A030192 -id:A030192 - cp's OEIS frontend

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 .

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

A161728 a(n) = ((3+sqrt(3))(4+sqrt(3))^n-(3-sqrt(3))(4-sqrt(3))^n)/sqrt(12).

Original entry on oeis.org

1, 7, 43, 253, 1465, 8431, 48403, 277621, 1591729, 9124759, 52305595, 299822893, 1718610409, 9851185663, 56467549987, 323674986277, 1855321740385, 10634799101479, 60959210186827, 349421293175389, 2002900612974361, 11480728092514831, 65808116771451955, 377215468968922837

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jun 17 2009

Fourth binomial transform of A162436, inverse binomial transform of A162272.

The inverse binomial transform yields A030192. The binomial transform yields A162272. - R. J. Mathar, Jul 07 2009

Index entries for linear recurrences with constant coefficients, signature (8,-13).

Cf. A162436, A162272, A030192, A161729.

F=nfinit(x^2-3); for(n=0, 20, print1(nfeltdiv(F, ((3+x)*(4+x)^n-(3-x)*(4-x)^n), (2*x))[1], ",")) \\ Klaus Brockhaus, Jun 19 2009

Vec((1-x)/(1-8*x+13*x^2)+O(x^25)) \\ M. F. Hasler, Dec 03 2014

a(n) = 8*a(n-1) - 13(n-2) for n > 1; a(0) = 1, a(1) = 7.
G.f.: (1 - x)/(1 - 8*x + 13*x^2). - Klaus Brockhaus, Jun 19 2009
E.g.f.: exp(4*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). - Stefano Spezia, Dec 31 2022

Extended beyond a(5) by Klaus Brockhaus, Jun 19 2009

Edited by Klaus Brockhaus, Jul 05 2009; M. F. Hasler, Dec 03 2014

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.

A110212 a(n+3) = 6a(n) - 5a(n+2), a(0) = -1, a(1) = 5, a(2) = -25.

Original entry on oeis.org

-1, 5, -25, 119, -565, 2675, -12661, 59915, -283525, 1341659, -6348805, 30042875, -142164421, 672729275, -3183389125, 15063959099, -71283419845, 337316764475, -1596200067781, 7553299819835, -35742598512325, 169135792154939, -800359161855685, 3787340218204475

Offset: 0

Creighton Dement, Jul 16 2005

Superseeker finds: a(n+1) - a(n) = ((-1)^n)*A030192(n+1) (Scaled Chebyshev U-polynomial evaluated at sqrt(6)/2)

Cf. A030192, A110210, A110211, A110213.

eriestolist(series(1/((x-1)*(6*x^2+6*x+1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 2basejsumseq[A*B] with A = + 'i + 'ii' + 'ij' + 'ik' and B = + .5'i - .5'j + .5'k + .5i' + .5j' - .5k' - .5'ij' - .5'ik' + .5'ji' + .5'ki' Sumtype is set to: sum[(Y[0], Y[1], Y[2]),mod(3)

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

A136526 Coefficients polynomials B(x, n) = ((1 + a + b)x - c)B(x, n-1) - abB(x, n-2) with a = 3, b = 2, and c = 0.

Original entry on oeis.org

1, 0, 1, -6, 0, 6, 0, -42, 0, 36, 36, 0, -288, 0, 216, 0, 468, 0, -1944, 0, 1296, -216, 0, 4536, 0, -12960, 0, 7776, 0, -4104, 0, 38880, 0, -85536, 0, 46656, 1296, 0, -51840, 0, 311040, 0, -559872, 0, 279936, 0, 32400, 0, -544320, 0, 2379456, 0, -3639168, 0, 1679616

Offset: 0

Roger L. Bagula, Mar 23 2008

			Triangle begins as:
     1;
     0,     1;
    -6,     0,      6;
     0,   -42,      0,    36;
    36,     0,   -288,     0,    216;
     0,   468,      0, -1944,      0,   1296;
  -216,     0,   4536,     0, -12960,      0,    7776;
     0, -4104,      0, 38880,      0, -85536,       0, 46656;
  1296,     0, -51840,     0, 311040,      0, -559872,     0, 279936;

Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 93

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000567, A002414, A002419, A016921, A027810, A030192, A034265, A051843, A054487, A055848, A081106, A136531.

f:= func< n,k | k eq 0 select (-1)^Floor(n/2) else (-1)^Floor((n-k)/2)*6^Floor((k-1)/2)*(1/k)*(6*Floor((n-k)/2) +k)*Binomial(Floor((n-k)/2) +k-1, k-1) >;
A136526:= func< n,k | ((n+k+1) mod 2)*6^Floor(n/2)*f(n,k) >;
[A136526(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 22 2022

(* First program *)
a= (b+1)/(b-1); c=0; b=2;
B[x_, n_]:= B[x, n]= If[n<2, x^n, ((1+a+b)*x -c)*B[x, n-1] -a*b*B[x, n-2]];
Table[CoefficientList[B[x,n], x], {n,0,10}]//Flatten
(* Second program *)
B[x_, n_]:= 6^(n/2)*(ChebyshevU[n, Sqrt[3/2]*x] -(5*x/Sqrt[6])*ChebyshevU[n-1, Sqrt[3/2]*x]);
Table[CoefficientList[B[x, n], x]/6^Floor[n/2], {n,0,16}]//Flatten (* G. C. Greubel, Sep 22 2022 *)

def f(n,k):
    if (k==0): return (-1)^(n//2)
    else: return (-1)^((n-k)//2)*6^((k-1)//2)*(1/k)*(6*((n-k)//2) + k)*binomial(((n-k)//2) +k-1, k-1)
def A136526(n,k): return ((n+k+1)%2)*6^(n//2)*f(n,k)
flatten([[A136526(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 22 2022

T(n, k) = coefficients of the polynomials defined by B(x, n) = ((1 + a + b)*x - c)*B(x, n - 1) - a*b*B(x, n - 2) with B(x, 0) = 1, B(x, 1) = x, a = 3, b = 2, and c = 0.
From G. C. Greubel, Sep 22 2022: (Start)
T(n, k) = coefficients of the polynomials defined by B(x, n) = 6^(n/2)*(ChebyshevU(n, sqrt(3/2)*x) - (5*x/sqrt(6))*ChebyshevU(n-1, sqrt(3/2)*x)).
T(n, k) = (1/2)*(1+(-1)^(n+k))*6^floor(n/2)*f(n, k), where f(n, k) = (-1)^floor((n -k)/2)*6^floor((k-1)/2)*(1/k)*(6*floor((n-k)/2) + k)*binomial(floor((n-k)/2) + k -1, k-1), for k >= 1, and (-1)^floor(n/2) for k = 0.
T(n, 0) = (1/2)*(1+(-1)^n)*(-6)^floor(n/2).
T(n, 1) = (1/2)*(1-(-1)^n)*(-6)^floor((n-1)/2)*A016921(floor((n-1)/2)), n >= 1.
T(n, 2) = (1/2)*(1+(-1)^n)*(-1)^(1+Floor((n+1)/2))*6^floor((n+1)/2)*A000567(floor( (n+1)/2)), n >= 2.
T(n, 3) = (1/2)*(1-(-1)^n)*(-6)^floor((n+1)/2)*A002414(floor((n-1)/2)), n >= 3.
T(n, 4) = (3/2)*(1+(-1)^n)*(-6)^floor((n+1)/2)*A002419(floor((n-1)/2)), n >= 4.
T(n, 5) = 18*(1-(-1)^n)*(-6)^floor((n-1)/2)*A051843(floor((n-3)/2)), n >= 5.
T(n, n) = 6^(n-1) + (5/6)*[n=0].
T(n, n-2) = -6*A081106(n-2), n >= 2.
Sum_{k=0..n} T(n, k) = -6*A030192(n-3), n>= 0.
Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] - 5*[n=2].
G.f.: (1 - 5*x*y)/(1 - 6*x*y + 6*y^2). (End)

Edited by G. C. Greubel, Sep 22 2022

A140766 a(n) = 6a(n-1) - 6a(n-2).

Original entry on oeis.org

1, 5, 24, 114, 540, 2556, 12096, 57240, 270864, 1281744, 6065280, 28701216, 135815616, 642686400, 3041224704, 14391229824, 68100030720, 322252805376, 1524916647936, 7215983055360, 34146398444544, 161582492335104, 764616563343360, 3618204426049536

Offset: 1

Gary W. Adamson and Roger L. Bagula, May 28 2008

Companion sequence = A030192 beginning (1, 6, 30, 144, 684,...).

			a(5) = 540 = 6*a(4) - 6*a(3) = 6*(114) - 6*24.
a(5) = 540 = term (1,1) of X^5.

Index entries for linear recurrences with constant coefficients, signature (6,-6).

Cf. A030192, A030523.

LinearRecurrence[{6,-6},{1,5},30] (* Harvey P. Dale, Oct 01 2014 *)

a(n) = 6*a(n-1) - 6*a(n-2); a(1) = 1, a(2) = 5.
Term (1,1) of M^n, where M = the 3x3 matrix [1,1,1; 1,2,1; 3,1,3].
From R. J. Mathar, May 31 2008: (Start)
O.g.f.: x*(1 - x)/(1 - 6*x + 6*x^2).
a(n) = A030192(n-1) - A030192(n-2). (End)
a(n) = Sum_{1<=k<=n} A030523(n,k)*2^(k-1). - Philippe Deléham, Feb 19 2013
a(n) = (sqrt(3)/36)*((3 + sqrt(3))^(n+1) - (3 - sqrt(3))^(n+1)). - Taras Goy, Jan 03 2025
E.g.f.: (exp(3*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) - 1)/6. - Stefano Spezia, Jan 04 2025

More terms from R. J. Mathar, May 31 2008

More terms from Harvey P. Dale, Oct 01 2014

A167925 Triangle, T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2), read by rows.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 0, 2, 6, 12, -1, 0, 9, 32, 75, -1, -4, 9, 80, 275, 684, 0, -8, 0, 192, 1000, 3240, 8232, 1, -8, -27, 448, 3625, 15336, 47677, 122368, 1, 0, -81, 1024, 13125, 72576, 276115, 835584, 2158569, 0, 16, -162, 2304, 47500, 343440, 1599066, 5705728, 16953624, 44010000

Offset: 0

Roger L. Bagula, Nov 15 2009

			Triangle begins as:
   0;
   1,  1;
   1,  2,   3;
   0,  2,   6,   12;
  -1,  0,   9,   32,    75;
  -1, -4,   9,   80,   275,   684;
   0, -8,   0,  192,  1000,  3240,   8232;
   1, -8, -27,  448,  3625, 15336,  47677, 122368;
   1,  0, -81, 1024, 13125, 72576, 276115, 835584, 2158569;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A009545, A030191, A030192, A030240, A057083, A057084, A057085, A057086, A099087, A128834, A190871, A190873.

A167925:= func< n,k | Round((Sqrt(k+1))^(n-1)*Evaluate(ChebyshevSecond(n), Sqrt(k+1)/2)) >;
[A167925(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 11 2023

(* First program *)
m[k_]= {{k,1}, {-1,1}};
v[0, k_]:= {0,1};
v[n_, k_]:= v[n, k]= m[k].v[n-1,k];
T[n_, k_]:= v[n, k][[1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
A167925[n_, k_]:= (Sqrt[k+1])^(n-1)*ChebyshevU[n-1, Sqrt[k+1]/2];
Table[A167925[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 11 2023 *)

def A167925(n,k): return (sqrt(k+1))^(n-1)*chebyshev_U(n-1, sqrt(k+1)/2)
flatten([[A167925(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 11 2023

T(n, k) = (-1)^(n+1) * [x^(n-1)]( 1/(1 + (k+1)*x + (k+1)*x^2) ). - Francesco Daddi, Aug 04 2011 (modified by G. C. Greubel, Sep 11 2023)
From G. C. Greubel, Sep 11 2023: (Start)
T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2).
T(n, 0) = A128834(n).
T(n, 1) = A009545(n) = A099087(n-1).
T(n, 2) = A057083(n-1).
T(n, 3) = A001787(n).
T(n, 4) = A030191(n-1).
T(n, 5) = A030192(n-1).
T(n, 6) = A030240(n-1).
T(n, 7) = A057084(n-1).
T(n, 8) = A057085(n).
T(n, 9) = A057086(n-1).
T(n, 10) = A190871(n).
T(n, 11) = A190873(n). (End)

Edited by G. C. Greubel, Sep 11 2023

A361290 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor((n-1)/2)} k^(n-1-j) * binomial(n,2*j+1).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 4, 4, 0, 0, 1, 6, 14, 8, 0, 0, 1, 8, 30, 48, 16, 0, 0, 1, 10, 52, 144, 164, 32, 0, 0, 1, 12, 80, 320, 684, 560, 64, 0, 0, 1, 14, 114, 600, 1936, 3240, 1912, 128, 0, 0, 1, 16, 154, 1008, 4400, 11648, 15336, 6528, 256, 0

Offset: 0

Seiichi Manyama, Mar 11 2023

			Square array begins:
  0,  0,   0,   0,    0,    0, ...
  1,  1,   1,   1,    1 ,   1, ...
  0,  2,   4,   6,    8,   10, ...
  0,  4,  14,  30,   52,   80, ...
  0,  8,  48, 144,  320,  600, ...
  0, 16, 164, 684, 1936, 4400, ...

Column k=1..10 give A131577, A007070(n-1), A030192(n-1), A016129(n-1), A093145, A154237, A154248, A154348(n-1), A016175(n-1), A361293.
Main diagonal gives A360766.
Cf. A361432.

T(n, k) = polcoef(lift(Mod('x, ('x-k)^2-k)^n), 1);

T(0,k) = 0, T(1,k) = 1; T(n,k) = 2 * k * T(n-1,k) - (k-1) * k * T(n-2,k).
T(n,k) = ((k + sqrt(k))^n - (k - sqrt(k))^n)/(2 * sqrt(k)) for k > 0.
G.f. of column k: x/(1 - 2 * k * x + (k-1) * k * x^2).
E.g.f. of column k: exp(k * x) * sinh(sqrt(k) * x) / sqrt(k) for k > 0.

A103280 Array read by antidiagonals, generated by the matrix M = [1,1,1;1,N,1;1,1,1].

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 9, 16, 1, 5, 14, 27, 44, 1, 6, 21, 48, 81, 120, 1, 7, 30, 85, 164, 243, 328, 1, 8, 41, 144, 341, 560, 729, 896, 1, 9, 54, 231, 684, 1365, 1912, 2187, 2448, 1, 10, 69, 352, 1289, 3240, 5461, 6528, 6561, 6688, 1, 11, 86, 513, 2276, 7175, 15336, 21845

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 27 2005

Consider the matrix M = [1,1,1;1,N,1;1,1,1];
Characteristic polynomial of M is x^3 + (-N - 2)*x^2 + (2*N - 2)*x.
Now (M^n)[1,2] is equivalent to the recursion a(1) = 1, a(2) = N+2, a(n) = (N+2)a(n-1)+(2N-2)a(n-2). (This also holds for negative N and fractional N.)
a(n+1)/a(n) converges to the upper root of the characteristic polynomial ((N + 2) + sqrt((N - 2)^2 + 8))/2 for n to infinity.
Columns of array follow the polynomials:
0
1
N + 2
N^2 + 2*N + 6
N^3 + 2*N^2 + 8*N + 16
N^4 + 2*N^3 + 10*N^2 + 24*N + 44
N^5 + 2*N^4 + 12*N^3 + 32*N^2 + 76*N + 120
N^6 + 2*N^5 + 14*N^4 + 40*N^3 + 112*N^2 + 232*N + 328
N^7 + 2*N^6 + 16*N^5 + 48*N^4 + 152*N^3 + 368*N^2 + 704*N + 896
N^8 + 2*N^7 + 18*N^6 + 56*N^5 + 196*N^4 + 528*N^3 + 1200*N^2 + 2112*N + 2448
etc.

			Array begins:
1,2,6,16,44,120,328,896,2448,6688,...
1,3,9,27,81,243,729,2187,6561,19683, ...
1,4,14,48,164,560,1912,6528,22288,76096,...
1,5,21,85,341,1365,5461,21845,87381,349525,...
1,6,30,144,684,3240,15336,72576,343440,1625184,...
1,7,41,231,1289,7175,39913,221991,1234633,6866503,...
...

Cf. A103279 (for (M^n)[1, 1]), A002605 (for N=0), A000244 (for N=1), A007070 (for N=2), A002450 (for N=3), A030192 (for N=4), A152268 (for N=5), A006131 (for N=-1), A000400 (bisection for N=-2), A015443 (for N=-3), A083102 (for N=-4).

T12(N, n) = if(n==1,1,if(n==2,N+2,(N+2)*T12(N,n-1)-(2*N-2)*T12(N,n-2)))
for(k=0,10,print1(k,": ");for(i=1,10,print1(T12(k,i),","));print())

T(N, 1)=1, T(N, 2)=N+2, T(N, n)=(N+2)*T(N, n-1)-(2*N-2)*T(N, n-2).

A162273 a(n) = ((2+sqrt(3))(3+sqrt(3))^n + (2-sqrt(3))(3-sqrt(3))^n)/2.

Original entry on oeis.org

2, 9, 42, 198, 936, 4428, 20952, 99144, 469152, 2220048, 10505376, 49711968, 235239552, 1113165504, 5267555712, 24926341248, 117952713216, 558158231808, 2641233111552, 12498449278464, 59143297001472, 279869086338048

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009

Binomial transform of A001075 without initial term 1, inverse binomial transform of A162274.

The INVERTi transform yields A007051 without A007051(0). - R. J. Mathar, Jul 07 2009

Index entries for linear recurrences with constant coefficients, signature (6,-6).

Cf. A001075, A162274.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((2+r)*(3+r)^n+(2-r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 05 2009

seq(simplify(((2+sqrt(3))*(3+sqrt(3))^n+(2-sqrt(3))*(3-sqrt(3))^n)*1/2), n = 0 .. 22); # Emeric Deutsch, Jul 11 2009

LinearRecurrence[{6,-6},{2,9},30] (* Harvey P. Dale, Dec 17 2019 *)

a(n) = 6*a(n-1) - 6*a(n-2) for n > 1; a(0) = 2, a(1) = 9.
G.f.: (2-3*x)/(1-6*x+6*x^2).
a(n) = 2*A030192-3*A030192(n-1). - R. J. Mathar, Feb 04 2021

Edited and extended beyond a(5) by R. J. Mathar and Klaus Brockhaus, Jul 05 2009

cp's OEIS Frontend

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 .

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

Extensions

A161728 a(n) = ((3+sqrt(3))*(4+sqrt(3))^n-(3-sqrt(3))*(4-sqrt(3))^n)/sqrt(12).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

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).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A110212 a(n+3) = 6*a(n) - 5*a(n+2), a(0) = -1, a(1) = 5, a(2) = -25.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Formula

A136526 Coefficients polynomials B(x, n) = ((1 + a + b)*x - c)*B(x, n-1) - a*b*B(x, n-2) with a = 3, b = 2, and c = 0.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A140766 a(n) = 6*a(n-1) - 6*a(n-2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A167925 Triangle, T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2), read by rows.

Views

Author

Keywords

A161728 a(n) = ((3+sqrt(3))(4+sqrt(3))^n-(3-sqrt(3))(4-sqrt(3))^n)/sqrt(12).

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).

A110212 a(n+3) = 6a(n) - 5a(n+2), a(0) = -1, a(1) = 5, a(2) = -25.

A136526 Coefficients polynomials B(x, n) = ((1 + a + b)x - c)B(x, n-1) - abB(x, n-2) with a = 3, b = 2, and c = 0.

A140766 a(n) = 6a(n-1) - 6a(n-2).

A162273 a(n) = ((2+sqrt(3))(3+sqrt(3))^n + (2-sqrt(3))(3-sqrt(3))^n)/2.