A144139 -id:A144139 - cp's OEIS frontend

A323182 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Chebyshev polynomial of the second kind U_{n}(x), evaluated at x=k.

1, 1, 0, 1, 2, -1, 1, 4, 3, 0, 1, 6, 15, 4, 1, 1, 8, 35, 56, 5, 0, 1, 10, 63, 204, 209, 6, -1, 1, 12, 99, 496, 1189, 780, 7, 0, 1, 14, 143, 980, 3905, 6930, 2911, 8, 1, 1, 16, 195, 1704, 9701, 30744, 40391, 10864, 9, 0, 1, 18, 255, 2716, 20305, 96030, 242047, 235416, 40545, 10, -1

Offset: 0

Seiichi Manyama, Jan 06 2019

			Square array begins:
   1, 1,    1,     1,      1,      1,       1, ...
   0, 2,    4,     6,      8,     10,      12, ...
  -1, 3,   15,    35,     63,     99,     143, ...
   0, 4,   56,   204,    496,    980,    1704, ...
   1, 5,  209,  1189,   3905,   9701,   20305, ...
   0, 6,  780,  6930,  30744,  96030,  241956, ...
  -1, 7, 2911, 40391, 242047, 950599, 2883167, ...

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

Mirror of A228161.
Columns 0-19 give A056594, A000027(n+1), A001353(n+1), A001109(n+1), A001090(n+1), A004189(n+1), A004191, A007655(n+2), A077412, A049660(n+1), A075843(n+1), A077421, A077423, A097309, A097311, A097313, A029548, A029547, A144128(n+1), A078987.
Rows 0-10 give A000012, A005843, A000466, A144138, A144139, A242850, A242851, A242852, A242853, A242854, A243130.
Main diagonal gives A323118.
Cf. A179943, A322836 (Chebyshev polynomial of the first kind).

T(n,k)  = polchebyshev(n, 2, k);
matrix(7, 7, n, k, T(n-1,k-1)) \\ Michel Marcus, Jan 07 2019

T(n, k) = sum(j=0, n, (2*k-2)^j*binomial(n+1+j, 2*j+1)); \\ Seiichi Manyama, Mar 03 2021

T(0,k) = 1, T(1,k) = 2 * k and T(n,k) = 2 * k * T(n-1,k) - T(n-2,k) for n > 1.

T(n, k) = Sum_{j=0..n} (2*k-2)^j * binomial(n+1+j,2*j+1). - Seiichi Manyama, Mar 03 2021

A144138 Chebyshev polynomial of the second kind U(3,n).

Original entry on oeis.org

0, 4, 56, 204, 496, 980, 1704, 2716, 4064, 5796, 7960, 10604, 13776, 17524, 21896, 26940, 32704, 39236, 46584, 54796, 63920, 74004, 85096, 97244, 110496, 124900, 140504, 157356, 175504, 194996, 215880, 238204, 262016, 287364, 314296, 342860, 373104, 405076

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 (4,-6,4,-1).

Cf. A100551, A123027, A144139.

[8*n^3-4*n: n in [0..40]]; // Vincenzo Librandi, May 29 2014

Table[ChebyshevU[3,n], {n,0,100}] (* and *) Table[4n*(2*n^2-1), {n,0,100}]
CoefficientList[Series[4 x (1 + 10 x + x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, May 29 2014 *)
LinearRecurrence[{4,-6,4,-1},{0,4,56,204},40] (* Harvey P. Dale, Dec 23 2022 *)

G.f.: 4*x*(1 + 10*x + x^2)/(1 - x)^4. - Vincenzo Librandi, May 29 2014

a(n) = 4*n*(2*n^2-1). - Vincenzo Librandi, May 29 2014

A242850 32n^5 - 32n^3 + 6*n.

Original entry on oeis.org

0, 6, 780, 6930, 30744, 96030, 241956, 526890, 1032240, 1866294, 3168060, 5111106, 7907400, 11811150, 17122644, 24192090, 33423456, 45278310, 60279660, 79015794, 102144120, 130395006, 164575620, 205573770, 254361744, 312000150, 379641756, 458535330, 550029480, 655576494

Offset: 0

Vincenzo Librandi, May 29 2014

Chebyshev polynomial of the second kind U(5,n).

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

Cf. A000466, A144138, A144139, A242851.

Magma
```
[32*n^5-32*n^3+6*n: n in [0..40]];
```

Table[32 n^5 - 32 n^3 + 6 n, {n, 0, 40}] (* or *) Table[ChebyshevU[5, n], {n, 0, 40}]

G.f.: x*(6 + 744*x + 2340*x^2 + 744*x^3 + 6*x^4)/(1 - x)^6.

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

Edited by Bruno Berselli, May 29 2014

A242851 64n^6 - 80n^4 + 24*n^2 - 1.

Original entry on oeis.org

-1, 7, 2911, 40391, 242047, 950599, 2883167, 7338631, 16451071, 33489287, 63202399, 112211527, 189447551, 306634951, 478821727, 724955399, 1068505087, 1538129671, 2168392031, 3000519367, 4083209599, 5473483847, 7237584991, 9451922311, 12204062207, 15593764999

Offset: 0

Vincenzo Librandi, May 29 2014

Chebyshev polynomial of the second kind U(6,n).

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

Cf. A000466, A144138, A144139, A242850.

Magma
```
[64*n^6-80*n^4+24*n^2-1: n in [0..40]];
```

Table[64 n^6 - 80 n^4 + 24 n^2 - 1, {n, 0, 40}] (* or *) Table[ChebyshevU[6, n], {n, 0, 40}]

G.f.: (-1 + 14*x + 2841*x^2 + 20196*x^3 + 20161*x^4 + 2862*x^5 + 7*x^6)/(1 - x)^7.

a(n) = (8*n^3-4*n^2-4*n+1)*(8*n^3+4*n^2-4*n-1).

Edited by Bruno Berselli, May 29 2014

A242852 128n^7-192n^5+80n^3-8n.

Original entry on oeis.org

0, 8, 10864, 235416, 1905632, 9409960, 34356048, 102213944, 262184896, 600940872, 1260879920, 2463542488, 4538833824, 7960697576, 13389885712, 21724469880, 34158739328, 52251130504, 78001833456, 113940720152, 163226239840, 229755926568, 318289163984, 434582852536

Offset: 0

Vincenzo Librandi, May 30 2014

Chebyshev polynomial of the second kind U(7,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A144138, A144139, A242850, A242851.

[128*n^7-192*n^5+80*n^3-8*n: n in [0..30]];

Table[ChebyshevU[7, n], {n, 0, 30}] (* or *) Table[128 n^7 - 192 n^5 + 80 n^3 - 8 n, {n, 0, 30}]

G.f.: (8*x + 10800*x^2 + 148728*x^3 + 326048*x^4 + 148728*x^5 + 10800*x^6 + 8*x^7)/(1-x)^8.

a(n) = 8*n*(2*n^2-1)*(8*n^4-8*n^2+1).

A242853 256n^8 - 448n^6 + 240n^4 - 40n^2 + 1.

Original entry on oeis.org

1, 9, 40545, 1372105, 15003009, 93149001, 409389409, 1423656585, 4178507265, 10783446409, 25154396001, 54085723209, 108742564225, 206671502025, 374437978209, 651009141001, 1092011153409, 1775000307465, 2805897612385, 4326746846409, 6524966384001, 9644275432009

Offset: 0

Vincenzo Librandi, May 30 2014

Chebyshev polynomial of the second kind U(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. A144138, A144139, A242850, A242851.

[256*n^8-448*n^6+240*n^4-40*n^2+1: n in [0..30]];

Table[ChebyshevU[8, n], {n, 0, 30}] (* or *) Table[256 n^8 - 448 n^6 + 240 n^4 - 40 n^2 + 1, {n, 0, 30}]

G.f.: (1 + 40500*x^2 + 1007440*x^3 + 4113054*x^4 + 4112928*x^5 + 1007524*x^6 + 40464*x^7 + 9*x^8)/(1 - x)^9.

a(n) = (2*n - 1)*(2*n + 1)*(8*n^3 - 6*n - 1) (8*n^3 - 6*n + 1).

A242854 a(n) = 512n^9 - 1024n^7 + 672n^5 - 160n^3 + 10*n.

Original entry on oeis.org

0, 10, 151316, 7997214, 118118440, 922080050, 4878316860, 19828978246, 66593931344, 193501094490, 501827040100, 1187422368110, 2605282707576, 5365498355074, 10470873504140, 19508549760150, 34910198169760, 60297759323306, 100934312212404, 164302439443390

Offset: 0

Vincenzo Librandi, May 30 2014

Chebyshev polynomial of the second kind U(9,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A144138, A144139, A242850, A242851.

[512*n^9-1024*n^7+672*n^5-160*n^3+10*n: n in [0..20]];

A242854:=n->512*n^9 - 1024*n^7 + 672*n^5 - 160*n^3 + 10*n: seq(A242854(n), n=0..30); # Wesley Ivan Hurt, Feb 04 2017

Table[ChebyshevU[9, n], {n, 0, 20}] (* or *) Table[512 n^9 - 1024 n^7 + 672 n^5 - 160 n^3 + 10 n, {n, 0, 20}]

G.f.: x*(10 + 151216*x + 6484504*x^2 + 44954320*x^3 + 82614460*x^4 + 44954320*x^5 + 6484504*x^6 + 151216*x^7 + 10*x^8)/(1 - x)^10.

a(n) = 2*n*(4*n^2-2*n-1)*(4*n^2+2*n-1)*(16*n^4-20*n^2+5).

A243130 1024n^10 - 2304n^8 + 1792n^6 - 560n^4 + 60*n^2 - 1.

Original entry on oeis.org

-1, 11, 564719, 46611179, 929944511, 9127651499, 58130412911, 276182038859, 1061324394239, 3472236254411, 10011386405999, 26069206375211, 62418042417599, 139296285729899, 292810020137711, 584605483663499, 1116034330278911, 2048348816684939, 3630829342034159

Offset: 0

Vincenzo Librandi, May 30 2014

Chebyshev polynomial of the second kind U(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. A144138, A144139, A242850, A242851.

[1024*n^10-2304*n^8+1792*n^6-560*n^4+60*n^2-1: n in [0..20]];

Table[ChebyshevU[10, n], {n, 0, 20}] (* or *) Table[1024 n^10 - 2304 n^8 + 1792 n^6 - 560 n^4 + 60 n^2 - 1, {n, 0, 20}]
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{-1,11,564719,46611179,929944511,9127651499,58130412911,276182038859,1061324394239,3472236254411,10011386405999},20] (* Harvey P. Dale, Dec 10 2023 *)

G.f.: (-1 + 22*x + 564543*x^2 + 40400040*x^3 + 448278942*x^4 + 1368702180*x^5 + 1368701718*x^6 + 448279272*x^7 + 40399875*x^8 + 564598*x^9 + 11*x^10)/(1 - x)^11.

a(n) = (32*n^5 - 16*n^4 - 32*n^3 +12*n^2 + 6*n - 1)*(32*n^5 + 16*n^4 - 32*n^3 -12*n^2 + 6*n + 1).

cp's OEIS Frontend

A323182 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Chebyshev polynomial of the second kind U_{n}(x), evaluated at x=k.

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A144138 Chebyshev polynomial of the second kind U(3,n).

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A242850 32*n^5 - 32*n^3 + 6*n.

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A242851 64*n^6 - 80*n^4 + 24*n^2 - 1.

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A242852 128*n^7-192*n^5+80*n^3-8*n.

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A242853 256*n^8 - 448*n^6 + 240*n^4 - 40*n^2 + 1.

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A242854 a(n) = 512*n^9 - 1024*n^7 + 672*n^5 - 160*n^3 + 10*n.

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A242850 32n^5 - 32n^3 + 6*n.

A242851 64n^6 - 80n^4 + 24*n^2 - 1.

A242852 128n^7-192n^5+80n^3-8n.

A242853 256n^8 - 448n^6 + 240n^4 - 40n^2 + 1.

A242854 a(n) = 512n^9 - 1024n^7 + 672n^5 - 160n^3 + 10*n.

A243130 1024n^10 - 2304n^8 + 1792n^6 - 560n^4 + 60*n^2 - 1.