A144129 -id:A144129 - cp's OEIS frontend

A153641 Nonzero coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent, and Bernoulli numbers (triangle read by rows).

1, 1, 1, -1, 1, -3, 1, -6, 5, 1, -10, 25, 1, -15, 75, -61, 1, -21, 175, -427, 1, -28, 350, -1708, 1385, 1, -36, 630, -5124, 12465, 1, -45, 1050, -12810, 62325, -50521, 1, -55, 1650, -28182, 228525, -555731, 1, -66, 2475, -56364, 685575, -3334386, 2702765, 1

Offset: 0

Peter Luschny, Dec 29 2008

In the following the expression [n odd] is 1 if n is odd, 0 otherwise.
(+) W_n(0) = E_n are the Euler (or secant) numbers A122045.
(+) W_n(1) = T_n are the signed tangent numbers, see A009006.
(+) W_{n-1}(1) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli number A027641/A027642.
(+) W_n(-1) 2^{-n}(n+1) = G_n the Genocchi number A036968.
(+) W_n(1/2) 2^{n} are the signed generalized Euler (Springer) number, see A001586.
(+) | W_n([n odd]) | the number of alternating permutations A000111.
(+) | W_n([n odd]) / n! | for 0<=n the Euler zeta number A099612/A099617 (see Wikipedia on Bernoulli number). - Peter Luschny, Dec 29 2008
The diagonals in the full triangle (with zero coefficients) of the polynomials have the general form E(k)*binomial(n+k,k) (k>=0 fixed, n=0,1,...) where E(n) are the Euler numbers in the enumeration A122045. For k=2 we find the triangular numbers A000217 and for k=4 A154286. - Peter Luschny, Jan 06 2009
From Peter Bala, Jun 10 2009: (Start)
The Swiss-Knife polynomials W_n(x) may be expressed in terms of the Bernoulli polynomials B(n,x) as
... W_n(x) = 4^(n+1)/(2*n+2)*[B(n+1,(x+3)/4) - B(n+1,(x+1)/4)].
The Swiss-Knife polynomials are, apart from a multiplying factor, examples of generalized Bernoulli polynomials.
Let X be the Dirichlet character modulus 4 defined by X(4*n+1) = 1, X(4*n+3) = -1 and X(2*n) = 0. The generalized Bernoulli polynomials B(X;n,x), n = 1,2,..., associated with the character X are defined by means of the generating function
... t*exp(x*t)*(exp(t)-exp(3*t))/(exp(4*t)-1) = sum {n = 1..inf} B(X;n,x)*t^n/n!.
The first few values are B(X;1,x) = -1/2, B(X;2,x) = -x, B(X,3,x) = -3/2*(x^2-1) and B(X;4,x) = -2*(x^3-3*x).
In general, W_n(x) = -2/(n+1)*B(X;n+1,x).
For the theory of generalized Bernoulli polynomials associated to a periodic arithmetical function see [Cohen, Section 9.4].
The generalized Bernoulli polynomials may be used to evaluate twisted sums of k-th powers. For the present case the result is
sum{n = 0..4*N-1} X(n)*n^k = 1^k - 3^k + 5^k - 7^k + ... - (4*N-1)^k
= [B(X;k+1,4*N) - B(X;k+1,0)]/(k+1) = [W_k(0) - W_k(4*N)]/2.
For the proof apply [Cohen, Corollary 9.4.17 with m = 4 and x = 0].
The generalized Bernoulli polynomials and the Swiss-Knife polynomials are also related to infinite sums of powers through their Fourier series - see the formula section below. For a table of the coefficients of generalized Bernoulli polynomials attached to a Dirichlet character modulus 8 see A151751.
(End)
The Swiss-Knife polynomials provide a general formula for alternating sums of powers similar to the formula which are provided by the Bernoulli polynomials for non-alternating sums of powers (see the Luschny link). Sequences covered by this formula include A001057, A062393, A062392, A011934, A144129, A077221, A137501, A046092. - Peter Luschny, Jul 12 2009
The greatest common divisor of the nonzero coefficients of the decapitated Swiss-Knife polynomials is exp(Lambda(n)), where Lambda(n) is the von Mangoldt function for odd primes, symbolically:
gcd(coeffs(SKP_{n}(x) - x^n)) = A155457(n) (n>1). - Peter Luschny, Dec 16 2009
Another version is at A119879. - Philippe Deléham, Oct 26 2013

			1
x
x^2  -1
x^3  -3x
x^4  -6x^2   +5
x^5 -10x^3  +25x
x^6 -15x^4  +75x^2  -61
x^7 -21x^5 +175x^3 -427x

H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag. [From Peter Bala, Jun 10 2009]

G. C. Greubel, Table of n, a(n) for the first 76 rows, flattened
Ayse Yilmaz Ceylan and Yilmaz Simsek, Formulae for Generalization of Touchard Polynomials with Their Generating Functions, Symmetry (2025) Vol. 17, Issue 7, Art. No. 1126. See Eq. 28 and after.
Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
Suyoung Choi and Hanchul Park, A new graph invariant arises in toric topology, arXiv preprint arXiv:1210.3776 [math.AT], 2012.
Leonhard Euler (1735), De summis serierum reciprocarum, Opera Omnia I.14, E 41, 73-86; On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A Oct. 2007 vol 463 no. 463 2086 2401-2414. [Added by Tom Copeland, Aug 31 2015]
Peter Luschny, The Swiss-Knife polynomials.
Peter Luschny, Swiss-Knife polynomials and Euler numbers
Wikipedia, Bernoulli number
J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

Cf. A151751, A162590, A000111, A001586, A009006, A027641/A027642, A036968, A099612/A099617, A119879, A104033.
W_n(k), k=0,1,...
W_0:  1,  1,  1,  1,   1,   1, ........ A000012
W_1:  0,  1,  2,  3,   4,   5, ........ A001477
W_2: -1,  0,  3,  8,  15,  24, ........ A067998
W_3:  0, -2,  2, 18,  52, 110, ........ A121670
W_4:  5,  0, -3, 32, 165, 480, ........
W_n(k), n=0,1,...
k=0:  1,  0, -1,  0,   5,   0, -61, ... A122045
k=1:  1,  1,  0, -2,   0,  16,   0, ... A155585
k=2:  1,  2,  3,  2,  -3,   2,  63, ... A119880
k=3:  1,  3,  8, 18,  32,  48, 128, ... A119881
k=4:  1,  4, 15, 52, 165, 484, ........         [Peter Luschny, Jul 07 2009]

w := proc(n,x) local v,k,pow,chen; pow := (a,b) -> if a = 0 and b = 0 then 1 else a^b fi; chen := proc(m) if irem(m+1,4) = 0 then RETURN(0) fi; 1/((-1)^iquo(m+1,4) *2^iquo(m,2)) end; add(add((-1)^v*binomial(k,v)*pow(v+x+1,n)*chen(k),v=0..k), k=0..n) end:
# Coefficients with zeros:
seq(print(seq(coeff(i!*coeff(series(exp(x*t)*sech(t),t,16),t,i),x,i-n),n=0..i)), i=0..8);
# Recursion
W := proc(n,z) option remember; local k,p;
if n = 0 then 1 else p := irem(n+1,2);
z^n - p + add(`if`(irem(k,2)=1,0,
W(k,0)*binomial(n,k)*(power(z,n-k)-p)),k=2..n-1) fi end:
# Peter Luschny, edited and additions Jul 07 2009, May 13 2010, Oct 24 2011

max = 9; rows = (Reverse[ CoefficientList[ #, x]] & ) /@ CoefficientList[ Series[ Exp[x*t]*Sech[t], {t, 0, max}], t]*Range[0, max]!; par[coefs_] := (p = Partition[ coefs, 2][[All, 1]]; If[ EvenQ[ Length[ coefs]], p, Append[ p, Last[ coefs]]]); Flatten[ par /@ rows] (* Jean-François Alcover, Oct 03 2011, after g.f. *)
sk[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; Table[CoefficientList[sk[n, x], x] // Reverse // Select[#, # =!= 0 &] &, {n, 0, 13}] // Flatten (* Jean-François Alcover, May 21 2013 *)
Flatten@Table[Binomial[n, 2k] EulerE[2k], {n, 0, 12}, {k, 0, n/2}](* Oliver Seipel, Jan 14 2025 *)

def A046978(k):
    if k % 4 == 0:
        return 0
    return (-1)**(k // 4)
def A153641_poly(n, x):
    return expand(add(2**(-(k // 2))*A046978(k+1)*add((-1)**v*binomial(k,v)*(v+x+1)**n for v in (0..k)) for k in (0..n)))
for n in (0..7): print(A153641_poly(n, x))  # Peter Luschny, Oct 24 2011

W_n(x) = Sum_{k=0..n}{v=0..k} (-1)^v binomial(k,v)*c_k*(x+v+1)^n where c_k = frac((-1)^(floor(k/4))/2^(floor(k/2))) [4 not div k] (Iverson notation).
From Peter Bala, Jun 10 2009: (Start)
E.g.f.: 2*exp(x*t)*(exp(t)-exp(3*t))/(1-exp(4*t))= 1 + x*t + (x^2-1)*t^2/2! + (x^3-3*x)*t^3/3! + ....
W_n(x) = 1/(2*n+2)*Sum_{k=0..n+1} 1/(k+1)*Sum_{i=0..k} (-1)^i*binomial(k,i)*((x+4*i+3)^(n+1) - (x+4*i+1)^(n+1)).
Fourier series expansion for the generalized Bernoulli polynomials:
B(X;2*n,x) = (-1)^n*(2/Pi)^(2*n)*(2*n)! * {sin(Pi*x/2)/1^(2*n) - sin(3*Pi*x/2)/3^(2*n) + sin(5*Pi*x/2)/5^(2*n) - ...}, valid for 0 <= x <= 1 when n >= 1.
B(X;2*n+1,x) = (-1)^(n+1)*(2/Pi)^(2*n+1)*(2*n+1)! * {cos(Pi*x/2)/1^(2*n+1) - cos(3*Pi*x/2)/3^(2*n+1) + cos(5*Pi*x/2)/5^(2*n+1) - ...}, valid for 0 <= x <= 1 when n >= 1 and for 0 <= x < 1 when n = 0.
(End)
E.g.f.: exp(x*t) * sech(t). - Peter Luschny, Jul 07 2009
O.g.f. as a J-fraction: z/(1-x*z+z^2/(1-x*z+4*z^2/(1-x*z+9*z^2/(1-x*z+...)))) = z + x*z^2 + (x^2-1)*z^3 + (x^3-3*x)*z^4 + .... - Peter Bala, Mar 11 2012
Conjectural o.g.f.: Sum_{n >= 0} (1/2^((n-1)/2))*cos((n+1)*Pi/4)*( Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - (k + x)*t) ) = 1 + x*t + (x^2 - 1)*t^2 + (x^3 - 3*x)*t^3 + ... (checked up to O(t^13)), which leads to W_n(x) = Sum_{k = 0..n} 1/2^((k - 1)/2)*cos((k + 1)*Pi/4)*( Sum_{j = 0..k} (-1)^j*binomial(k, j)*(j + x)^n ). - Peter Bala, Oct 03 2016

A144130 a(n) = ChebyshevT(4, n).

Original entry on oeis.org

1, 1, 97, 577, 1921, 4801, 10081, 18817, 32257, 51841, 79201, 116161, 164737, 227137, 305761, 403201, 522241, 665857, 837217, 1039681, 1276801, 1552321, 1870177, 2234497, 2649601, 3120001, 3650401, 4245697, 4910977, 5651521

Offset: 0

Vladimir Joseph Stephan Orlovsky, Sep 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A144129.

[8*n^4-8*n^2+1: n in [0..30]]; // Vincenzo Librandi, May 30 2014

A144130:=n->8*n^4-8*n^2+1: seq(A144130(n), n=0..60); # Wesley Ivan Hurt, Apr 10 2017

Table[ChebyshevT[4, n], {n, 0, 50}] (* Vincenzo Librandi, May 30 2014 *)

From Paul Barry, Nov 17 2009: (Start)
G.f.: (1-4x+102x^2+92x^3+x^4)/(1-x)^5;
a(n) = 8*n^4-8*n^2+1. (End)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Mar 15 2023

Changed offset from 1 to 0 by Vincenzo Librandi, May 30 2014

A322836 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{n}(x), evaluated at x=k.

Original entry on oeis.org

1, 1, 0, 1, 1, -1, 1, 2, 1, 0, 1, 3, 7, 1, 1, 1, 4, 17, 26, 1, 0, 1, 5, 31, 99, 97, 1, -1, 1, 6, 49, 244, 577, 362, 1, 0, 1, 7, 71, 485, 1921, 3363, 1351, 1, 1, 1, 8, 97, 846, 4801, 15124, 19601, 5042, 1, 0, 1, 9, 127, 1351, 10081, 47525, 119071, 114243, 18817, 1, -1

Offset: 0

Seiichi Manyama, Dec 28 2018

			Square array begins:
   1, 1,    1,     1,      1,      1,       1, ...
   0, 1,    2,     3,      4,      5,       6, ...
  -1, 1,    7,    17,     31,     49,      71, ...
   0, 1,   26,    99,    244,    485,     846, ...
   1, 1,   97,   577,   1921,   4801,   10081, ...
   0, 1,  362,  3363,  15124,  47525,  120126, ...
  -1, 1, 1351, 19601, 119071, 470449, 1431431, ...

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

Mirror of A101124.
Columns 0-20 give A056594, A000012, A001075, A001541, A001091, A001079, A023038, A011943(n+1), A001081, A023039, A001085, A077422, A077424, A097308, A097310, A068203, A322888, A056771, A322889, A078986, A322890.
Rows 0-10 give A000012, A001477, A056220, A144129, A144130, A243131, A243132, A243133, A243134, A243135, A243136.
Main diagonal gives A115066.
Cf. A323182 (Chebyshev polynomial of the second kind).

Table[ChebyshevT[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Dec 28 2018 *)

T(n,k) = polchebyshev(n,1,k);
matrix(7, 7, n, k, T(n-1,k-1)) \\ Michel Marcus, Dec 28 2018

T(n, k) = round(cos(n*acos(k)));\\ Seiichi Manyama, Mar 05 2021

T(n, k) = if(n==0, 1, n*sum(j=0, n, (2*k-2)^j*binomial(n+j, 2*j)/(n+j))); \\ Seiichi Manyama, Mar 05 2021

A(0,k) = 1, A(1,k) = k and A(n,k) = 2 * k * A(n-1,k) - A(n-2,k) for n > 1.
A(n,k) = cos(n*arccos(k)). - Seiichi Manyama, Mar 05 2021
A(n,k) = n * Sum_{j=0..n} (2*k-2)^j * binomial(n+j,2*j)/(n+j) for n > 0. - Seiichi Manyama, Mar 05 2021

A154560 a(n) = (n+3)^2*n/2 + 1.

Original entry on oeis.org

1, 9, 26, 55, 99, 161, 244, 351, 485, 649, 846, 1079, 1351, 1665, 2024, 2431, 2889, 3401, 3970, 4599, 5291, 6049, 6876, 7775, 8749, 9801, 10934, 12151, 13455, 14849, 16336, 17919, 19601, 21385, 23274, 25271, 27379, 29601, 31940, 34399, 36981

Offset: 0

Klaus Brockhaus, Jan 12 2009

8*a(n) is the y value of a solution (x, y) to the Diophantine equation 2*x^3+12*x^2 = y^2. The corresponding x value is A152811(n+1).

			a(5) = (5+3)^2*5/2+1 = 64*5/2+1 = 161.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A058794 (row 3 of A007754), A117560 (n*(n^2-1)/2-1), A144129 (4*n^3-3*n), A141530, A152811 (2*(n^2+2*n-2)).

[(n+3)^2*n/2 + 1: n in [0..50]]; // Vincenzo Librandi, Sep 06 2011

{for(n=0,40,print1((n+3)^2*n/2+1,","))}

G.f.: (1+5*x-4*x^2+x^3)/(1-x)^4.
a(n) = A058794(n)/2.
a(n) = A117560(n+2) - n - 1.
a(2*n) = A144129(n+1).
a(2*n-1) = A141530(n+1).  a(n) = -a(-n-4). - Bruno Berselli, Sep 05 2011
a(n) = ((n+2-i)^3+(n+2+i)^3)/4, where i is the imaginary unit. - Nicolas Bělohoubek, Jul 03 2025

A239608 Sin( arcsin(n)- 2*arccos(n) )^2.

Original entry on oeis.org

0, 1, 676, 9801, 59536, 235225, 715716, 1825201, 4096576, 8346321, 15760900, 27994681, 47279376, 76545001, 119552356, 181037025, 266864896, 384199201, 541679076, 749609641, 1020163600, 1367594361, 1808460676, 2361862801, 3049690176, 3896880625, 4931691076

Offset: 0

José María Grau Ribas, Mar 22 2014

nonn

The terms are integers.

This is assuming the "standard branch" of arcsin and arccos, where sin(arccos(n)) = cos(arcsin(n)) = sqrt(1-n^2). - Robert Israel, May 25 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A239607, A239609, A239610.

[n^2*(3-4*n^2)^2 : n in [0..50]]; // Vincenzo Librandi, May 30 2014

G[n_, a_, b_] := G[n, a, b] = Sin[a ArcSin[ n] + b ArcCos[n]]^2 // ComplexExpand // FullSimplify; Table[G[n, 1, -2], {n, 0, 43}]
CoefficientList[Series[- x (x + 1) (x^4 + 668 x^3 + 4422 x^2 + 668 x + 1)/(x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2014 *)
Table[n^2*(3-4*n^2)^2,{n,0,30}] (* Harvey P. Dale, Aug 05 2016 *)

vector(100, n, round(sin(asin(n-1) - 2*acos(n-1))^2)) \\ Colin Barker, May 24 2014

a(n) = n^2*(3-4*n^2)^2. G.f.: -x*(x+1)*(x^4+668*x^3+4422*x^2+668*x+1) / (x-1)^7. - Colin Barker, May 24 2014

a(n) = A144129(n)^2. - Robert Israel, May 25 2014

A243132 32n^6 - 48n^4 + 18*n^2 - 1.

Original entry on oeis.org

-1, 1, 1351, 19601, 119071, 470449, 1431431, 3650401, 8193151, 16692641, 31521799, 55989361, 94558751, 153090001, 239104711, 362074049, 533729791, 768398401, 1083358151, 1499219281, 2040327199, 2735188721, 3616921351, 4723725601, 6099380351, 7793761249

Offset: 0

Vincenzo Librandi, May 31 2014

Chebyshev polynomial of the first kind T(6,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A056220, A144129, A144130.

Magma
```
[32*n^6-48*n^4+18*n^2-1: n in [0..40]];
```

Table[ChebyshevT[6, n], {n, 0, 40}] (* or *) Table[32 n^6 - 48 n^4 + 18 n^2 - 1, {n, 0, 20}]

G.f.: (-1 + 8*x + 1323*x^2 + 10200*x^3 + 10165*x^4 + 1344*x^5 + x^6)/(1 - x)^7.

a(n) = (2*n^2 - 1)*(16*n^4 - 16*n^2 + 1).

A243134 128n^8 - 256n^6 + 160n^4 - 32n^2 + 1.

Original entry on oeis.org

1, 1, 18817, 665857, 7380481, 46099201, 203253121, 708158977, 2081028097, 5374978561, 12545596801, 26986755841, 54276558337, 103182433537, 186979578241, 325142092801, 545471324161, 886731088897, 1401864610177, 2161873163521, 3260441587201, 4819400974081

Offset: 0

Vincenzo Librandi, May 31 2014

Chebyshev polynomial of the first kind T(8,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A056220, A144129, A144130.

[128*n^8-256*n^6+160*n^4-32*n^2+1: n in [0..40]];

Table[ChebyshevT[8, n], {n, 0, 40}] (* or *) Table[128 n^8 - 256 n^6 + 160 n^4 - 32 n^2 + 1, {n, 0, 20}]
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,1,18817,665857,7380481,46099201,203253121,708158977,2081028097},30] (* Harvey P. Dale, Nov 01 2015 *)

G.f.: (1 - 8*x + 18844*x^2 + 496456*x^3 + 2065222*x^4 + 2065096*x^5 + 496540*x^6 + 18808*x^7 + x^8)/(1 - x)^9.

a(0)=1, a(1)=1, a(2)=18817, a(3)=665857, a(4)=7380481, a(5)=46099201, a(6)=203253121, a(7)=708158977, a(8)=2081028097, a(n)=9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)- 9*a(n-8)+ a(n-9). - Harvey P. Dale, Nov 01 2015

A243136 a(n) = 512n^10 - 1280n^8 + 1120n^6 - 400n^4 + 50*n^2 - 1.

Original entry on oeis.org

-1, 1, 262087, 22619537, 457470751, 4517251249, 28860511751, 137379191137, 528572943487, 1730726404001, 4993116004999, 13007560326001, 31154649926687, 69544807113937, 146217791079751, 291977237261249, 557471159562751, 1023286908188737, 1814011722210887

Offset: 0

Vincenzo Librandi, May 31 2014

Chebyshev polynomial of the first kind T(10,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A056220, A144129, A144130.

[512*n^10-1280*n^8+1120*n^6-400*n^4+50*n^2-1: n in [0..30]];

a:= n-> simplify(ChebyshevT(10, n)):
seq(a(n), n=0..30);  # Alois P. Heinz, May 31 2014

Table[ChebyshevT[10, n], {n, 0, 30}] (* or *) Table[512 n^10 - 1280 n^8 + 1120 n^6 - 400 n^4 + 50 n^2 - 1, {n, 0, 30}]

G.f.: (-1 + 12*x + 262021*x^2 + 19736800*x^3 + 223070134*x^4 + 685903960*x^5 + 685903498*x^6 + 223070464*x^7 + 19736635*x^8 + 262076*x^9 + x^10)/(1 - x)^11.
a(n) = (2*n^2 - 1)*(256*n^8 - 512*n^6 + 304*n^4 - 48*n^2 + 1).
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11). - Wesley Ivan Hurt, May 04 2024

cp's OEIS Frontend

A153641 Nonzero coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent, and Bernoulli numbers (triangle read by rows).

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Maple

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Sage

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A144130 a(n) = ChebyshevT(4, n).

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Magma

Maple

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A322836 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{n}(x), evaluated at x=k.

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Mathematica

PARI

PARI

PARI

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A154560 a(n) = (n+3)^2*n/2 + 1.

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Magma

PARI

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A239608 Sin( arcsin(n)- 2*arccos(n) )^2.

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Magma

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PARI

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A243132 32*n^6 - 48*n^4 + 18*n^2 - 1.

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Magma

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A243134 128*n^8 - 256*n^6 + 160*n^4 - 32*n^2 + 1.

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A243132 32n^6 - 48n^4 + 18*n^2 - 1.

A243134 128n^8 - 256n^6 + 160n^4 - 32n^2 + 1.

A243136 a(n) = 512n^10 - 1280n^8 + 1120n^6 - 400n^4 + 50*n^2 - 1.

A243131 a(n) = 16n^5 - 20n^3 + 5*n.