a(n) = 4a(n-1) + 4a(n-2), n>2, a(0)=1, a(1)=3, a(2)=15.

A155119 a(n) = 5a(n-1) + 5a(n-2), n > 2, a(0)=1, a(1)=4, a(2)=24.

A155127 a(n) = 6a(n-1) + 6a(n-2), n>2, a(0)=1, a(1)=5, a(2)=35.

A155130 a(n) = 7a(n-1) + 7a(n-2), n>2, a(0)=1, a(1)=6, a(2)=48.

A155132 a(n) = 8a(n-1) + 8a(n-2), n > 2, a(0)=1, a(1)=7, a(2)=63.

A155144 a(n) = 9a(n-1) + 9a(n-2), n>2; a(0)=1, a(1)=8, a(2)=80.

A155157 a(n) = 10a(n-1) + 10a(n-2), with a(0)=1, a(1)=9, a(2)=99.

A180147 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 4x - 3x^2 + 6x^3).