A053123 -id:A053123 - cp's OEIS frontend

A053128 Binomial coefficients C(2*n-5,6).

7, 84, 462, 1716, 5005, 12376, 27132, 54264, 100947, 177100, 296010, 475020, 736281, 1107568, 1623160, 2324784, 3262623, 4496388, 6096454, 8145060, 10737573, 13983816, 18009460, 22957480, 28989675, 36288252, 45057474, 55525372, 67945521, 82598880, 99795696

Offset: 6

Wolfdieter Lang

a(n) = A053123(n,6), n >= 6; a(n) = 0, n=0..5, (seventh column of shifted Chebyshev's S-triangle, decreasing order).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

Vincenzo Librandi, Table of n, a(n) for n = 6..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Milan Janjić, Two Enumerative Functions, University of Banja Luka (Bosnia and Herzegovina, 2017).
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A053123, A053127.

[Binomial(2*n-5,6): n in [6..40]]; // Vincenzo Librandi, Oct 07 2011

Table[Binomial[2*n-5,6], {n,6,50}] (* G. C. Greubel, Aug 26 2018 *)

for(n=6,50, print1(binomial(2*n-5,6), ", ")) \\ G. C. Greubel, Aug 26 2018

a(n) = binomial(2*n-5, 6) if n >= 6 else 0.
G.f.: (7+35*x+21*x^2+x^3)/(1-x)^7.
E.g.f.: (18900 - 16380*x + 6975*x^2 - 1935*x^3 + 390*x^4 - 60*x^5 + 8*x^6)*exp(x)/90. - G. C. Greubel, Aug 26 2018
a(n) = (n-5)*(n-4)*(n-3)*(2*n-9)*(2*n-7)*(2*n-5)/90. - Wesley Ivan Hurt, Mar 25 2020
From Amiram Eldar, Oct 21 2022: (Start)
Sum_{n>=6} 1/a(n) = 667/10 - 96*log(2).
Sum_{n>=6} (-1)^n/a(n) = 273/10 - 6*Pi - 12*log(2). (End)

A053129 Binomial coefficients C(2*n-6,7).

Original entry on oeis.org

8, 120, 792, 3432, 11440, 31824, 77520, 170544, 346104, 657800, 1184040, 2035800, 3365856, 5379616, 8347680, 12620256, 18643560, 26978328, 38320568, 53524680, 73629072, 99884400, 133784560, 177100560, 231917400, 300674088, 386206920, 491796152, 621216192

Offset: 7

Wolfdieter Lang

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

Vincenzo Librandi, Table of n, a(n) for n = 7..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Milan Janjić, Two Enumerative Functions, University of Banja Luka (Bosnia and Herzegovina, 2017).
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A053123, A053128, A000973.

[Binomial(2*n-6,7): n in [7..40]]; // Vincenzo Librandi, Oct 07 2011

A053129:=n->binomial(2*n-6,7); seq(A053129(n), n=7..50); # Wesley Ivan Hurt, Nov 14 2013

Table[Binomial[2 n - 6, 7], {n, 7, 50}] (* Wesley Ivan Hurt, Nov 14 2013 *)

for(n=7, 50, print1(binomial(2*n-6,7), ", ")) \\ G. C. Greubel, Aug 26 2018

a(n) = binomial(2*n-6, 7) if n >= 7 else 0.
a(n) = -A053123(n,7), n >= 7; a(n) := 0, n=0..6, (eighth column of shifted Chebyshev's S-triangle, decreasing order).
a(n) = 8*A000973(n).
G.f.: (8+56*x+56*x^2+8*x^3)/(1-x)^8.
a(n) = (n-6)*(n-5)*(n-4)*(n-3)*(2*n-11)*(2*n-9)*(2*n-7)/315. - Wesley Ivan Hurt, Mar 25 2020
From Amiram Eldar, Oct 21 2022: (Start)
Sum_{n>=7} 1/a(n) = 777/5 - 224*log(2).
Sum_{n>=7} (-1)^(n+1)/a(n) = 441/10 - 14*Pi. (End)

A053130 Binomial coefficients C(2*n-7,8).

Original entry on oeis.org

9, 165, 1287, 6435, 24310, 75582, 203490, 490314, 1081575, 2220075, 4292145, 7888725, 13884156, 23535820, 38608020, 61523748, 95548245, 145008513, 215553195, 314457495, 450978066, 636763050, 886322710, 1217566350, 1652411475, 2217471399, 2944827765, 3872894697

Offset: 8

Wolfdieter Lang

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

Vincenzo Librandi, Table of n, a(n) for n = 8..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Milan Janjić, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A053123, A053129.

[Binomial(2*n-7, 8): n in [8..50]]; // Vincenzo Librandi, Apr 07 2011

Table[Binomial[2*n-7,8], {n,8,50}] (* G. C. Greubel, Aug 26 2018 *)

for(n=8,50, print1(binomial(2*n-7,8), ", ")) \\ G. C. Greubel, Aug 26 2018

a(n) = binomial(2*n-7, 8) if n >= 8 else 0.
G.f.: (9+84*x+126*x^2+36*x^3+x^4)/(1-x)^9.
a(n) = A053123(n,8), n >= 8; a(n) := 0, n=0..7, (ninth column of shifted Chebyshev's S-triangle, decreasing order).
From Amiram Eldar, Oct 21 2022: (Start)
Sum_{n>=8} 1/a(n) = 37276/105 - 512*log(2).
Sum_{n>=8} (-1)^n/a(n) = 592/21 - 16*Pi + 32*log(2). (End)

A053131 Binomial coefficients C(2*n-8,9).

Original entry on oeis.org

10, 220, 2002, 11440, 48620, 167960, 497420, 1307504, 3124550, 6906900, 14307150, 28048800, 52451256, 94143280, 163011640, 273438880, 445891810, 708930508, 1101716330, 1677106640, 2505433700, 3679075400, 5317936260, 7575968400, 10648873950, 14783142660

Offset: 9

Wolfdieter Lang

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

Vincenzo Librandi, Table of n, a(n) for n = 9..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Milan Janjić, Two Enumerative Functions.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45, 10,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A053123, A053130, A053133.

[Binomial(2*n-8,9): n in [9..40]]; // Vincenzo Librandi, Oct 07 2011

Binomial[2*Range[9,40]-8,9] (* Harvey P. Dale, Mar 19 2012 *)

for(n=9,50, print1(binomial(2*n-8,9), ", ")) \\ G. C. Greubel, Aug 26 2018

a(n) = binomial(2*n-8, 9) if n >= 9 else 0.
G.f.: (10+120*x+252*x^2+120*x^3+10*x^4)/(1-x)^10.
a(n) = 2*A053133(n).
a(n) = -A053123(n,9), n >= 9; a(n) := 0, n=0..8 (tenth column of shifted Chebyshev's S-triangle, decreasing order).
From Amiram Eldar, Oct 21 2022: (Start)
Sum_{n>=9} 1/a(n) = 223611/280 - 1152*log(2).
Sum_{n>=9} (-1)^(n+1)/a(n) = 72*log(2) - 13947/280. (End)

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

Original entry on oeis.org

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

A126570 Triangle read by rows: row n gives coefficients (ignoring the alternating signs) of the characteristic polynomial of the n X n matrix with 2's in the main diagonal, 1's in the super- and subdiagonals, and 1 in the upper-right corner, with other elements zeros.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 10, 5, 1, 8, 21, 20, 4, 1, 10, 36, 56, 35, 7, 1, 12, 55, 120, 126, 56, 6, 1, 14, 78, 220, 330, 252, 84, 9, 1, 16, 105, 364, 715, 792, 462, 120, 8, 1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 11, 1, 20, 171, 816, 2380, 4368, 5005, 3432, 1287, 220, 10

Offset: 0

Gary W. Adamson, Dec 28 2006

			First few rows of the triangle are:
  1;
  1, 2;
  1, 4, 3;
  1, 6, 10, 5;
  1, 8, 21, 20, 4;
  1, 10, 36, 56, 35, 7;
  1, 12, 55, 120, 126, 56, 6;
  ...
Charpoly of the 4 X 4 matrix [2,1,0,1; 1,2,1,0; 0,1,2,1; 0,0,1,2] = x^4 - 8*x^3 + 21*x^2 - 20*x + 4; with a root sqrt(3)+2.

William G. Harter, Physics Department, University of Arkansas; personal communication.

Cf. A152060, A053123.

M[i_,j_,n_]:=If[i==j,2,If[Abs[i-j]==1,1,If[j==n&&i==1,1,0]]]; row[0]=1; row[n_]:=Reverse[Abs[CoefficientList[CharacteristicPolynomial[Table[M[i,j,n],{i,n},{j,n}],x],x]]]; Array[row,11,0]//Flatten (* Stefano Spezia, Jun 30 2025 *)

It seems that for n > 2, T(n, k) = binomial(2n+1-k, k) - [k=n] * (-1)^n. - Andrei Zabolotskii, Jun 29 2025

Edited by N. J. A. Sloane, Aug 10 2019

Edited and extended by Andrei Zabolotskii, Jun 29 2025

A360282 Triangle read by rows. T(n, k) = (1/2) * binomial(2(n - k + 1), n - k + 1) binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.

Original entry on oeis.org

1, 0, 1, 0, 3, 2, 0, 10, 12, 3, 0, 35, 60, 30, 4, 0, 126, 280, 210, 60, 5, 0, 462, 1260, 1260, 560, 105, 6, 0, 1716, 5544, 6930, 4200, 1260, 168, 7, 0, 6435, 24024, 36036, 27720, 11550, 2520, 252, 8

Offset: 0

Peter Luschny, Feb 11 2023

Inspired by Vladimir Kruchinin's A360546.

			Triangle T(n, k) starts:
[0] 1;
[1] 0,     1;
[2] 0,     3,      2;
[3] 0,    10,     12,      3;
[4] 0,    35,     60,     30,      4;
[5] 0,   126,    280,    210,     60,     5;
[6] 0,   462,   1260,   1260,    560,   105,     6;
[7] 0,  1716,   5544,   6930,   4200,  1260,   168,    7;
[8] 0,  6435,  24024,  36036,  27720, 11550,  2520,  252,   8;
[9] 0, 24310, 102960, 180180, 168168, 90090, 27720, 4620, 360, 9;

Cf. A135503, A088218 (and A001700), A182626 (row sums apart from sign).

Family: A172431 and unsigned A053123 (case 1), this sequence is case 2, A360546 (case 3).

T := proc(n, k) if n = 0 then 1 else m := n - k + 1; (1/2) * binomial(2*m, m) * binomial(m + n - 1, k - 1) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..8);
# With Vladimir Kruchinin's g.f.:
gf := 1 + (y - x*y^2)/(2*sqrt((x*y - 1)^2 - 4*x)) ;
serx := series(gf, x, 10): poly := n -> simplify(coeff(serx, x, n)):
seq(print(seq(coeff(poly(n), y, k), k = 0..n)), n = 0..9); # Peter Luschny, Feb 14 2023

The sequence is member of a family of sequences defined by T(0, 0, m) = 1 and for m > 0, n > 0 as T(n, k, m) = (1/m)*binomial(m*u, u)*binomial(u + n - 1, k - 1), where u = n - k + 1. Here the case m = 2 is considered.

G.f.: (y*(1 - x*y)) / (2*sqrt(x*(y*(x*y - 2) - 4) + 1)) - y/2 + 1. - Vladimir Kruchinin, Feb 14 2023

cp's OEIS Frontend

A053128 Binomial coefficients C(2*n-5,6).

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A053129 Binomial coefficients C(2*n-6,7).

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A053130 Binomial coefficients C(2*n-7,8).

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A053131 Binomial coefficients C(2*n-8,9).

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

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A126570 Triangle read by rows: row n gives coefficients (ignoring the alternating signs) of the characteristic polynomial of the n X n matrix with 2's in the main diagonal, 1's in the super- and subdiagonals, and 1 in the upper-right corner, with other elements zeros.

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A360282 Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.

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A360282 Triangle read by rows. T(n, k) = (1/2) * binomial(2(n - k + 1), n - k + 1) binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.