A100257 -id:A100257 - cp's OEIS frontend

A008311 Triangle of expansions of powers of x in terms of Chebyshev polynomials T_n (x).

1, 1, 1, 1, 3, 1, 3, 4, 1, 10, 5, 1, 10, 15, 6, 1, 35, 21, 7, 1, 35, 56, 28, 8, 1, 126, 84, 36, 9, 1, 126, 210, 120, 45, 10, 1, 462, 330, 165, 55, 11, 1, 462, 792, 495, 220, 66, 12, 1, 1716, 1287, 715, 286, 78, 13, 1, 1716, 3003, 2002, 1001, 364, 91, 14, 1, 6435, 5005, 3003

Offset: 0

N. J. A. Sloane

This triangle is the right half of Pascal's triangle (A007318), but with each number along the center of Pascal's triangle (except the 1 at the top) divided by 2. - Benjamin Schak (schak(AT)math.upenn.edu), Dec 02 2005

For n>=2 found in A002378, a(n)=A034869(n)/2, for all others a(n)=A034869(n). - R. J. Mathar, May 13 2006

			Triangle begins:
1;
-, 1;
1, -, 1;
-, 3, -, 1;
3, -, 4, -, 1;
-, 10, -, 5, -, 1;
...
From _Philippe Deléham_, Mar 09 2013: (Start)
cos(x)      = 1*cos(x),
2*cos(x)^2  = 1 + cos(2x),
4*cos(x)^3  = 3*cos(x) + cos(3x),
8*cos(x)^4  = 3 + 4*cos(2x) + cos(4x),
16*cos(x)^5 = 10*cos(x) + 5*cos(3x) + cos(5x), etc. (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.

Vincenzo Librandi, Table of n, a(n) for n = 0..5775
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].
H. J. Brothers, Pascal's Prism: Supplementary Material.
Index entries for sequences related to Chebyshev polynomials.

With zeros: A100257.

printf("1,") ; for n from 1 to 20 do for j from n mod 2 to n by 2 do if j = 0 then printf("%d,",binomial(n,(n-j)/2)/2) ; else printf("%d,",binomial(n,(n-j)/2)) ; fi ; od ; od ; # R. J. Mathar, May 13 2006

row[n_] := If[n == 0, {1}, Table[If[j == 0, Binomial[n, (n - j)/2]/2, Binomial[n, (n - j)/2]], {j, Mod[n, 2], n, 2}]];
Table[row[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, May 05 2017, after R. J. Mathar *)

Sum_{k, 0<=k}T(n,k)*cos(kx) = 2^(n-1)*cos(x)^n. - Philippe Deléham, Mar 09 2013

Corrected by Philippe Deléham, Nov 12 2005

More terms from R. J. Mathar, May 13 2006

A030054 a(n) = binomial(2n+1,n-4).

Original entry on oeis.org

1, 11, 78, 455, 2380, 11628, 54264, 245157, 1081575, 4686825, 20030010, 84672315, 354817320, 1476337800, 6107086800, 25140840660, 103077446706, 421171648758, 1715884494940, 6973199770790, 28277527346376, 114456658306760, 462525733568080, 1866442158555975

Offset: 4

N. J. A. Sloane

nonn

Robert Israel, Table of n, a(n) for n = 4..1661
Milan Janjic, Two Enumerative Functions.

Diagonal 10 of triangle A100257.
Fifth unsigned column (s=4) of A113187. - Wolfdieter Lang, Oct 19 2012
Cf. A001622.
Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A002694 (m = 4), A003516 (m = 5), A002696 (m = 6), A030053 - A030056, A004310 - A004318.

seq(binomial(2*n+1,n-4),n=4..50); # Robert Israel, Jun 11 2019

Table[Binomial[2n+1,n-4],{n,4,40}]  (* Harvey P. Dale, Mar 31 2011 *)

vector(30, n, m=n+4; binomial(2*m+1,m-4)) \\ Michel Marcus, Aug 11 2015

G.f.: x^4*512/((1-sqrt(1-4*x))^9*sqrt(1-4*x))+(-1/x^5+7/x^4-15/x^3+10/x^2-1/x). - Vladimir Kruchinin, Aug 11 2015
From Robert Israel, Jun 11 2019: (Start)
(54 + 36*n)*a(n) + (-438 - 129*n)*a(n + 1) + (714 + 138*n)*a(n + 2) + (-432 - 63*n)*a(n + 3) + (110 + 13*n)*a(n + 4) + (-10 - n)*a(n + 5) = 0.
a(n) ~ 2^(2*n+1)/sqrt(n*Pi). (End)
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=4} 1/a(n) = 317/210 - 2*Pi/(9*sqrt(3)).
Sum_{n>=4} (-1)^n/a(n) = 2908*log(phi)/(5*sqrt(5)) - 8697/70, where phi is the golden ratio (A001622). (End)
G.f.: 2F1([11/2,5],[10],4*x). - Karol A. Penson, Apr 24 2024
From Peter Bala, Oct 13 2024: (Start)
a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * sqrt(x)*(x^4 - 9*x^3 + 27*x^2 - 30*x + 9)/sqrt((4 - x)).
G.f. x^4 * B(x) * C(x)^9, where B(x) = 1/sqrt(1 - 4*x) is the g.f. of the central binomial coefficients A000984 and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
D-finite with recurrence -(n+5)*(n-4)*a(n) +2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A030055 a(n) = binomial(2*n+1, n-5).

Original entry on oeis.org

1, 13, 105, 680, 3876, 20349, 100947, 480700, 2220075, 10015005, 44352165, 193536720, 834451800, 3562467300, 15084504396, 63432274896, 265182149218, 1103068603890, 4568648125690, 18851684897584

Offset: 5

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 5..1000
Milan Janjic, Two Enumerative Functions.

Diagonal 12 of triangle A100257.

Cf. A001622.

List([5..25],n->Binomial(2*n+1,n-5)); # Muniru A Asiru, Oct 24 2018

[Binomial(2*n+1, n-5): n in [5..30]]; // G. C. Greubel, Oct 23 2018

seq(binomial(2*n+1,n-5),n=5..25); # Muniru A Asiru, Oct 24 2018

Table[Binomial[2*n+1, n-5], {n, 5, 30}] (* G. C. Greubel, Oct 23 2018 *)

vector(30, n, m=n+4; binomial(2*m+1,m-5)) \\ Michel Marcus, Aug 11 2015

G.f.: x^5*2048/((1-sqrt(1-4*x))^11*sqrt(1-4*x))+(-1/x^6+9/x^5-28/x^4+35/x^3-15/x^2+1/x). - Vladimir Kruchinin, Aug 11 2015
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=5} 1/a(n) = 9497/1260 - 32*Pi/(9*sqrt(3)).
Sum_{n>=5} (-1)^(n+1)/a(n) = 9392*log(phi)/(5*sqrt(5)) - 508169/1260, where phi is the golden ratio (A001622). (End)

A004311 Binomial coefficient C(2n,n-5).

Original entry on oeis.org

1, 12, 91, 560, 3060, 15504, 74613, 346104, 1562275, 6906900, 30045015, 129024480, 548354040, 2310789600, 9669554100, 40225345056, 166509721602, 686353797976, 2818953098830, 11541847896480, 47129212243960, 191991813933920, 780512175396135, 3167295784216200

Offset: 5

N. J. A. Sloane

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=5. - Herbert Kociemba, May 24 2004

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Seiichi Manyama, Table of n, a(n) for n = 5..1000
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 Janjic, Two Enumerative Functions
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Diagonal 11 of triangle A100257.

Cf. A001622.

[ Binomial(2*n,n-5): n in [5..150] ]; // Vincenzo Librandi, Apr 13 2011

Table[Binomial[2*n, n-5], {n, 5, 30}] (* Amiram Eldar, Aug 27 2022 *)

first(m)=vector(m,i,binomial(2*(i+4),i-1)) \\ Anders Hellström, Aug 17 2015

a(n) = Sum{k=0..n} C(n, k)*C(n, k+5). - Hermann Stamm-Wilbrandt, Aug 17 2015
-(n-5)*(n+5)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jan 24 2018
E.g.f.: BesselI(5,2*x) * exp(2*x). - Ilya Gutkovskiy, Jun 27 2019
From Amiram Eldar, Aug 27 2022: (Start)
Sum_{n>=5} 1/a(n) = 6169/840 - 31*Pi/(9*sqrt(3)).
Sum_{n>=5} (-1)^(n+1)/a(n) = 5254*log(phi)/(5*sqrt(5)) - 63059/280, where phi is the golden ratio (A001622). (End)
G.f.: 2F1([11/2,6],[11],4*x). - Karol A. Penson, Apr 24 2024

A004312 Binomial coefficient C(2n,n-6).

Original entry on oeis.org

1, 14, 120, 816, 4845, 26334, 134596, 657800, 3108105, 14307150, 64512240, 286097760, 1251677700, 5414950296, 23206929840, 98672427616, 416714805914, 1749695026860, 7309837001104, 30405943383200, 125994627894135, 520341450264090, 2142582442263900, 8799226775309880

Offset: 6

N. J. A. Sloane

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=6. - Herbert Kociemba, May 24 2004

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Seiichi Manyama, Table of n, a(n) for n = 6..1000
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 Janjic, Two Enumerative Functions
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.

Diagonal 13 of triangle A100257.

Cf. A001622.

[ Binomial(2*n,n-6): n in [6..150] ]; // Vincenzo Librandi, Apr 13 2011

Table[Binomial[2*n, n-6], {n, 6, 30}] (* Amiram Eldar, Aug 27 2022 *)

a(n)=binomial(2*n,n-6) \\ Charles R Greathouse IV, Oct 23 2023

G.f.: ((1/(sqrt(1-4*x)*x)-(1-sqrt(1-4*x))/(2*x^2))*x)/((1-sqrt(1-4*x))/(2*x)-1)^7+6/x-35/x^2+56/x^3-36/x^4+10/x^5-1/x^6. - Vladimir Kruchinin, Aug 11 2015
-(n-6)*(n+6)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jan 24 2018
E.g.f.: BesselI(6,2*x) * exp(2*x). - Ilya Gutkovskiy, Jun 27 2019
From Amiram Eldar, Aug 27 2022: (Start)
Sum_{n>=6} 1/a(n) = 2*Pi/(9*sqrt(3)) + 1709/2520.
Sum_{n>=6} (-1)^n/a(n) = 16636*log(phi)/(5*sqrt(5)) - 1802033/2520, where phi is the golden ratio (A001622). (End)

A004313 a(n) = binomial coefficient C(2n, n-7).

Original entry on oeis.org

1, 16, 153, 1140, 7315, 42504, 230230, 1184040, 5852925, 28048800, 131128140, 600805296, 2707475148, 12033222880, 52860229080, 229911617056, 991493848554, 4244421484512, 18053528883775, 76360380541900, 321387366339585, 1346766106565880, 5621728217559090

Offset: 7

N. J. A. Sloane

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Seiichi Manyama, Table of n, a(n) for n = 7..1000
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 Janjic, Two Enumerative Functions
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.

Diagonal 15 of triangle A100257.

Cf. A001622.

[ Binomial(2*n,n-7): n in [7..150] ]; // Vincenzo Librandi, Apr 13 2011

Table[Binomial[2n,n-7],{n,7,30}] (* Harvey P. Dale, Nov 27 2013 *)

a(n)=binomial(2*n,n-7) \\ Charles R Greathouse IV, Oct 23 2023

-(n-7)*(n+7)*a(n) + 2*n*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Jan 24 2018
E.g.f.: BesselI(7,2*x)*exp(2*x). - Ilya Gutkovskiy, Jun 27 2019
From Amiram Eldar, Aug 27 2022: (Start)
Sum_{n>=7} 1/a(n) = 41*Pi/(9*sqrt(3)) - 24923/3465.
Sum_{n>=7} (-1)^(n+1)/a(n) = 51094*log(phi)/(5*sqrt(5)) - 7616722/3465, where phi is the golden ratio (A001622). (End)

A286096 Triangle read by rows giving numerators of the Fourier expansion of cos^n(x).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 4, 0, 1, 0, 10, 0, 5, 0, 1, 10, 0, 15, 0, 6, 0, 1, 0, 35, 0, 21, 0, 7, 0, 1, 35, 0, 56, 0, 28, 0, 8, 0, 1, 0, 126, 0, 84, 0, 36, 0, 9, 0, 1, 126, 0, 210, 0, 120, 0, 45, 0, 10, 0, 1, 0, 462, 0, 330, 0, 165, 0, 55, 0, 11, 0, 1, 462, 0, 792, 0, 495, 0, 220, 0, 66, 0, 12, 0, 1

Offset: 0

Landry Salle, May 02 2017

Doubling the initial term of each line and dropping the 0's transforms this triangle to the right half of Pascal's triangle (A007318).

Row sums are A011782. - Omar E. Pol, May 02 2017

			Triangle begins:
1;
0,   1;
1,   0,   1;
0,   3,   0,   1;
3,   0,   4,   0,   1;
0,  10,   0,   5,   0,   1;
10,  0,  15,   0,   6,   0,   1;
0,  35,   0,  21,   0,   7,   0,   1;
35,  0,  56,   0,  28,   0,   8,   0,   1;
0, 126,   0,  84,   0,  36,   0,   9,   0,   1;
126, 0, 210,   0, 120,   0,  45,   0,  10,   0,   1;
0, 462,   0, 330,   0, 165,   0,  55,   0,  11,   0,   1;
462, 0, 792,   0, 495,   0, 220,   0,  66,   0,  12,   0,   1;
...

Cf. A007318, A100257 (same sequence with rows reversed).

row[n_] := If[n==0, {1}, 2^(n-1)*TrigReduce[Cos[x]^n] /. Cos[Times[k_., x]] -> x^k // CoefficientList[#, x]&]; Table[row[n], {n, 0, 12}] // Flatten
(* Second program: *)
T[n_, n_] = 1; T[n_, k_] /; k == n-1 || k>n = 0; T[n_, 1] := 2 T[n-1, 0] + T[n-1, 2]; T[n_, 0] := T[n-1, 1]; T[n_, k_] /; 1 < k <= n := T[n, k] = T[n-1, k-1] + T[n-1, k+1]; T[, ] = 0; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 02 2017 *)

cos^n(x) = (1/2^(n-1)) * Sum_{k=0..n} T(n,k) * cos(k*x).

T(n,k) = T(n-1,k-1) + T(n-1,k+1) if k != 1, T(n,1) = 2*T(n-1,0) + T(n-1,2), T(n,k) = 0 if k < 0 or k > n.

cp's OEIS Frontend

A008311 Triangle of expansions of powers of x in terms of Chebyshev polynomials T_n (x).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A030054 a(n) = binomial(2n+1,n-4).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A030055 a(n) = binomial(2*n+1, n-5).

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A004311 Binomial coefficient C(2n,n-5).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A004312 Binomial coefficient C(2n,n-6).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A004313 a(n) = binomial coefficient C(2n, n-7).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A286096 Triangle read by rows giving numerators of the Fourier expansion of cos^n(x).

Views

Author