A053125 -id:A053125 - cp's OEIS frontend

A054322 Fourth unsigned column of Lanczos triangle A053125 (decreasing powers).

4, 80, 896, 7680, 56320, 372736, 2293760, 13369344, 74711040, 403701760, 2122317824, 10905190400, 54962159616, 272461987840, 1331439861760, 6425271074816, 30666066493440, 144929376436224, 678948430151680

Offset: 0

Wolfdieter Lang

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (16,-96,256,-256).

Cf. A053125, A053123, A002700, A054329.

List([0..20], n-> 4^n*Binomial(2*n+4, 3)); # G. C. Greubel, Jul 22 2019

[4^n*Binomial(2*n+4, 3): n in [0..20]]; // G. C. Greubel, Jul 22 2019

Table[4^n*Binomial[2*n+4, 3], {n,0,20}] (* G. C. Greubel, Jul 22 2019 *)
LinearRecurrence[{16,-96,256,-256},{4,80,896,7680},20] (* Harvey P. Dale, Mar 27 2023 *)

vector(20, n, n--; 4^n*binomial(2*n+4, 3)) \\ G. C. Greubel, Jul 22 2019

[4^n*binomial(2*n+4, 3) for n in (0..20)] # G. C. Greubel, Jul 22 2019

a(n) = 4^n*binomial(2*n+4, 3) = -A053125(n+3, 3) = 4*A054329(n).
G.f.: 4*(1+4*x)/(1-4*x)^4.
E.g.f.: (4/3)*(3 +48*x +120*x^2 +64*x^3)*exp(4*x). - G. C. Greubel, Jul 22 2019

A054324 Sixth unsigned column of Lanczos triangle A053125 (decreasing powers).

Original entry on oeis.org

6, 224, 4032, 50688, 512512, 4472832, 35094528, 254017536, 1725825024, 11142168576, 68975329280, 412216197120, 2390853943296, 13514114596864, 74693776244736, 404792077713408, 2155824474488832, 11304491362025472

Offset: 0

Wolfdieter Lang

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (24, -240, 1280, -3840, 6144, -4096).

Cf. A053125, A054323, A054330.

List([0..20], n-> 4^n*Binomial(2*n+6, 5)); # G. C. Greubel, Jul 22 2019

[4^n*Binomial(2*n+6, 5): n in [0..20]]; // G. C. Greubel, Jul 22 2019

Table[4^n Binomial[2n+6,5],{n,0,20}] (* or *) LinearRecurrence[{24,-240, 1280,-3840,6144,-4096},{6,224,4032,50688,512512,4472832},20] (* Harvey P. Dale, Jul 02 2017 *)

vector(20, n, n--; 4^n*binomial(2*n+6, 5)) \\ G. C. Greubel, Jul 22 2019

[4^n*binomial(2*n+6, 5) for n in (0..20)] # G. C. Greubel, Jul 22 2019

a(n) = 4^n*binomial(2*n+6, 5) = -A053125(n+5, 5)= 2*A054330(n).
G.f.: 2*(3+4*x)*(1+12*x)/(1-4*x)^6.
E.g.f.: (2/15)*(45 +1500*x +8760*x^2 +15840*x^3 +10240*x^4 +2048*x^5) * exp(4*x). - G. C. Greubel, Jul 22 2019

A054326 Eighth unsigned column of Lanczos triangle A053125 (decreasing powers).

Original entry on oeis.org

8, 480, 12672, 219648, 2928640, 32587776, 317521920, 2794192896, 22682271744, 172438323200, 1241555927040, 8538764083200, 56469693136896, 361019918516224, 2240813287342080, 13550896696786944, 80073480481013760

Offset: 0

Wolfdieter Lang

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (32, -448, 3584, -17920, 57344, -114688, 131072, -65536).

Cf. A053125, A054325, A054331.

List([0..20], n-> 4^n*Binomial(2*n+8,7)); # G. C. Greubel, Jul 22 2019

[4^n*Binomial(2*n+8,7): n in [0..20]]; // G. C. Greubel, Jul 22 2019

Table[4^n Binomial[2n+8,7],{n,0,20}] (* or *) LinearRecurrence[{32,-448, 3584,-17920,57344,-114688,131072,-65536},{8,480,12672,219648,2928640, 32587776,317521920,2794192896},20] (* Harvey P. Dale, Oct 23 2012 *)

vector(20, n, n--; 4^n*binomial(2*n+8,7)) \\ G. C. Greubel, Jul 22 2019

[4^n*binomial(2*n+8,7) for n in (0..20)] # G. C. Greubel, Jul 22 2019

a(n) = 4^n*binomial(2*n+8, 7) = -A053125(n+7, 7) = 8*A054331(n).
G.f.: 8*(4*x+1)*(16*x^2+24*x+1)/(1-4*x)^8.
a(0)=8, a(1)=480, a(2)=12672, a(3)=219648, a(4)=2928640, a(5)=32587776, a(6)=317521920, a(7)=2794192896, a(n) = 32*a(n-1) - 448*a(n-2) + 3584*a(n-3) - 17920*a(n-4) + 57344*a(n-5) - 114688*a(n-6) + 131072*a(n-7) - 65536*a(n-8). - Harvey P. Dale, Oct 23 2012

A054323 Fifth column of Lanczos triangle A053125 (decreasing powers).

Original entry on oeis.org

5, 140, 2016, 21120, 183040, 1397760, 9748480, 63504384, 392232960, 2321285120, 13264486400, 73610035200, 398475657216, 2111580405760, 10984378859520, 56221121904640, 283661115064320, 1413061420253184, 6959221409054720

Offset: 0

Wolfdieter Lang

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (20, -160, 640, -1280, 1024).

Cf. A053125, A054322.

Bisection of A080951.

List([0..20], n-> 4^n*Binomial(2*n+5, 4)); # G. C. Greubel, Jul 22 2019

[4^n*Binomial(2*n+5, 4): n in [0..20]]; // G. C. Greubel, Jul 22 2019

Table[4^n Binomial[2n+5,4],{n,0,20}] (* or *) LinearRecurrence[{20,-160, 640,-1280,1024},{5,140,2016,21120,183040},20] (* Harvey P. Dale, Mar 03 2018 *)

vector(20, n, n--; 4^n*binomial(2*n+5, 4)) \\ G. C. Greubel, Jul 22 2019

[4^n*binomial(2*n+5, 4) for n in (0..20)] # G. C. Greubel, Jul 22 2019

a(n) = 4^n*binomial(2*n+5, 4) = 4^n*A053126(n+4) = A053125(n+4, 4).
G.f.: (5 +40*x +16*x^2)/(1-4*x)^5.
E.g.f.: (15 +360*x +1464*x^2 +1664*x^3 +512*x^4)*exp(4*x)/3. - G. C. Greubel, Jul 22 2019
a(n) = 20*a(n-1)-160*a(n-2)+640*a(n-3)-1280*a(n-4)+1024*a(n-5). - Wesley Ivan Hurt, May 02 2021

A054325 Seventh column of Lanczos triangle A053125 (decreasing powers).

Original entry on oeis.org

7, 336, 7392, 109824, 1281280, 12673024, 111132672, 889061376, 6615662592, 46425702400, 310388981760, 1992378286080, 12352745373696, 74327630282752, 435713694760960, 2496217812566016, 14012859084177408, 77247357640507392

Offset: 0

Wolfdieter Lang

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (28, -336, 2240, -8960, 21504, -28672, 16384).

Cf. A053125, A054324.

List([0..20], n-> 4^n*Binomial(2*n+7, 6)); # G. C. Greubel, Jul 22 2019

[4^n*Binomial(2*n+7, 6): n in [0..20]]; // G. C. Greubel, Jul 22 2019

Table[4^n*Binomial[2*n+7, 6], {n,0,20}] (* G. C. Greubel, Jul 22 2019 *)

vector(20, n, n--; 4^n*binomial(2*n+7, 6)) \\ G. C. Greubel, Jul 22 2019

[4^n*binomial(2*n+7, 6) for n in (0..20)] # G. C. Greubel, Jul 22 2019

a(n) = 4^n*binomial(2*n+7, 6) = A053125(n+6, 6).

G.f.: (7 +140*x +336*x^2 +64*x^3)/(1-4*x)^7.

A054327 Ninth column of Lanczos triangle A053125 (decreasing powers).

Original entry on oeis.org

9, 660, 20592, 411840, 6223360, 77395968, 833495040, 8033304576, 70882099200, 581979340800, 4500640235520, 33087710822400, 232937484189696, 1579462143508480, 10363761453957120, 66060621396836352

Offset: 0

Wolfdieter Lang

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (36, -576, 5376, -32256, 129024, -344064, 589824, -589824, 262144).

Cf. A053125, A054326.

List([0..20], n-> 4^n*Binomial(2*n+9,8)); # G. C. Greubel, Jul 22 2019

[4^n*Binomial(2*n+9,8): n in [0..20]]; // G. C. Greubel, Jul 22 2019

Table[4^n*Binomial[2*n+9, 8], {n,0,20}] (* G. C. Greubel, Jul 22 2019 *)

vector(20, n, n--; 4^n*binomial(2*n+9,8)) \\ G. C. Greubel, Jul 22 2019

[4^n*binomial(2*n+9,8) for n in (0..20)] # G. C. Greubel, Jul 22 2019

a(n) = 4^n*binomial(2*n+9, 8) = A053125(n+8, 8).

G.f.: (4*x+3)*(64*x^3+528*x^2+108*x+3)/(1-4*x)^9.

A054328 Tenth unsigned column of Lanczos triangle A053125 (decreasing powers).

Original entry on oeis.org

10, 880, 32032, 732160, 12446720, 171991040, 2037432320, 21422145536, 204770508800, 1810602393600, 15002134118400, 117645194035200, 879986051383296, 6317848574033920, 43758103916707840, 293602761763717120

Offset: 0

Wolfdieter Lang

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (40, -720, 7680, -53760, 258048, -860160, 1966080, -2949120, 2621440, -1048576).

Cf. A053125, A054327, A054332.

List([0..20], n-> 4^n*Binomial(2*n+10,9)); # G. C. Greubel, Jul 22 2019

[4^n*Binomial(2*n+10,9): n in [0..20]]; // G. C. Greubel, Jul 22 2019

CoefficientList[Series[2(1+40x+80x^2)(5+40x+16x^2)/(1-4x)^10,{x,0,20}],x]  (* Harvey P. Dale, Feb 28 2011 *)
Table[4^n*Binomial[2*n+10, 9], {n,0,20}] (* G. C. Greubel, Jul 22 2019 *)

vector(20, n, n--; 4^n*binomial(2*n+10,9)) \\ G. C. Greubel, Jul 22 2019

[4^n*binomial(2*n+10,9) for n in (0..20)] # G. C. Greubel, Jul 22 2019

a(n) = 4^n*binomial(2*n+10, 9)= -A053125(n+9, 9) = 2* A054332(n).

G.f. 2*(1+40*x+80*x^2)*(5+40*x+16*x^2)/(1-4*x)^10.

A054329 One quarter of fourth unsigned column of Lanczos' triangle A053125.

Original entry on oeis.org

1, 20, 224, 1920, 14080, 93184, 573440, 3342336, 18677760, 100925440, 530579456, 2726297600, 13740539904, 68115496960, 332859965440, 1606317768704, 7666516623360, 36232344109056, 169737107537920, 788899592929280

Offset: 0

Wolfdieter Lang

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (16,-96,256,-256).

Cf. A054322, A053125.

List([0..30], n-> 4^(n-1)*Binomial(2*n+4,3)); # G. C. Greubel, Jul 22 2019

[4^(n-1)*Binomial(2*n+4,3): n in [0..30]]; // G. C. Greubel, Jul 22 2019

Table[4^(n-1)*Binomial[2*n+4, 3], {n,0,30}] (* G. C. Greubel, Jul 22 2019 *)

vector(30, n, n--; 4^(n-1)*binomial(2*n+4,3)) \\ G. C. Greubel, Jul 22 2019

[4^(n-1)*binomial(2*n+4,3) for n in (0..30)] # G. C. Greubel, Jul 22 2019

a(n)= 4^(n-1)*binomial(2*n+4, 3)= -A053125(n+3, 3)/4 = A054322(n)/4.
G.f.: (1+4*x)/(1-4*x)^4.
E.g.f.: (3 + 48*x + 120*x^2 + 64*x^3)*exp(4*x)/3. - G. C. Greubel, Jul 22 2019

A054330 One half of sixth unsigned column of Lanczos' triangle A053125.

Original entry on oeis.org

3, 112, 2016, 25344, 256256, 2236416, 17547264, 127008768, 862912512, 5571084288, 34487664640, 206108098560, 1195426971648, 6757057298432, 37346888122368, 202396038856704, 1077912237244416, 5652245681012736

Offset: 0

Wolfdieter Lang

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (24, -240, 1280, -3840, 6144, -4096).

Cf. A054324, A053125.

List([0..20], n-> 2^(2*n-1)*Binomial(2*n+6,5)); # G. C. Greubel, Jul 22 2019

[2^(2*n-1)*Binomial(2*n+6,5): n in [0..20]]; // G. C. Greubel, Jul 22 2019

Table[2^(2*n-1)*Binomial[2*n+6, 5], {n,0,20}] (* G. C. Greubel, Jul 22 2019 *)

vector(20, n, n--; 2^(2*n-1)*binomial(2*n+6,5)) \\ G. C. Greubel, Jul 22 2019

[2^(2*n-1)*binomial(2*n+6,5) for n in (0..20)] # G. C. Greubel, Jul 22 2019

a(n)= 2^(2*n-1)*binomial(2*n+6, 5) = -A053125(n+5, 5)/2 = A054324(n)/2.
G.f.: (4*x+3)*(12*x+1)/(1-4*x)^6.
E.g.f.: (90 + 3000*x + 17520*x^2 + 31680*x^3 + 20480*x^4 + 4096*x^5)* exp(4*x)/30. - G. C. Greubel, Jul 22 2019

A054331 One eighth of eighth unsigned column of Lanczos' triangle A053125.

Original entry on oeis.org

1, 60, 1584, 27456, 366080, 4073472, 39690240, 349274112, 2835283968, 21554790400, 155194490880, 1067345510400, 7058711642112, 45127489814528, 280101660917760, 1693862087098368, 10009185060126720, 57935518230380544

Offset: 0

Wolfdieter Lang

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (32,-448,3584,-17920,57344,-114688,131072,-65536).

Cf. A054326, A053125.

List([0..20], n-> 2^(2*n-3)*Binomial(2*n+8, 7)); # G. C. Greubel, Jul 22 2019

[2^(2*n-3)*Binomial(2*n+8, 7): n in [0..20]]; // G. C. Greubel, Jul 22 2019

Table[4^n Binomial[2n+8,7]/8,{n,0,20}] (* Harvey P. Dale, Nov 03 2011 *)
LinearRecurrence[{32,-448,3584,-17920,57344,-114688,131072,-65536},{1,60,1584,27456,366080,4073472,39690240,349274112},20] (* Harvey P. Dale, Feb 25 2022 *)

vector(20, n, n--; 2^(2*n-3)*binomial(2*n+8, 7)) \\ G. C. Greubel, Jul 22 2019

[2^(2*n-3)*binomial(2*n+8, 7) for n in (0..20)] # G. C. Greubel, Jul 22 2019

a(n) = 2^(2*n-3)*binomial(2*n+8, 7) = -A053125(n+7, 7)/8 = A054326(n)/8.

G.f. (1+4*x)*(1+24*x+16*x^2)/(1-4*x)^8.

cp's OEIS Frontend

A054322 Fourth unsigned column of Lanczos triangle A053125 (decreasing powers).

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A054324 Sixth unsigned column of Lanczos triangle A053125 (decreasing powers).

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A054326 Eighth unsigned column of Lanczos triangle A053125 (decreasing powers).

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A054323 Fifth column of Lanczos triangle A053125 (decreasing powers).

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A054325 Seventh column of Lanczos triangle A053125 (decreasing powers).

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A054327 Ninth column of Lanczos triangle A053125 (decreasing powers).

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