A002605 -id:A002605 - cp's OEIS frontend

A368152 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = up(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 3 - x^2.

1, 1, 3, 4, 6, 8, 7, 27, 25, 21, 19, 66, 126, 90, 55, 40, 204, 392, 504, 300, 144, 97, 522, 1363, 1884, 1851, 954, 377, 217, 1425, 4065, 7281, 8011, 6435, 2939, 987, 508, 3642, 12332, 24606, 34044, 31446, 21524, 8850, 2584, 1159, 9441, 35236, 82020, 127830

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    3
    4    6    8
    7   27   25   21
   19   66  126   90   55
   40  204  392  504  300  144
   97  522 1363 1884 1851  954  377
  217 1425 4065 7281 8011 6435 2939 987
Row 4 represents the polynomial p(4,x) = 7 + 27*x + 25*x^2 + 21*x^3, so (T(4,k)) = (7,27,25,21), 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. A006130 (column 1); A001906 (p(n,n-1)); A090017 (row sums), (p(n,1)); A002605 (alternating row sums), (p(n,-1)); A004187, (p(n,2)); A004254, (p(n,-2)); A190988, (p(n,3)); A190978 (unsigned), (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151.

p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 3 - 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 + 3*x, u = p(2,x), and v = 3 - x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(13 + 6*x + 5*x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(3*x + 1 + 1/k).

A099177 a(n)=2a(n-1)+4a(n-2)-4a(n-3)-4a(n-4).

Original entry on oeis.org

0, 1, 2, 8, 20, 60, 160, 448, 1216, 3344, 9120, 24960, 68160, 186304, 508928, 1390592, 3799040, 10379520, 28357120, 77473792, 211661824, 578272256, 1579868160, 4316282880, 11792302080, 32217174016, 88018952192, 240472260608

Offset: 0

Paul Barry, Oct 02 2004

Form the 6 node graph with matrix A=[1,1,1,1,0,0; 1,1,0,0,1,1; 1,0,0,0,0,0; 1,0,0,0,0,0; 0,1,0,0,0,0; 0,1,0,0,0,0]. Then A099177 counts walks of length n between the degree 5 vertices.

Index entries for linear recurrences with constant coefficients, signature (2,4,-4,-4).

Cf. A099176.

LinearRecurrence[{2,4,-4,-4},{0,1,2,8},30] (* Harvey P. Dale, Feb 12 2023 *)

G.f.: x/((1-2x^2)(1-2x-2x^2)); a(n)=(3+sqrt(3))(1+sqrt(3))^n/12+(3-sqrt(3))(1-sqrt(3))^n/12-2^((n-4)/2)(1+(-1)^n); a(n)=A002605(n)/2-2^((n-4)/2)(1+(-1)^n).

a(n)=sum{k=0..floor((n+1)/2), binomial(n-k+1, k-1)2^(n-k)} - Paul Barry, Oct 23 2004

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,8).

Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446
.A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102
.A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528
.A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226
.A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250
.A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005
.A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568
.A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953
.A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585
A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)

a(n)=([0,1; 8,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: x/(1 - 2*x*(3+4*x)). - Harvey P. Dale, Jul 26 2011

A293005 Number of associative, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n}.

Original entry on oeis.org

0, 1, 4, 12, 34, 94, 258, 706, 1930, 5274, 14410, 39370, 107562, 293866, 802858, 2193450, 5992618, 16372138, 44729514, 122203306, 333865642, 912137898, 2492007082, 6808289962, 18600594090, 50817768106, 138836724394, 379308985002, 1036291418794, 2831200807594

Offset: 0

J. Devillet, Sep 28 2017

Colin Barker, Table of n, a(n) for n = 0..1000
M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA] (2017).
Jimmy Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA], 2017.
Index entries for linear recurrences with constant coefficients, signature (3,0,-2).

Cf. A002605, A028860, A293006, A293007.

concat(0, Vec(x*(1 + x) / ((1 - x)*(1 - 2*x - 2*x^2)) + O(x^30))) \\ Colin Barker, Sep 28 2017

G.f.: x(x+1) / (2x^3-3*x+1).
a(0) = 0, a(1) = 1, a(n+2)-2*a(n+1)-2*a(n) = 2.
3*a(n)+2 = Sum_{k>=0} (2*binomial(n,2*k)+3*binomial(n,2*k+1))*3^k.
From Colin Barker, Sep 28 2017: (Start)
a(n) = (-4 - (1-sqrt(3))^n*(-2+sqrt(3)) + (1+sqrt(3))^n*(2+sqrt(3))) / 6.
a(n) = 3*a(n-1) - 2*a(n-2) for n>2. (End)
E.g.f.: exp(x)*(2*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x) - 2)/3. - Stefano Spezia, Apr 11 2025

A293006 Expansion of 2x^2(x+1) / (2x^3-3x+1).

Original entry on oeis.org

0, 0, 2, 8, 24, 68, 188, 516, 1412, 3860, 10548, 28820, 78740, 215124, 587732, 1605716, 4386900, 11985236, 32744276, 89459028, 244406612, 667731284, 1824275796, 4984014164, 13616579924, 37201188180, 101635536212, 277673448788, 758617970004, 2072582837588

Offset: 0

J. Devillet, Sep 28 2017

Number of associative, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n} that have annihilator elements.

Robert Israel, Table of n, a(n) for n = 0..2289
M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA] (2017).
Index entries for linear recurrences with constant coefficients, signature (3,0,-2).

Cf. A002605, A028860, A293005, A293007.

f:= gfun:-rectoproc({a(n) = 3*a(n-1) - 2*a(n-3),a(0)=0,a(1)=0,a(2)=2,a(3)=8},a(n),remember):
map(f, [$0..100]); # Robert Israel, Sep 28 2017

Join[{0}, LinearRecurrence[{3, 0, -2}, {0, 2, 8}, 30]] (* Jean-François Alcover, Sep 19 2018 *)

concat(vector(2), Vec(2*x^2*(1 + x) / ((1 - x)*(1 - 2*x - 2*x^2)) + O(x^30))) \\ Colin Barker, Sep 28 2017

a(n) = 2*A293005(n-1), a(0) = 0.
From Colin Barker, Sep 28 2017: (Start)
a(n) = (-8 + (1-sqrt(3))^(1+n) + (1+sqrt(3))^(1+n)) / 6 for n>0.
a(n) = 3*a(n-1) - 2*a(n-2) for n>3.
(End)

A293007 Expansion of 2x^2 / (1-2x-2*x^2).

Original entry on oeis.org

0, 0, 2, 4, 12, 32, 88, 240, 656, 1792, 4896, 13376, 36544, 99840, 272768, 745216, 2035968, 5562368, 15196672, 41518080, 113429504, 309895168, 846649344, 2313089024, 6319476736, 17265131520, 47169216512, 128868696064, 352075825152, 961889042432

Offset: 0

J. Devillet, Sep 28 2017

Number of associative, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n} that have neutral and annihilator elements.

Colin Barker, Table of n, a(n) for n = 0..1000
M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA] (2017).
Index entries for linear recurrences with constant coefficients, signature (2,2).

Cf. A002605, A293005, A293006.

Essentially the same as A028860 and A152035.

concat(vector(2), Vec(2*x^2 / (1-2*x-2*x^2) + O(x^50))) \\ Colin Barker, Sep 28 2017

a(n) = 2*A002605(n-1), a(0) = 0.
a(n) = A028860(n+1), a(0) = 0.
From Colin Barker, Sep 28 2017: (Start)
a(n) = ((1-sqrt(3))^n*(1+sqrt(3)) + (-1+sqrt(3))*(1+sqrt(3))^n) / (2*sqrt(3)) for n>0.
a(n) = 2*a(n-1) + 2*a(n-2) for n>2. (End)
E.g.f.: exp(x)*(cosh(sqrt(3)*x) - sinh(sqrt(3)*x)/sqrt(3)) - 1. - Stefano Spezia, Sep 05 2025

A342120 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, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 16, 5, 0, 1, 5, 20, 45, 44, 8, 0, 1, 6, 30, 96, 171, 120, 13, 0, 1, 7, 42, 175, 464, 648, 328, 21, 0, 1, 8, 56, 288, 1025, 2240, 2457, 896, 34, 0, 1, 9, 72, 441, 1980, 6000, 10816, 9315, 2448, 55, 0

Offset: 0

Seiichi Manyama, Feb 28 2021

			Square array begins:
  1, 1,   1,   1,    1,    1, ...
  0, 1,   2,   3,    4,    5, ...
  0, 2,   6,  12,   20,   30, ...
  0, 3,  16,  45,   96,  175, ...
  0, 5,  44, 171,  464, 1025, ...
  0, 8, 120, 648, 2240, 6000, ...

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

Columns 0..10 give A000007, A000045(n+1), A002605(n+1), A030195(n+1), A057087, A057088, A057089, A057090, A057091, A057092, A057093.
Rows 0..2 give A000012, A001477, A002378.
Main diagonal gives A109516(n+1).
Cf. A342129, A342133, A342134.

T:= (n, k)-> (<<0|1>, >^(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_] := Sum[If[k == j == 0, 1, k^j] * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)

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

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

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

A355320 Irregular triangle T(n, k), n >= 0, -2n <= k <= 2n, read by rows; T(0, 0) = 1; for n > 0, T(n, k) is the sum of all terms in previous rows at one knight's move away.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 0, 0, 2, 3, 2, 0, 2, 3, 2, 0, 0, 1, 1, 0, 0, 3, 4, 3, 1, 6, 8, 6, 1, 3, 4, 3, 0, 0, 1, 1, 0, 0, 4, 5, 4, 3, 12, 16, 12, 6, 12, 16, 12, 3, 4, 5, 4, 0, 0, 1, 1, 0, 0, 5, 6, 5, 6, 20, 27, 21, 18, 33, 44, 33, 18, 21, 27, 20, 6, 5, 6, 5, 0, 0, 1

Offset: 0

Rémy Sigrist, Jun 28 2022

See A096608 for the right half of the triangle.

Odd terms form fractal patterns (see illustrations in Links section).

			Triangle T(n, k) begins:
                                 1
                           1  0  0  0  1
                     1  0  0  1  2  1  0  0  1
               1  0  0  2  3  2  0  2  3  2  0  0  1
         1  0  0  3  4  3  1  6  8  6  1  3  4  3  0  0  1
   1  0  0  4  5  4  3 12 16 12  6 12 16 12  3  4  5  4  0  0  1

Paolo Xausa, Table of n, a(n) for n = 0..10010 (rows 0..70 of triangle, flattened)
Rémy Sigrist, Representation of the odd terms for n = 0..2^11 (only the right half of the triangle is represented)
Rémy Sigrist, Representation of the odd terms for n = 0..2^10

Cf. A002605 (row sums), A096608, A096609, A096610, A096611, A096612, A355339.

A355320[rowmax_]:=Module[{T},T[0,0]=1;T[n_,k_]:=T[n,k]=If[k<=2n,T[n-1,Abs[k-2]]+T[n-2,Abs[k-1]]+T[n-1,k+2]+T[n-2,k+1],0];Table[T[n,Abs[k]],{n,0,rowmax},{k,-2n,2n}]]; A355320[10] (* Generates 11 rows *) (* Paolo Xausa, May 09 2023 *)

row(n) = { my (rr=0, r=1); for (k=1, n, [rr,r]=[r,r*(1+'X^4)+rr*('X^3+'X^5)]); Vec(r) }

T(n, k) = A096608(n, abs(k)).
T(n, 0) = A096609(n).
T(n, 1) = A096610(n).
T(n, 2) = A096611(n).
T(n, n) = A096612(n).
T(n, 2*n) = 1.
T(n, 2*n-1) = T(n, 2*n-2) = 0 for any n > 0.
T(n, k) = T'(n-1, k-2) + T'(n-1, k+2) + T'(n-2, k-1) + T'(n-2, k+1) for n > 0 (where T' extends T with 0's outside its domain of definition).
T(n, -k) = T(n, k).
Sum_{k = -2*n..2*n} T(n, k) = A002605(n+1).

A074359 Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(2,2).

Original entry on oeis.org

0, 0, 0, 0, 12, 64, 280, 1088, 3968, 13856, 46912, 155136, 503616, 1610496, 5086336, 15895552, 49229312, 151275008, 461662208, 1400356864, 4224703488, 12683452416, 37911164928, 112865394688, 334788444160, 989756825600

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

nonn

Coefficient of q^0 is A002605.

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=6, nu(3)=16+4q, nu(4)=44+20q+12q^2, nu(5)=120+80q+64q^2+40q^3+8q^4, so the coefficients of q^2 are 0,0,0,0,12,64.

M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
Index entries for linear recurrences with constant coefficients, signature (6, -6, -16, 12, 24, 8).

Coefficient of q^0, q^1 and q^3 are in A002605, A074358 and A074360. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074357, A074361-A074363.

nu := proc(n,b,lambda) if n = 0 then 1 ; elif n = 1 then b ; else b*nu(n-1,b,lambda)+lambda*nu(n-2,b,lambda)*add(q^i,i=0..n-2) ; fi ; end: A074359 := proc(n) local b,lambda,thisnu ; b := 2 ; lambda := 2 ; thisnu := nu(n,b,lambda) ; RETURN( coeftayl(thisnu,q=0,2) ) ; end: for n from 0 to 40 do printf("%d, ",A074359(n) ) ; od ; # R. J. Mathar, Mar 20 2007

Join[{0, 0}, LinearRecurrence[{6, -6, -16, 12, 24, 8}, {0, 0, 12, 64, 280, 1088}, 24]] (* Jean-François Alcover, Sep 23 2017 *)

Conjecture: O.g.f: 4*x^4*(-3+2*x+8*x^2+4*x^3)/(2*x^2+2*x-1)^3. - R. J. Mathar, Jul 22 2009

More terms from R. J. Mathar, Mar 20 2007

A099173 Array, A(k,n), read by diagonals: g.f. of k-th row x/(1-2x-(k-1)x^2).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 4, 4, 0, 1, 2, 5, 8, 5, 0, 1, 2, 6, 12, 16, 6, 0, 1, 2, 7, 16, 29, 32, 7, 0, 1, 2, 8, 20, 44, 70, 64, 8, 0, 1, 2, 9, 24, 61, 120, 169, 128, 9, 0, 1, 2, 10, 28, 80, 182, 328, 408, 256, 10, 0, 1, 2, 11, 32, 101, 256, 547, 896, 985, 512, 11

Offset: 0

Ralf Stephan, Oct 13 2004

			Square array, A(n, k), begins as:
  0, 1, 2,  3,  4,   5,    6,    7,     8, ... A001477;
  0, 1, 2,  4,  8,  16,   32,   64,   128, ... A000079;
  0, 1, 2,  5, 12,  29,   70,  169,   408, ... A000129;
  0, 1, 2,  6, 16,  44,  120,  328,   896, ... A002605;
  0, 1, 2,  7, 20,  61,  182,  547,  1640, ... A015518;
  0, 1, 2,  8, 24,  80,  256,  832,  2688, ... A063727;
  0, 1, 2,  9, 28, 101,  342, 1189,  4088, ... A002532;
  0, 1, 2, 10, 32, 124,  440, 1624,  5888, ... A083099;
  0, 1, 2, 11, 36, 149,  550, 2143,  8136, ... A015519;
  0, 1, 2, 12, 40, 176,  672, 2752, 10880, ... A003683;
  0, 1, 2, 13, 44, 205,  806, 3457, 14168, ... A002534;
  0, 1, 2, 14, 48, 236,  952, 4264, 18048, ... A083102;
  0, 1, 2, 15, 52, 269, 1110, 5179, 22568, ... A015520;
  0, 1, 2, 16, 56, 304, 1280, 6208, 27776, ... A091914;
Antidiagonal triangle, T(n, k), begins as:
  0;
  0,  1;
  0,  1,  2;
  0,  1,  2,  3;
  0,  1,  2,  4,  4;
  0,  1,  2,  5,  8,  5;
  0,  1,  2,  6, 12, 16,   6;
  0,  1,  2,  7, 16, 29,  32,   7;
  0,  1,  2,  8, 20, 44,  70,  64,   8;
  0,  1,  2,  9, 24, 61, 120, 169, 128,   9;
  0,  1,  2, 10, 28, 80, 182, 328, 408, 256,  10;

G. C. Greubel, Antidiagonals n = 0..50, flattened
Ralf Stephan, Prove or disprove. 100 Conjectures from the OEIS, #16, arXiv:math/0409509 [math.CO], 2004.

Rows m: A001477 (m=0), A000079 (m=1), A000129 (m=2), A002605 (m=3), A015518 (m=4), A063727 (m=5), A002532 (m=6), A083099 (m=7), A015519 (m=8), A003683 (m=9), A002534 (m=10), A083102 (m=11), A015520 (m=12), A091914 (m=13).
Columns q: A000004 (q=0), A000012 (q=1), A009056 (q=2), A008586 (q=3).
Main diagonal gives A357502.

A099173:= func< n,k | (&+[n^j*Binomial(k,2*j+1): j in [0..Floor(k/2)]]) >;
[A099173(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2023

A[k_, n_]:= Which[k==0, n, n==0, 0, True, ((1+Sqrt[k])^n - (1-Sqrt[k])^n)/(2 Sqrt[k])]; Table[A[k-n, n]//Simplify, {k, 0, 12}, {n, 0, k}]//Flatten (* Jean-François Alcover, Jan 21 2019 *)

A(k,n)=sum(i=0,n\2,k^i*binomial(n,2*i+1))

def A099173(n,k): return sum( n^j*binomial(k, 2*j+1) for j in range((k//2)+1) )
flatten([[A099173(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 17 2023

A(n, k) = Sum_{i=0..floor(k/2)} n^i * C(k, 2*i+1) (array).
Recurrence: A(n, k) = 2*A(n, k-1) + (n-1)*A(n, k-2), with A(n, 0) = 0, A(n, 1) = 1.
T(n, k) = A(n-k, k) (antidiagonal triangle).
T(2*n, n) = A357502(n).
A(n, k) = ((1+sqrt(n))^k - (1-sqrt(n))^k)/(2*sqrt(n)). - Jean-François Alcover, Jan 21 2019

cp's OEIS Frontend

A368152 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 3 - x^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A099177 a(n)=2a(n-1)+4a(n-2)-4a(n-3)-4a(n-4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A189800 a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A293005 Number of associative, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n}.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A293006 Expansion of 2*x^2*(x+1) / (2*x^3-3*x+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A293007 Expansion of 2*x^2 / (1-2*x-2*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A342120 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).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A355320 Irregular triangle T(n, k), n >= 0, -2*n <= k <= 2*n, read by rows; T(0, 0) = 1; for n > 0, T(n, k) is the sum of all terms in previous rows at one knight's move away.

Views

A368152 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = up(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 3 - x^2.

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

A293006 Expansion of 2x^2(x+1) / (2x^3-3x+1).

A293007 Expansion of 2x^2 / (1-2x-2*x^2).

A342120 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).

A355320 Irregular triangle T(n, k), n >= 0, -2n <= k <= 2n, read by rows; T(0, 0) = 1; for n > 0, T(n, k) is the sum of all terms in previous rows at one knight's move away.

A074359 Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(2,2).

A099173 Array, A(k,n), read by diagonals: g.f. of k-th row x/(1-2x-(k-1)x^2).