A111125 -id:A111125 - cp's OEIS frontend

A132460 Irregular triangle read by rows of the initial floor(n/2) + 1 coefficients of 1/C(x)^n, where C(x) is the g.f. of the Catalan sequence (A000108).

1, 1, 1, -2, 1, -3, 1, -4, 2, 1, -5, 5, 1, -6, 9, -2, 1, -7, 14, -7, 1, -8, 20, -16, 2, 1, -9, 27, -30, 9, 1, -10, 35, -50, 25, -2, 1, -11, 44, -77, 55, -11, 1, -12, 54, -112, 105, -36, 2, 1, -13, 65, -156, 182, -91, 13, 1, -14, 77, -210, 294, -196, 49, -2

Offset: 0

Paul D. Hanna, Aug 21 2007

The length of row n is A008619(n).
Essentially equals a signed version of A034807, the triangle of Lucas polynomials. The initial n coefficients of 1/C(x)^n consist of row n followed by floor((n-1)/2) zeros for n > 0.
For the following formula for 1/C(x)^n see the W. Lang reference, proposition 1 on p. 411:
1/C(x)^n = (sqrt(x))^n*(S(n,1/sqrt(x)) - sqrt(x)*S(n-1,1/sqrt(x))*C(x)), n >= 0, with the Chebyshev polynomials S(n,x) with coefficients given in A049310. See also the coefficient array A115139 for P(n,x) = (sqrt(x)^(n-1))*S(n-1, 1/sqrt(x)). - Wolfdieter Lang, Sep 14 2013
This triangular array is composed of interleaved rows of reversed, A127677 (cf. A156308, A217476, A263916) and reversed, signed A111125. - Tom Copeland, Nov 07 2015
It seems that the n-th row lists the coefficients of the HOMFLYPT (HOMFLY) polynomial reduced to one variable for link family n, see Jablan's slide 38. - Andrey Zabolotskiy, Jan 16 2018
For n >= 1 row n gives the coefficients of the Girard-Waring formula for the sum of x1^n + x2^n in terms of the elementary symmetric functions e_1(x1,x2) = x1 + x2 and e_2(x1,x2) = x1*x2. This is an array using the partitions of n, in the reverse Abramowitz-Stegun order, with all partitions with parts larger than 2 eliminated. E.g., n = 4: x1^4 + x2^4 = 1*e1^4 - 4*e1^3*e2 + 2*e1*e2^2. See also A115131, row n = 4, with the mentioned partitions omitted. - Wolfdieter Lang, May 03 2019
Row n lists the coefficients of the n-th Faber polynomial for the replicable function given in A154272 with offset -1. - Ben Toomey, May 12 2020

			The irregular triangle T(n,k) begins:
n\k 0    1    2    3    4    5    6   7 ...
-------------------------------------------------
0:  1
1:  1
2:  1   -2
3:  1   -3
4:  1   -4    2
5:  1   -5    5
6:  1   -6    9   -2
7:  1   -7   14   -7
8:  1   -8   20  -16    2
9:  1   -9   27  -30    9
10: 1  -10   35  -50   25   -2
11: 1  -11   44  -77   55  -11
12: 1  -12   54 -112  105  -36    2
13: 1  -13   65 -156  182  -91   13
14: 1  -14   77 -210  294 -196   49  -2
... (reformatted - _Wolfdieter Lang_, May 03 2019)

Michael De Vlieger, Table of n, a(n) for n = 0..10200 (rows 0 <= n <= 200, flattened)
Tom Copeland, Addendum to Elliptic Lie Triad
G. Dattoli, E. Di Palma and E. Sabia, Cardan Polynomials, Chebyshev Exponents, Ultra-Radicals and Generalized Imaginary Units, Advances in Applied Clifford Algebras, 2014.
Pentti Haukkanen, Jorma Merikoski and Seppo Mustonen, Some polynomials associated with regular polygons, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.
S. Jablan, Knots, computers, conjectures
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq. (23) with n -> -n and eq. (20).
Jorma K. Merikoski, Regular polygons, Morgan-Voyce polynomials, and Chebyshev polynomials, Notes on Num. Theor. and Disc. Math. (2021) Vol. 27, No. 2, 79-87.

Cf. A000108, A008619, A034807 (Lucas polynomials), A111125, A115131 (Waring numbers), A127677, A132461 (row squared sums), A156308, A217476, A263916.

T[0, 0] = 1; T[n_, k_] := (-1)^k (Binomial[n-k, k] + Binomial[n-k-1, k-1]);
Table[T[n, k], {n, 0, 14}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Jun 04 2018 *)

{T(n,k)=if(k>n\2,0,(-1)^k*(binomial(n-k, k)+binomial(n-k-1, k-1)))}

T(n,k) = (-1)^k*( C(n-k,k) + C(n-k-1,k-1) ) for n >= 0, 0 <= k <= floor(n/2).

T(0,0) = 1; T(n,k) = (-1)^k*n*binomial(n-k,k)/(n-k), k = 0..floor(n/2). - Wolfdieter Lang, May 03 2019

A156308 Inverse of triangle S(n,m) defined by sequence A156290, n >= 1, 1 <= m <= n.

Original entry on oeis.org

1, 4, 1, 9, 6, 1, 16, 20, 8, 1, 25, 50, 35, 10, 1, 36, 105, 112, 54, 12, 1, 49, 196, 294, 210, 77, 14, 1, 64, 336, 672, 660, 352, 104, 16, 1, 81, 540, 1386, 1782, 1287, 546, 135, 18, 1, 100, 825, 2640, 4290, 4004, 2275, 800, 170, 20, 1

Offset: 1

Hartmut F. W. Hoft, Feb 07 2009

From Wolfdieter Lang, Jun 26 2011: (Start)
This triangle S(n,m) appears as U_m(n) in the Knuth reference on p. 285. It is related to the Riordan triangle T_m(n) = A111125(n,m) by S(n,m) = A111125(n,m) - A111125(n-1,m), n >= m >= 1 (identity on p. 286).
Also, S(n,m)-S(n-1,m) = A111125(n-1,m-1), n >= 2, m >= 1 (identity on p. 286). (End)
These polynomials may be expressed in terms of the Faber polynomials of A263916 and are embedded in A127677 and A208513. - Tom Copeland, Nov 06 2015

			Triangle starts:
  n=1:  1;
  n=2:  4,  1;
  n=3:  9,  6,  1;
  n=4: 16, 20,  8,  1;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.
Johann Cigler, A bivariate version of the spread polynomials, arXiv:2508.04751 [math.CO], 2025.
D. E. Knuth, Johann Faulhaber and sums of powers, arXiv:math/9207222 [math.CA], 1992; Math. Comp. 61 (1993), no. 203, 277-294.

Same as triangle A208513 with the first column truncated.
Columns: A000290 (m=1), A002415 (m=2), A040977 (m=3), A053347 (m=4), A054334 (m=5).
Cf. A078812, A111125, A127677, A208513, A263916.

[(n/k)*Binomial(n+k-1, 2*k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 01 2022

S[m_] := Flatten[Table[k/j Binomial[k + j - 1, 2 j - 1], {k, 1, m}, {j, 1, k}]]

flatten([[(n/k)*binomial(n+k-1, 2*k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 01 2022

S(n, m) = (n/m) * binomial(n + m - 1, 2*m - 1).
From Peter Bala, May 01 2014: (Start)
The n-th row o.g.f. is polynomial R(n,x) = 2/x*( T(n,(x + 2)/2) - 1 ), where T(n,x) is Chebyshev polynomial of the first kind. They form a divisibility sequence: if n divides m then R(n,x) divides R(m,x) in the ring Z[x].
R(2*n,x) = (x + 4)*U(n-1,(x + 2)/2)^2;
R(2*n + 1,x) = ( U(n,(x + 2)/2) + U(n-1,(x + 2)/2) )^2.
O.g.f.: Sum_{n >= 0} R(n,x)*z^n = z*(1 + z)/( (1 - z)*(1 - (x + 2)*z + z^2) ). (End)
The polynomial R(n,x) defined above satisfies (x + 1/x - 2) * R(n, x + 1/x - 2) = x^n + 1/x^n - 2. - Alexander Burstein, May 23 2021

Edited by Max Alekseyev, Mar 05 2018

A049667 a(n) = Fibonacci(7*n)/13.

Original entry on oeis.org

0, 1, 29, 842, 24447, 709805, 20608792, 598364773, 17373187209, 504420793834, 14645576208395, 425226130837289, 12346203370489776, 358465123875040793, 10407834795746672773, 302185674200528551210, 8773792386611074657863

Offset: 0

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 0..650
Tanya Khovanova, Recursive Sequences
Shaoxiong Yuan, Generalized Identities of Certain Continued Fractions, arXiv:1907.12459 [math.NT], 2019.
Index entries for linear recurrences with constant coefficients, signature (29, 1).

A column of array A028412.

Cf. A000045, A134498.

[Fibonacci(7*n)/13: n in [0..30]]; // G. C. Greubel, Dec 02 2017

a:= n-> (<<0|1>, <1|29>>^n)[1, 2]:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 20 2017

Fibonacci[(7*Range[0,20])]/13 (* or *) LinearRecurrence[{29,1},{0,1},20] (* Harvey P. Dale, Sep 17 2017 *)

numlib::fibonacci(7*n)/13 $ n = 0..25; // Zerinvary Lajos, May 09 2008

a(n)=fibonacci(7*n)/13 \\ Charles R Greathouse IV, Oct 07 2016

[fibonacci(7*n)/13 for n in range(0, 17)] # Zerinvary Lajos, May 15 2009

G.f.: x/(1 - 29*x - x^2).
a(n) = A134498(n)/13.
a(n) = F(n, 29), the n-th Fibonacci polynomial evaluated at x=29. - T. D. Noe, Jan 19 2006
a(n) = 29*a(n-1) + a(n-2), n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Nov 22 2008
For n >= 1, a(n) equals the denominator of the continued fraction [29, 29, ..., 29] (with n copies of 29). The numerator of that continued fraction is a(n+1). - Greg Dresden and Shaoxiong Yuan, Jul 26 2019
a(n) = ((-1)^n*7*F(n) + 14*5*F(n)^3 + (-1)^n*7*5^2*F(n)^5 + 5^3*F(n)^7)/13, n >= 0. See the general D. Jennings formula given in comment on triangle A111125, where also the reference is given. Here the fourth row (k=3) applies. - Wolfdieter Lang, Sep 01 2012
G.f.: G(0)*x/(2-29*x), where G(k)= 1 + 1/(1 - (x*(845*k-841))/((x*(845*k+4)) - 58/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 15 2013
O.g.f.: x*exp(Sum_{n >= 1} Lucas(7*n)*x^n/n) = x + 29*x^2 + 842*x^3 + .... - Peter Bala, Oct 11 2019

A100218 Riordan array ((1-2*x)/(1-x), (1-x)).

Original entry on oeis.org

1, -1, 1, -1, -2, 1, -1, 0, -3, 1, -1, 0, 2, -4, 1, -1, 0, 0, 5, -5, 1, -1, 0, 0, -2, 9, -6, 1, -1, 0, 0, 0, -7, 14, -7, 1, -1, 0, 0, 0, 2, -16, 20, -8, 1, -1, 0, 0, 0, 0, 9, -30, 27, -9, 1, -1, 0, 0, 0, 0, -2, 25, -50, 35, -10, 1, -1, 0, 0, 0, 0, 0, -11, 55, -77, 44, -11, 1, -1, 0, 0, 0, 0, 0, 2, -36, 105, -112, 54, -12, 1

Offset: 0

Paul Barry, Nov 08 2004

			Triangle begins as:
   1;
  -1,  1;
  -1, -2,  1;
  -1,  0, -3,  1;
  -1,  0,  2, -4,  1;
  -1,  0,  0,  5, -5,   1;
  -1,  0,  0, -2,  9,  -6,   1;
  -1,  0,  0,  0, -7,  14,  -7,  1;
  -1,  0,  0,  0,  2, -16,  20, -8,  1;
  -1,  0,  0,  0,  0,   9, -30, 27, -9,  1;

G. C. Greubel, Rows n = 0..49 of the triangle, flattened, flattened
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

Cf. A000007, A000027, A077961, A080956, A098601.
Cf. A154955, A157142, A280560, A355021.
Row sums are A100219.
Matrix inverse of A100100.
Apart from signs, same as A098599.
Very similar to triangle A111125.

A100218:= func< n,k | n eq 0 select 1 else (-1)^(n+k)*(Binomial(k,n-k) + Binomial(k-1,n-k-1)) >;
[A100218(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Mar 28 2024

T[0,0]:= 1; T[1,1]:= 1; T[1,0]:= -1; T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, T[n- 1,k] +T[n-1,k-1] -2*T[n-2,k-1] +T[n-3,k-1]]; Table[T[n,k], {n,0,14}, {k,0,n} ]//Flatten (* G. C. Greubel, Mar 13 2017 *)

def A100218(n,k): return 1 if n==0 else (-1)^(n+k)*(binomial(k,n-k) + binomial(k-1,n-k-1))
flatten([[A100218(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Mar 28 2024

Sum_{k=0..n} T(n, k) = A100219(n) (row sums).
Number triangle T(n, k) = (-1)^(n-k)*(binomial(k, n-k) + binomial(k-1, n-k-1)), with T(0, 0) = 1. - Paul Barry, Nov 09 2004
T(n,k) = T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k-1) + T(n-3,k-1), T(0,0)=1, T(1,0)=-1, T(1,1)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 09 2014
From G. C. Greubel, Mar 28 2024: (Start)
T(n, n-1) = A000027(n), n >= 1.
T(n, n-2) = -A080956(n-1), n >= 2.
T(2*n, n) = A280560(n).
T(2*n-1, n) = A157142(n-1), n >= 1.
T(2*n+1, n) = -A000007(n) = A154955(n+2).
T(3*n, n) = T(4*n, n) = A000007(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A355021(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A098601(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = -1 + 2*A077961(n) + A077961(n-2). (End)
From Peter Bala, Apr 28 2024: (Start)
This Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = x*(1 - x) and hence belongs to the hitting time subgroup of the Riordan group (see Peart and Woan for properties of this subgroup).
T(n,k) = [x^(n-k)] (1/c(x))^n, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. In general the (n, k)-th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

A098599 Riordan array ((1+2*x)/(1+x), (1+x)).

Original entry on oeis.org

1, 1, 1, -1, 2, 1, 1, 0, 3, 1, -1, 0, 2, 4, 1, 1, 0, 0, 5, 5, 1, -1, 0, 0, 2, 9, 6, 1, 1, 0, 0, 0, 7, 14, 7, 1, -1, 0, 0, 0, 2, 16, 20, 8, 1, 1, 0, 0, 0, 0, 9, 30, 27, 9, 1, -1, 0, 0, 0, 0, 2, 25, 50, 35, 10, 1, 1, 0, 0, 0, 0, 0, 11, 55, 77, 44, 11, 1, -1, 0, 0, 0, 0, 0, 2, 36, 105, 112, 54, 12, 1, 1, 0, 0, 0, 0, 0, 0, 13, 91, 182, 156, 65, 13, 1

Offset: 0

Paul Barry, Sep 17 2004

			Triangle begins as:
   1;
   1, 1;
  -1, 2, 1;
   1, 0, 3, 1;
  -1, 0, 2, 4, 1;
   1, 0, 0, 5, 5,  1;
  -1, 0, 0, 2, 9,  6,  1;
   1, 0, 0, 0, 7, 14,  7,  1;
  -1, 0, 0, 0, 2, 16, 20,  8, 1;
   1, 0, 0, 0, 0,  9, 30, 27, 9, 1;

G. C. Greubel, Table of n, a(n) for n = 0..1325

Cf. A000007, A005408, A029635, A040000, A079757, A098601.
Row sums are A098600.
Diagonal sums are A098601.
Apart from signs, same as A100218.
Very similar to triangle A111125.

A098599:= func< n,k | n eq 0 select 1 else Binomial(k, n-k) + Binomial(k-1, n-k-1) >;
[A098599(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 27 2024

T[n_, k_]:= If[n==0, 1, Binomial[k,n-k] +Binomial[k-1,n-k-1]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 27 2024 *)

def A098599(n,k): return 1 if n==0 else binomial(k, n-k) + binomial(k-1, n-k-1)
flatten([[A098599(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 27 2024

Triangle: T(n, k) = binomial(k, n-k) + binomial(k-1, n-k-1), with T(0, 0) = 1.
Sum_{k=0..n} T(n, k) = A098600(n) (row sums).
T(n,k) = T(n-1,k-1) - T(n-1,k) + 2*T(n-2,k-1) + T(n-3,k-1), T(0,0)=1, T(1,0)=1, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 09 2014
From G. C. Greubel, Mar 27 2024: (Start)
T(2*n, n) = A040000(n).
T(2*n+1, n) = A000007(n).
T(2*n-1, n) = A005408(n-1), n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = A079757(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A098601(n). (End)

A228785 Table of coefficients of the algebraic number s(2l+1) = 2sin(Pi/(2l+1)) as a polynomial in odd powers of rho(2(2l+1)) = 2cos(Pi/(2(2l+1))) (reduced version).

Original entry on oeis.org

1, -3, 1, 5, -5, 1, -4, 5, -1, 9, -30, 27, -9, 1, -11, 55, -77, 44, -11, 1, 4, -13, 7, -1, -15, 140, -378, 450, -275, 90, -15, 1, 17, -204, 714, -1122, 935, -442, 119, -17, 1, -4, 25, -26, 9, -1, 0, 21, -385, 2079, -5148, 7007, -5733, 2940, -952, 189, -21, 1, -8, 126, -539, 967, -870, 429, -118, 17, -1, 0

Offset: 1

Wolfdieter Lang, Oct 07 2013

In the regular (2*l+1)-gon, l >= 1, inscribed in a circle of radius R the length ratio side/R is s(2*l+1) = 2*sin(Pi/(2*l+1)). This can be written as a polynomial in the length ratio (smallest diagonal)/side in the (2*(2*l+1))-gon given by rho(2*(2*l+1)) = 2*cos(Pi/(2*(2*l+1))). This leads, in a first step, to the signed triangle A111125. Because of the minimal polynomial of the algebraic number rho(2*(2*l+1)) of degree delta(2*(2*l+1)) = A055034(2*(2*l+1)), called C(2*(2*l+1),x) (with coefficients given in A187360), one can eliminate all powers rho(2*(2*l+1))^k with k >= delta(2*(2*l+1)) by using C(2*(2*l+1),rho(2*(2*l+1))) = 0. This leads to the present table expressing s(2*(l+1)) in terms of odd powers of rho(2*(2*l+1)) with maximal exponent delta(2*(2*l+1))-1.
This table gives the coefficients of s(2*l+1), related to the (2*l+1)-gon, in the power basis of the algebraic number field Q(rho(2*(2*l+1))) of degree delta(2*(2*l+1)), related to rho from the (2*(2*l+1))-gon, provided one inserts zeros for the even powers, starting each row with a zero and filling zeros at the end in order to obtain the row length delta(2*(2*l+1)). Note that some trailing zeros in the present table (e.g., row l = 10) have been given such that the row length for the s(2*l+1) coefficients in the power basis Q(rho(2*(2*l+1))) becomes just twice the one of this table.
Thanks go to Seppo Mustonen for telling me about his findings regarding the square of the sum of all length in the regular n-gon, which led me to consider this entry (even though for odd n this is not needed because only s(2*l+1)^2  = 4 - rho(2*l+1)^2 enters).

			The table a(l,m), with n = 2*l+1, begins:
n,   l \m  0    1     2     3    4     5    6    7   8   9 10
3,   1:    1
5,   2:   -3    1
7,   3:    5   -5     1
9,   4:   -4    5    -1
11,  5:    9  -30    27    -9    1
13,  6:  -11   55   -77    44  -11     1
15,  7:    4  -13     7    -1
17,  8:  -15  140  -378   450 -275    90  -15    1
19,  9:   17 -204   714 -1122  935  -442  119  -17   1
21, 10:   -4   25   -26     9   -1     0
23, 11:   21 -385  2079 -5148 7007 -5733 2940 -952 189 -21  1
25, 12:   -8  126  -539   967 -870   429 -118   17  -1   0
27, 13:    4  -41    70   -43   11    -1    0    0   0
...
n = 29 l =  14:  -27, 819, -7371, 30888, -72930, 107406, -104652, 69768, -32319, 10395, -2277, 324, -27, 1.
n = 5, l=2: s(5) = -3*rho(10) + rho(10)^3 = (tau - 1)*sqrt(2 + tau), approximately 1.175570504, where tau = (1 + sqrt(5))/2 (golden section).
n = 17, l = 8: s(17) = -15*x + 140*x^3 - 378*x^5 + 450*x^7 - 275*x^9 + 90*x^11 - 15*x^13 + 1*x^15, with x = rho(34) = 2*cos(Pi/34). s(17) is approximately 0.3674990356. With the length row l = 8 the degree of the algebraic number s(17) = 2*sin(Pi/17) is therefore 2*8 = 16. See A228787 for the decimal expansion of s(17) and A228788 for the one of rho(34).

Cf. A055034, A187360, A228783 (even n case), A228786 (minimal polynomials).

Cf. A140882, A267633.

a(l,m) = [x^(2*m+1)](s(2*l+1,x)(mod C(2*(2l+1),x))), with s(2*l+1,x) = sum((-1)^(l-1-s)* A111125(l1,s)*x^(2*s+1), s=0..l-1), l >= 1, m=0, ..., (delta(2*(2*l+1))/2 - 1), with delta(n) = A055034(n).

Rows 9,15,21,27 are coefficients of polynomials in reciprocal powers of u for rows n=2,4,6,8 generated by the o.g.f. (u-4)/(u-ux+x^2) of A267633. These polynomials in u occur in a moving average of the polynomials of A140882 interlaced with these polynomials. - Tom Copeland, Jan 16 2016

A049669 a(n) = Fibonacci(9*n)/34.

Original entry on oeis.org

0, 1, 76, 5777, 439128, 33379505, 2537281508, 192866774113, 14660412114096, 1114384187445409, 84707858657965180, 6438911642192799089, 489441992665310695944, 37204030354205805690833

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..530
S. Falcon, Generalized Fibonacci Sequences Generated from a k-Fibonacci Sequence, Journal of Mathematics Research Vol. 4, No. 2; April 2012. - From _N. J. A. Sloane_, Sep 22 2012
Tanya Khovanova, Recursive Sequences
Shaoxiong Yuan, Generalized Identities of Certain Continued Fractions, arXiv:1907.12459 [math.NT], 2019.
Index entries for linear recurrences with constant coefficients, signature (76, 1).

A column of array A028412.

Cf. A000045.

[Fibonacci(9*n)/(34): n in [0..30]]; // G. C. Greubel, Dec 02 2017

with (combinat):seq(fibonacci(3*n,4)/17, n=0..13); # Zerinvary Lajos, Apr 20 2008

Fibonacci[9Range[0,20]]/34 (* or *) LinearRecurrence[{76,1},{0,1},20] (* Harvey P. Dale, Jan 20 2013 *)

numlib::fibonacci(9*n)/34 $ n = 0..25; // Zerinvary Lajos, May 09 2008

for(n=0,30, print1(fibonacci(9*n)/34, ", ")) \\ G. C. Greubel, Dec 02 2017

G.f.: x/(1-76*x-x^2), 76=L(9)=A000032(9) (Lucas).
a(n) = 76*a(n-1) + a(n-2), n>1, a(0)=0, a(1)=1. - Philippe Deléham, Nov 23 2008
a(n) = (9*F(n) + (-1)^n*30*5*F(n)^3 + 27*5^2*F(n)^5 + (-1)^n*9*5^3*F(n)^7 + 5^4*F(n)^9)/34, n >= 0. See the general D. Jennings formula given in a comment on the triangle A111125, where also the reference is given. Here the fifth row (k=4) applies. - Wolfdieter Lang, Sep 01 2012
For n >= 1, a(n) equals the denominator of the continued fraction [76, 76, ..., 76] (with n copies of 76). The numerator of that continued fraction is a(n+1). - Greg Dresden and Shaoxiong Yuan, Jul 26 2019
E.g.f.: exp(38*x)*sinh(17*sqrt(5)*x)/(17*sqrt(5)). - Stefano Spezia, Aug 05 2019

More terms from James Sellers, Jan 20 2000

A052227 a(n) = (4n+1)binomial(3n,n)/(2n+1).

Original entry on oeis.org

1, 5, 27, 156, 935, 5733, 35700, 224808, 1427679, 9126975, 58659315, 378658800, 2453288292, 15944020316, 103897691640, 678610095504, 4441369072335, 29120107628115, 191233066114545, 1257635016353100

Offset: 0

Barry E. Williams, Jan 29 2000

T(2n,n) for A111125. - Paul Barry, Apr 19 2007

a(n) = A182584(2*n+1). - Reinhard Zumkeller, May 06 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A045721, A016813, A005809, A005408, A227726.

a052227 n = (a016813 n) * (a005809 n) `div` (a005408 n)
-- Reinhard Zumkeller, May 06 2012

[(4*n+1)*Binomial(3*n,n)/(2*n+1) : n in [0..30]]; // Vincenzo Librandi, Nov 13 2011

Table[(4n + 1)Binomial[3n, n]/(2n + 1), {n, 0, 30}] (* Harvey P. Dale, Jan 31 2011 *)

makelist(binomial(3*n,n)*(4*n+1)/(2*n+1),n,0,100); /* Emanuele Munarini, Jun 06 2011 */

{a(n)=binomial(3*n+1, n)+binomial(3*n, n-1)}  /* Paul D. Hanna, Jul 22 2013 */

G.f.: 4*x*F(4/3,5/3;5/2;27*x/4) + 2*sin(1/3*arcsin((3*sqrt(3*x))/2))/sqrt(3*x), where F(a;b;z) is a hypergeometric series. - Emanuele Munarini, Jun 06 2011
G.f.: (g+1)/((3*g-1)*(g-1)) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011
Conjecture: 8*n*(2*n+1)*a(n) +6*(-8*n^2-25*n+13)*a(n-1) -45*(3*n-4)*(3*n-5)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
a(n) = binomial(3*n+1, n) + binomial(3*n, n-1) for n>=0. - Paul D. Hanna, Jul 22 2013
G.f.: G(x)*(2*G(x) - 1) / (3 - 2*G(x)), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764. - Paul D. Hanna, Jul 22 2013
a(n) is the coefficient of [x^n] in (1+x)/(1-x)^(2n+2) and forms the main diagonal in the following table of coefficients:
(1+x)/(1-x)^2: [1, 3, 5, 7, 9, 11, 13, 15, 17, ...];
(1+x)/(1-x)^4: [1, 5, 14, 30, 55, 91, 140, 204, 285, ...];
(1+x)/(1-x)^6: [1, 7, 27, 77, 182, 378, 714, 1254, ...];
(1+x)/(1-x)^8: [1, 9, 44, 156, 450, 1122, 2508, 5148, ...];
(1+x)/(1-x)^10:[1, 11, 65, 275, 935, 2717, 7007, 16445, ...];
(1+x)/(1-x)^12:[1, 13, 90, 442, 1729, 5733, 16744, 44200, ...];
(1+x)/(1-x)^14:[1, 15, 119, 665, 2940, 10948, 35700, 104652, ...];
(1+x)/(1-x)^16:[1, 17, 152, 952, 4692, 19380, 69768, 224808, ...]; ... - Paul D. Hanna, Jul 22 2013

More terms from Harvey P. Dale, Jan 31 2011

A211957 Triangle of coefficients of a polynomial sequence related to the Morgan-Voyce polynomials A085478.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 12, 4, 1, 16, 40, 32, 8, 1, 25, 100, 140, 80, 16, 1, 36, 210, 448, 432, 192, 32, 1, 49, 392, 1176, 1680, 1232, 448, 64, 1, 64, 672, 2688, 5280, 5632, 3328, 1024, 128, 1, 81, 1080, 5544, 14256, 20592, 17472, 8640, 2304, 256, 1, 100, 1650, 10560, 34320, 64064, 72800, 51200, 21760, 5120, 512

Offset: 0

Peter Bala, Apr 30 2012

Triangle formed from the even numbered rows of A211956.
The coefficients of the  Morgan-Voyce polynomials b(n,x) := sum {k = 0..n} binomial(n+k,2*k)*x^k are listed in A085478. The rational functions 1/2*(b(2*n,2*x) + 1)/b(n,2*x) turn out to be integer polynomials. Their coefficients are listed in this triangle. These polynomials occur as factors of the row polynomials R(n,x) of A211955.
This triangle appears to be the row reverse of the unsigned triangle |A204021|.

			Triangle begins
.n\k.|..0....1....2....3....4....5....6....7
= = = = = = = = = = = = = = = = = = = = = = =
..0..|..1
..1..|..1....1
..2..|..1....4....2
..3..|..1....9...12....4
..4..|..1...16...40...32....8
..5..|..1...25..100..140...80...16
..6..|..1...36..210..448..432..192...32
..7..|..1...49..392.1176.1680.1232..448...64

R. J. Mathar, Recurrence for the Atkinson-Steenwijk Integrals for Resistors in the Infinite Triangular Lattice, vixra:2208.0111
Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials

A085478, A111125, A204021, A211955, A211956.

T(n,0) = 1 and for k > 0, T(n,k) = n/k*2^(k-1)*binomial(n+k-1,2*k-1) = 2^(k-1)*A208513(n,k).
O.g.f.: ((1-t)-t*x)/((1-t)^2-2*t*x) = 1 + (1+x)*t + (1+4*x+2*x^2)*t^2 + ....
n-th row polynomial R(n,x) = 1/2*(b(2*n,2*x) + 1)/b(n,2*x) = T(2*n,u), where u = sqrt((x+2)/2) and T(n,u) denotes the Chebyshev polynomial of the first kind.
T(n,k) = 2*T(n-1,k)+2*T(n-1,k-1)-T(n-2,k), T(0,0)=T(1,0)=T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 16 2013

A228783 Table of coefficients of the algebraic number s(2l) = 2sin(Pi/2l) as a polynomial in powers of rho(2l) = 2cos(Pi/(2l)) if l is even and of rho(l) = 2*cos(Pi/l) if l is odd (reduced version).

Original entry on oeis.org

2, 0, 1, 1, 0, -3, 0, 1, -1, 1, 0, 4, 0, -1, -1, -1, 1, 0, -7, 0, 14, 0, -7, 0, 1, 2, 1, -1, 0, 8, 0, -18, 0, 8, 0, -1, 1, 2, -3, -1, 1, 0, -8, 0, 6, 0, -1, 0, 0, -1, 3, 3, -4, -1, 1, 0, 12, 0, -67, 0, 96, 0, -52, 0, 12, 0, -1, -2, 3, 1, -1, 0, -15, 0, 140, 0, -378, 0, 450, 0, -275, 0, 90, 0, -15, 0, 1

Offset: 1

Wolfdieter Lang, Oct 06 2013

In the regular (2*l)-gon inscribed in a circle of radius R the length ratio side/R is s(2*l) = 2*sin(Pi/(2*l)). This can be written as a polynomial in the length ratio (smallest diagonal)/side given by rho(2*l) = 2*cos(Pi/(2*l)). (For the 2-gon there is no such diagonal and rho(2) = 0). This leads, in a first step, to the triangle A127672 (see the Oct 05 2013 comment there referring also to the bisections signed A111125 and A127677). Because the minimal polynomial of the algebraic number rho(2*l) of degree delta(2*l) = A055034(2*l), called C(2*l,x) (with coefficients given in A187360) one can eliminate all powers rho(2*l)^k with k >= delta(2*l) by using C(2*l,rho(2*l)) = 0. Furthermore, because for odd l only even powers of rho(2*l) appear, but rho(2*l)^2 = 2 + rho(l), one will obtain a reduced table for s(2*l) with powers rho(2*l)^(2*k+1), k= 0, ..., (delta(2*l)-2)/2 if l is even, and with powers rho(l)^m, m=0, ... , delta(l)-1 if l is odd.
This leads to a reduction of the triangle A127672, when applied for the s(2*l) computation, giving the present table with row length delta(4*L) = A055034(4*L) = phi(8*L)/2 if l =2*L, if L >= 1, and  phi(2*L+1)/2 = A055035(4*L+2), if l = 2*L + 1,  L >= 1, where phi(n) = A000010(n) (Euler totient).
This table gives the coefficients of s(2*l) in the power basis of the algebraic number field Q(rho(2*l)) of degree delta(2*l) = A055034(2*l) if l is even, and in Q(rho(l)) of degree delta(2*l)/2 if l is odd. s(2) and s(6) are rational integers of degree 1.
Thanks go to Seppo Mustonen whose question about the square of the sum of all length in a regular n-gon, led me to this computation.
If l = 2*L+1, L >= 0, that is n == 2 (mod 4), one can obtain s(2*l) more directly in powers of rho(l) by s(2*l) = R(l-1, rho(l)) (mod C(l,rho(l))), with the monic (except for l=1) Chebyshev T-polynomials, called R, in A127672, and the C polynomials from A187360. - Wolfdieter Lang, Oct 10 2013

			The table a(l,m), with n = 2*l, begins:
n,  l \m  0   1   2    3   4   5   6    7   8   9  10  11 ...
2   1:    2
4   2:    0   1
6   3:    1
8   4:    0  -3   0    1
10  5:   -1   1
12  6:    0   4   0   -1
14  7:   -1  -1   1
16  8:    0  -7   0   14   0  -7   0    1
18  9:    2   1  -1
20 10:    0   8   0  -18   0   8   0   -1
22 11:    1   2  -3   -1   1
24 12:    0  -8   0    6   0  -1   0    0
26 13:   -1   3   3   -4  -1   1
28 14:    0  12   0  -67   0  96   0  -52  0  12  0  -1
30 15:   -2   3   1   -1
...
n = 8, l = 4:  s(8)  = -3*rho(8) + rho(8)^3 = -3*sqrt(2 + sqrt(2)) + (sqrt(2 + sqrt(2)))^3 = (sqrt(2) - 1)*sqrt(2 + sqrt(2)).
n = 10, l = 5:  s(10) =  -1 + rho(5), where rho(5) = tau = (1 + sqrt(5))/2, the golden section.

Cf. A127672, A111125, A127677, A055034, A187360, A228785 (odd n case), A228786 (minimal polynomials).

a(2*L,m) = [x^m](s(4*L,x)(mod C(4*L,x))), with s(4*L,x) = sum((-1)^(L-1-s)*A111125(L-1,s)*x^(2*s+1),s=0..L-1), L >= 1, m =0, ..., delta(4*L)-1, and

a(2*L+1,m) = [x^m](s(4*L+2,x)(mod C(2*L+1,x))), with s(4*L+2,x) = sum(A127677(L,s)*(2+x)^(L-s)),s=0..L) (with s(2,x) = 2 for L = 0), L >= 0, m = 0, ..., delta(4*L+2)/2, with delta(n) = A055034(2*l).

cp's OEIS Frontend

A132460 Irregular triangle read by rows of the initial floor(n/2) + 1 coefficients of 1/C(x)^n, where C(x) is the g.f. of the Catalan sequence (A000108).

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A156308 Inverse of triangle S(n,m) defined by sequence A156290, n >= 1, 1 <= m <= n.

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A049667 a(n) = Fibonacci(7*n)/13.

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A100218 Riordan array ((1-2*x)/(1-x), (1-x)).

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A098599 Riordan array ((1+2*x)/(1+x), (1+x)).

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A228785 Table of coefficients of the algebraic number s(2*l+1) = 2*sin(Pi/(2*l+1)) as a polynomial in odd powers of rho(2*(2*l+1)) = 2*cos(Pi/(2*(2*l+1))) (reduced version).

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A049669 a(n) = Fibonacci(9*n)/34.

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A228785 Table of coefficients of the algebraic number s(2l+1) = 2sin(Pi/(2l+1)) as a polynomial in odd powers of rho(2(2l+1)) = 2cos(Pi/(2(2l+1))) (reduced version).

A052227 a(n) = (4n+1)binomial(3n,n)/(2n+1).

A228783 Table of coefficients of the algebraic number s(2l) = 2sin(Pi/2l) as a polynomial in powers of rho(2l) = 2cos(Pi/(2l)) if l is even and of rho(l) = 2*cos(Pi/l) if l is odd (reduced version).