A242850 -id:A242850 - 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

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

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Magma

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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.