A111125 -id:A111125 - cp's OEIS frontend

A053122 Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).

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

Offset: 0

Wolfdieter Lang

Apart from signs, identical to A078812.
Another version with row-leading 0's and differing signs is given by A285072.
G.f. for row polynomials S(n,x-2) (signed triangle): 1/(1+(2-x)*z+z^2). Unsigned triangle |a(n,m)| has g.f. 1/(1-(2+x)*z+z^2) for row polynomials.
Row sums (signed triangle) A049347(n) (periodic(1,-1,0)). Row sums (unsigned triangle) A001906(n+1)=F(2*(n+1)) (even-indexed Fibonacci).
In the language of Shapiro et al. (see A053121 for the reference) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The (unsigned) column sequences are A000027, A000292, A000389, A000580, A000582, A001288 for m=0..5, resp. For m=6..23 they are A010966..(+2)..A011000 and for m=24..49 they are A017713..(+2)..A017763.
Riordan array (1/(1+x)^2,x/(1+x)^2). Inverse array is A039598. Diagonal sums have g.f. 1/(1+x^2). - Paul Barry, Mar 17 2005. Corrected by Wolfdieter Lang, Nov 13 2012.
Unsigned version is in A078812. - Philippe Deléham, Nov 05 2006
Also row n gives (except for an overall sign) coefficients of characteristic polynomial of the Cartan matrix for the root system A_n. - Roger L. Bagula, May 23 2007
From Wolfdieter Lang, Nov 13 2012: (Start)
The A-sequence for this Riordan triangle is A115141, and the Z-sequence is A115141(n+1), n>=0. For A- and Z-sequences for Riordan matrices see the W. Lang link under A006232 with details and references.
S(n,x^2-2) = sum(r(j,x^2),j=0..n) with Chebyshev's S-polynomials and r(j,x^2) := R(2*j+1,x)/x, where R(n,x) are the monic integer Chebyshv T-polynomials with coefficients given in A127672. Proof from comparing the o.g.f. of the partial sum of the r(j,x^2) polynomials (see a comment on the signed Riordan triangle A111125) with the present Riordan type o.g.f. for the row polynomials with x -> x^2.  (End)
S(n,x^2-2) = S(2*n+1,x)/x, n >= 0, from the odd part of the bisection of the o.g.f. - Wolfdieter Lang, Dec 17 2012
For a relation to a generator for the Narayana numbers A001263, see A119900, whose columns are unsigned shifted rows (or antidiagonals) of this array, referring to the tables in the example sections. - Tom Copeland, Oct 29 2014
The unsigned rows of this array are alternating rows of a mirrored A011973 and alternating shifted rows of A030528 for the Fibonacci polynomials. - Tom Copeland, Nov 04 2014
Boas-Buck type recurrence for column k >= 0 (see Aug 10 2017 comment in A046521 with references): a(n, m) =  (2*(m + 1)/(n - m))*Sum_{k = m..n-1} (-1)^(n-k)*a(k, m), with input a(n, n) = 1, and a(n,k) = 0 for n < k. - Wolfdieter Lang, Jun 03 2020
Row n gives the characteristic polynomial of the (n X n)-matrix M where M[i,j] = 2 if i = j, -1 if |i-j| = 1 and 0 otherwise. The matrix M is positive definite and has 2-condition number (cot(Pi/(2*n+2)))^2. - Jianing Song, Jun 21 2022
Also the convolution triangle of (-1)^(n+1)*n. - Peter Luschny, Oct 07 2022

			The triangle a(n,m) begins:
n\m   0    1    2     3     4     5     6    7    8  9
0:    1
1:   -2    1
2:    3   -4    1
3:   -4   10   -6     1
4:    5  -20   21    -8     1
5:   -6   35  -56    36   -10     1
6:    7  -56  126  -120    55   -12     1
7:   -8   84 -252   330  -220    78   -14    1
8:    9 -120  462  -792   715  -364   105  -16    1
9:  -10  165 -792  1716 -2002  1365  -560  136  -18  1
... Reformatted and extended by _Wolfdieter Lang_, Nov 13 2012
E.g., fourth row (n=3) {-4,10,-6,1} corresponds to the polynomial S(3,x-2) = -4+10*x-6*x^2+x^3.
From _Wolfdieter Lang_, Nov 13 2012: (Start)
Recurrence: a(5,1) = 35 = 1*5 + (-2)*(-20) -1*(10).
Recurrence from Z-sequence [-2,-1,-2,-5,...]: a(5,0) = -6 = (-2)*5 + (-1)*(-20) + (-2)*21 + (-5)*(-8) + (-14)*1.
Recurrence from A-sequence [1,-2,-1,-2,-5,...]: a(5,1) = 35 = 1*5  + (-2)*(-20) + (-1)*21 + (-2)*(-8) + (-5)*1.
(End)
E.g., the fourth row (n=3) {-4,10,-6,1} corresponds also to the polynomial S(7,x)/x = -4 + 10*x^2 - 6*x^4 + x^6. - _Wolfdieter Lang_, Dec 17 2012
Boas-Buck type recurrence: -56 = a(5, 2) = 2*(-1*1 + 1*(-6) - 1*21) = -2*28 = -56. - _Wolfdieter Lang_, Jun 03 2020

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.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 62.
Sigurdur Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S.: ISBN 0-8218-2848-7, 1978, p. 463.

T. D. Noe, Rows n=0..50 of triangle, flattened
Wolfdieter Lang, First rows of the triangle.
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].
P. Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Naiomi T. Cameron and Asamoah Nkwanta, On some (pseudo) involutions in the Riordan group, J. of Integer Sequences, 8(2005), 1-16.
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014 (p. 10). - _Tom Copeland_, Oct 11 2014
Pentti Haukkanen, Jorma Merikoski, Seppo Mustonen, Some polynomials associated with regular polygons, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.
Eric Weisstein's World of Mathematics, Cartan Matrix
Eric Weisstein's World of Mathematics, Dynkin Diagram
Index entries for sequences related to Chebyshev polynomials.

Cf. A005248, A127677, A053123, A049310.
Cf. A011973, A030528, A034867, A119900.
Cf. A285072 (version with row-leading 0's and differing signs). - Eric W. Weisstein, Apr 09 2017

seq(seq((-1)^(n+m)*binomial(n+m+1,2*m+1),m=0..n),n=0..10); # Robert Israel, Oct 15 2014
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> -(-1)^n*n); # Peter Luschny, Oct 07 2022

T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a = Join[M[1], Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a] (* Roger L. Bagula, May 23 2007 *)
(* Alternative code for the matrices from MathWorld: *)
sln[n_] := 2IdentityMatrix[n] - PadLeft[PadRight[IdentityMatrix[n - 1], {n, n - 1}], {n, n}] - PadLeft[PadRight[IdentityMatrix[n - 1], {n - 1, n}], {n, n}] (* Roger L. Bagula, May 23 2007 *)

@CachedFunction
def A053122(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    return A053122(n-1,k-1)-A053122(n-2,k)-2*A053122(n-1,k)
for n in (0..9): [A053122(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

a(n, m) := 0 if n

a(n, m) = -2*a(n-1, m) + a(n-1, m-1) - a(n-2, m), a(n, -1) := 0 =: a(-1, m), a(0, 0)=1, a(n, m) := 0 if n

O.g.f. for m-th column (signed triangle): ((x/(1+x)^2)^m)/(1+x)^2.

From Jianing Song, Jun 21 2022: (Start)

T(n,k) = [x^k]f_n(x), where f_{-1}(x) = 0, f_0(x) = 1, f_n(x) = (x-2)*f_{n-1}(x) - f_{n-2}(x) for n >= 2.

f_n(x) = (((x-2+sqrt(x^2-4*x))/2)^(n+1) - ((x-2-sqrt(x^2-4*x))/2)^(n+1))/sqrt(x^2-4x).

The roots of f_n(x) are 2 + 2*cos(k*Pi/(n+1)) = 4*(cos(k*Pi/(2*n+2)))^2 for 1 <= k <= n. (End)

A049666 a(n) = Fibonacci(5*n)/5.

Original entry on oeis.org

0, 1, 11, 122, 1353, 15005, 166408, 1845493, 20466831, 226980634, 2517253805, 27916772489, 309601751184, 3433536035513, 38078498141827, 422297015595610, 4683345669693537, 51939099382224517, 576013438874163224

Offset: 0

Clark Kimberling

For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010
For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 11's along the main diagonal and 1's along the subdiagonal and the superdiagonal. - John M. Campbell, Jul 08 2011
For n >= 1, a(n) equals the number of words of length n-1 on alphabet {0,1,...,11} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
For n >= 1, a(n) equals the denominator of the continued fraction [11, 11, ..., 11] (with n copies of 11). The numerator of that continued fraction is a(n+1). - Greg Dresden and Shaoxiong Yuan, Jul 26 2019
From Michael A. Allen, Mar 30 2023: (Start)
Also called the 11-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 11 kinds of squares available. (End)

			G.f. = x + 11*x^2 + 122*x^3 + 1353*x^4 + 15005*x^5 + 166408*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..950
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
Tanya Khovanova, Recursive Sequences
Kai Wang, On k-Fibonacci Sequences And Infinite Series List of Results and Examples, 2020.
Shaoxiong Yuan, Generalized Identities of Certain Continued Fractions, arXiv:1907.12459 [math.NT], 2019.
Index entries for linear recurrences with constant coefficients, signature (11,1).

A column of array A028412.
Cf. A000045, A102312, A243399.
Row n=11 of A073133, A172236 and A352361, and column k=11 of A157103.

[Fibonacci(5*n)/5: n in [0..30]]; // G. C. Greubel, Dec 02 2017

A049666 := proc(n)
    combinat[fibonacci](5*n)/5 ;
end proc: # R. J. Mathar, May 07 2024

Table[Fibonacci[5*n]/5, {n, 0, 100}] (* T. D. Noe, Oct 29 2009 *)
a[ n_] := Fibonacci[n, 11]; (* Michael Somos, May 28 2014 *)

numlib::fibonacci(5*n)/5 $ n = 0..25; // Zerinvary Lajos, May 09 2008

a(n)=fibonacci(5*n)/5 \\ Charles R Greathouse IV, Feb 03 2014

from sage.combinat.sloane_functions import recur_gen3
it = recur_gen3(0,1,11,11,1,0)
[next(it) for i in range(1,22)] # Zerinvary Lajos, Jul 09 2008

[lucas_number1(n,11,-1) for n in range(0, 19)] # Zerinvary Lajos, Apr 27 2009

[fibonacci(5*n)/5 for n in range(0, 19)] # Zerinvary Lajos, May 15 2009

G.f.: x/(1 - 11*x - x^2).
a(n) = A102312(n)/5.
a(n) = 11*a(n-1) + a(n-2) for n > 1, a(0)=0, a(1)=1. With a=golden ratio and b=1-a, a(n) = (a^(5n)-b^(5n))/(5*sqrt(5)). - Mario Catalani (mario.catalani(AT)unito.it), Jul 24 2003
a(n) = F(n, 11), the n-th Fibonacci polynomial evaluated at x=11. - T. D. Noe, Jan 19 2006
a(n) = ((11+sqrt(125))^n-(11-sqrt(125))^n)/(2^n*sqrt(125)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n) = 11*A049670(n), a(2n+1) = A097843(n).
a(3n+1) = A041227(5n), a(3n+2) = A041227(5n+3), a(3n+3) = 2*A041227(5n+4).
Limit_{k->oo} a(n+k)/a(k) = (A001946(n) + A049666(n)*sqrt(125))/2.
Limit_{n->oo} A001946(n)/A049666(n) = sqrt(125).
(End)
a(n) = F(n) + (-1)^n*5*F(n)^3 + 5*F(n)^5, n >= 0. See the D. Jennings formula given in a comment on A111125, where also the reference is given. - Wolfdieter Lang, Aug 31 2012
a(-n) = -(-1)^n * a(n). - Michael Somos, May 28 2014
E.g.f.: (exp((1/2)*(11-5*sqrt(5))*x)*(-1 + exp(5*sqrt(5)*x)))/(5*sqrt(5)). - Stefano Spezia, Aug 02 2019

A111418 Right-hand side of odd-numbered rows of Pascal's triangle.

Original entry on oeis.org

1, 3, 1, 10, 5, 1, 35, 21, 7, 1, 126, 84, 36, 9, 1, 462, 330, 165, 55, 11, 1, 1716, 1287, 715, 286, 78, 13, 1, 6435, 5005, 3003, 1365, 455, 105, 15, 1, 24310, 19448, 12376, 6188, 2380, 680, 136, 17, 1, 92378, 75582, 50388

Offset: 0

Philippe Deléham, Nov 13 2005

Riordan array (c(x)/sqrt(1-4*x),x*c(x)^2) where c(x) is g.f. of A000108. Unsigned version of A113187. Diagonal sums are A014301(n+1).
Triangle T(n,k),0<=k<=n, read by rows defined by :T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=3*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+2*T(n-1,k)+T(n-1,k+1) for k>=1. - Philippe Deléham, Mar 22 2007
Reversal of A122366. - Philippe Deléham, Mar 22 2007
Column k has e.g.f. exp(2x)(Bessel_I(k,2x)+Bessel_I(k+1,2x)). - Paul Barry, Jun 06 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
Diagonal sums are A014301(n+1). - Paul Barry, Mar 08 2011
This triangle T(n,k) appears in the expansion of odd powers of Fibonacci numbers F=A000045 in terms of F-numbers with multiples of odd numbers as indices. See the Ozeki reference, p. 108, Lemma 2. The formula is: F_l^(2*n+1) = sum(T(n,k)*(-1)^((n-k)*(l+1))* F_{(2*k+1)*l}, k=0..n)/5^n, n >= 0, l >= 0. - Wolfdieter Lang, Aug 24 2012
Central terms give A052203. - Reinhard Zumkeller, Mar 14 2014
This triangle appears in the expansion of (4*x)^n in terms of the polynomials Todd(n, x):= T(2*n+1, sqrt(x))/sqrt(x) = sum(A084930(n,m)*x^m), n >= 0. This follows from the inversion of the lower triangular Riordan matrix built from A084930 and comparing the g.f. of the row polynomials. - Wolfdieter Lang, Aug 05 2014
From Wolfdieter Lang, Aug 15 2014: (Start)
This triangle is the inverse of the signed Riordan triangle (-1)^(n-m)*A111125(n,m).
This triangle T(n,k) appears in the expansion of x^n in terms of the polynomials todd(k, x):= T(2*k+1, sqrt(x)/2)/(sqrt(x)/2) = S(k, x-2) - S(k-1, x-2) with the row polynomials T and S for the triangles A053120 and A049310, respectively: x^n = sum(T(n,k)*todd(k, x), k=0..n). Compare this with the preceding comment.
The A- and Z-sequences for this Riordan triangle are [1, 2, 1, repeated 0] and [3, 1, repeated 0]. For A- and Z-sequences for Riordan triangles see the W. Lang link under A006232. This corresponds to the recurrences given in the Philippe Deléham, Mar 22 2007 comment above. (End)

			From _Wolfdieter Lang_, Aug 05 2014: (Start)
The triangle T(n,k) begins:
n\k      0      1      2      3     4     5    6    7   8  9  10 ...
0:       1
1:       3      1
2:      10      5      1
3:      35     21      7      1
4:     126     84     36      9     1
5:     462    330    165     55    11     1
6:    1716   1287    715    286    78    13    1
7:    6435   5005   3003   1365   455   105   15    1
8:   24310  19448  12376   6188  2380   680  136   17   1
9:   92378  75582  50388  27132 11628  3876  969  171  19  1
10: 352716 293930 203490 116280 54264 20349 5985 1330 210 21   1
...
Expansion examples (for the Todd polynomials see A084930 and a comment above):
(4*x)^2 = 10*Todd(n,  0) + 5*Todd(n, 1) + 1*Todd(n, 2) = 10*1 + 5*(-3 + 4*x) + 1*(5 - 20*x + 16*x^2).
(4*x)^3 =  35*1 + 21*(-3 + 4*x) + 7*(5 - 20*x + 16*x^2) + (-7 + 56*x - 112*x^2 +64*x^3)*1. (End)
---------------------------------------------------------------------
Production matrix is
3, 1,
1, 2, 1,
0, 1, 2, 1,
0, 0, 1, 2, 1,
0, 0, 0, 1, 2, 1,
0, 0, 0, 0, 1, 2, 1,
0, 0, 0, 0, 0, 1, 2, 1,
0, 0, 0, 0, 0, 0, 1, 2, 1,
0, 0, 0, 0, 0, 0, 0, 1, 2, 1
- _Paul Barry_, Mar 08 2011
Application to odd powers of Fibonacci numbers F, row n=2:
F_l^5 = (10*(-1)^(2*(l+1))*F_l + 5*(-1)^(1*(l+1))*F_{3*l} + 1*F_{5*l})/5^2, l >= 0. - _Wolfdieter Lang_, Aug 24 2012

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Paul Barry, On the Hurwitz Transform of Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.
E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Article 12.3.3, 2012.
A. Nkwanta, A. Tefera, Curious Relations and Identities Involving the Catalan Generating Function and Numbers, Journal of Integer Sequences, 16 (2013), #13.9.5.
K. Ozeki, On Melham's sum, The Fibonacci Quart. 46/47 (2008/2009), no. 2, 107-110.
Sun, Yidong; Ma, Luping Minors of a class of Riordan arrays related to weighted partial Motzkin paths. Eur. J. Comb. 39, 157-169 (2014), Table 2.2.

Cf. A000108, A113187.

Columns are : A001700, A002054, A003516, A030053, A030054, A030055, A030056.

a111418 n k = a111418_tabl !! n !! k
a111418_row n = a111418_tabl !! n
a111418_tabl = map reverse a122366_tabl
-- Reinhard Zumkeller, Mar 14 2014

Table[Binomial[2*n+1, n-k], {n,0,10}, {k,0,n}] (* G. C. Greubel, May 22 2017 *)
T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0,
T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 3, 2], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)

T(n, k) = C(2*n+1, n-k).
Sum_{k=0..n} T(n, k) = 4^n.
Sum_{k, 0<=k<=n}(-1)^k *T(n,k) = binomial(2*n,n) = A000984(n). - Philippe Deléham, Mar 22 2007
T(n,k) = sum{j=k..n, C(n,j)*2^(n-j)*C(j,floor((j-k)/2))}. - Paul Barry, Jun 06 2007
Sum_{k, k>=0} T(m,k)*T(n,k) = T(m+n,0)= A001700(m+n). - Philippe Deléham, Nov 22 2009
G.f. row polynomials: ((1+x) - (1-x)/sqrt(1-4*z))/(2*(x - (1+x)^2*z))
(see the Riordan property mentioned in a comment above). - Wolfdieter Lang, Aug 05 2014

A263916 Coefficients of the Faber partition polynomials.

Original entry on oeis.org

-1, -2, 1, -3, 3, -1, -4, 4, 2, -4, 1, -5, 5, 5, -5, -5, 5, -1, -6, 6, 6, -6, 3, -12, 6, -2, 9, -6, 1, -7, 7, 7, -7, 7, -14, 7, -7, -7, 21, -7, 7, -14, 7, -1, -8, 8, 8, -8, 8, -16, 8, 4, -16, -8, 24, -8, -8, 12, 24, -32, 8, 2, -16, 20, -8, 1

Offset: 1

Tom Copeland, Oct 29 2015

The coefficients of the Faber polynomials F(n,b(1),b(2),...,b(n)) (Bouali, p. 52) in the order of the partitions of Abramowitz and Stegun. Compare with A115131 and A210258.
These polynomials occur in discussions of the Virasoro algebra, univalent function spaces and the Schwarzian derivative, symmetric functions, and free probability theory. They are intimately related to symmetric functions, free probability, and Appell sequences through the raising operator R = x - d log(H(D))/dD for the Appell sequence inverse pair associated to the e.g.f.s H(t)e^(xt) (cf. A094587) and (1/H(t))e^(xt) with H(0)=1.
Instances of the Faber polynomials occur in discussions of modular invariants and modular functions in the papers by Asai, Kaneko, and Ninomiya, by Ono and Rolen, and by Zagier. - Tom Copeland, Aug 13 2019
The Faber polynomials, denoted by s_n(a(t)) where a(t) is a formal power series defined by a product formula, are implicitly defined by equation 13.4 on p. 62 of Hazewinkel so as to extract the power sums of the reciprocals of the zeros of a(t). This is the Newton identity expressing the power sum symmetric polynomials in terms of the elementary symmetric polynomials/functions. - Tom Copeland, Jun 06 2020
From Tom Copeland, Oct 15 2020: (Start)
With a_n = n! * b_n = (n-1)! * c_n for n  > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
B) ordinary generating function (o.g.f.), or formal power series: f(x)  = 1/(1-b.x) = 1 + Sum_{n > 0}  b_n * x^n
C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0}  c_n * x^n /n.
Expansions of log(f(x)) are given in
I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] =  e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
II) A263916 for the o.g.f.: log[ 1/(1-b.x) ] =  log[ 1 - F.(b_1,b_2,...)x ] = -Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
Expansions of exp(f(x)-1) are given in
III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
IV) A130561 for an o.g.f.: exp[ b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
V) A036039 for an l.g.f.: exp[ -log(1-c.x) ] =  e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)

			F(1,b1) = - b1
F(2,b1,b2) = -2 b2 + b1^2
F(3,b1,b2,b3) = -3 b3 + 3 b1 b2 - b1^3
F(4,b1,...) = -4 b4 + 4 b1 b3 + 2 b2^2  - 4 b1^2 b2 + b1^4
F(5,...) = -5 b5 + 5 b1 b4 + 5 b2 b3 - 5 b1^2 b3 - 5 b1 b2^2 + 5 b1^3 b2 - b1^5
------------------------------
IF(1,b1) = -b1
IF(2,b1,,b2) = -b2 + b1^2
IF(3,b1,b2,b3) = -2 b3 + 3 b1 b2 - b1^3
IF(4,b1,...) = -6 b4 + 8 b1 b3 + 3 b2^2  - 6 b1^2 b2 + b1^4
IF(5,...) = -24 b5 + 30 b1 b4 + 20 b2 b3 - 20 b1^2 b3 - 15 b1 b2^2 + 10 b1^3 b2 - b1^5
------------------------------
For 1/(1+x)^2 = 1- 2x + 3x^2 - 4x^3 + 5x^4 - ..., F(n,-2,3,-4,...) = (-1)^(n+1) 2.
------------------------------
F(n,x,2x,...,nx), F(n,-x,2x,-3x,...,(-1)^n n*x), and F(n,(2-x),1,0,0,...) are related to the Chebyshev polynomials through A127677 and A111125. See also A110162, A156308, A208513, A217476, and A220668.
------------------------------
For b1 = p, b2 = q, and all other indeterminates 0, see A113279 and A034807.
For b1 = -y, b2 = 1 and all other indeterminates 0, see A127672.

H. Airault, "Symmetric sums associated to the factorization of Grunsky coefficients," in Groups and Symmetries: From Neolithic Scots to John McKay, CRM Proceedings and Lecture Notes: Vol. 47, edited by J. Harnad and P. Winternitz, American Mathematical Society, 2009.
D. Bleeker and B. Booss, Index Theory with Applications to Mathematics and Physics, International Press, 2013, (see section 16.7 Characteristic Classes and Curvature).
M. Hazewinkel, Formal Groups and Applications, Academic Press, New York San Francisco London, 1978, p. 120.
F. Hirzebruch, Topological methods in algebraic geometry. Second, corrected printing of the third edition. Die Grundlehren der Mathematischen Wissenschaften, Band 131 Springer-Verlag, Berlin Heidelberg New York, 1978, p. 11 and 92.
D. Knutson, λ-Rings and the Representation Theory of the Symmetric Group, Lect. Notes in Math. 308, Springer-Verlag, 1973, p. 35.
D. Yau, Lambda-Rings, World Scientific Publishing Co., Singapore, 2010, p. 45.

Vincenzo Librandi, Table of n, a(n) for n = 1..5762
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. Airault, Remarks on Faber polynomials, International Mathematical Forum, 3, no. 9, 2008, pages 449-456.
H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bulletin des Sciences Mathématiques, Volume 130, Issue 3, April-May 2006, pages 179-222.
H. Airault and Y. Neretin, On the action of the Virasoro algebra on the space of univalent functions, arXiv:0704.2149 [math.RT], 2007.
F. Ardila, Algebraic and geometric methods in enumerative combinatorics, arXiv:1409.2562 [math.CO], 2015.
T. Asai, M. Kaneko, and H. Ninomiya, Zeros of certain modular functions and an application, 1997.
A. Bouali, Faber polynomials Cayley-Hamilton equation and Newton symmetric functions, Bulletin des Sciences Mathématiques, Volume 130, Issue 1, Jan-Feb 2006, pages 49-70.
J. M. Campbell, Combinatorial interpretations of primitivity in the algebra of symmetric functions, Sarajevo J. Math., 17(30) (2021), 151--165.
P. Cartier, Mathemagics: A tribute to L. Euler and R. Feynman, Séminaire Lotharingien de Combinatoire 44 (2000), Article B44d, 2000, p. 53.
P. Cartier, A primer of Hopf algebras, preprint, Institut des Hautes Etudes Scientifiques, France, 2006, pp. 56 and 57.
V. Chan, Topological K-theory of complex projective spaces, senior's thesis UC Davis (p. 11 on Chern characters of complex vector bundles), 2013.
Gi-Sang Cheon, Hana Kim, and Louis W. Shapiro, An algebraic structure for Faber polynomials, Lin. Alg. Applic. 433 (2010) 1170-1179.
Tom Copeland, Lagrange a la Lah, 2011.
Tom Copeland, Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants, MathOverflow question, 2012.
Tom Copeland, Addendum to Elliptic Lie Triad, 2015.
Tom Copeland, Connection between the Chebyshev polynomials and the Faber polynomials, MathOverflow question, 2015.
Tom Copeland, The Faber Appells, 2020.
Tom Copeland, Appells and Roses: Newton, Leibniz, Euler, Riemann and Symmetric Polynomials, 2020.
A. Dress and C. Siebeneicher, The Burnside ring of the infinite cyclic group and its relations to the necklace algebra, lambda-rings, and the universal ring of Witt vectors, Adv. in Math., Vol. 78, Issue 1, Nov. 1989, pages 1-41.
D. Dugger, A Geometric Introduction to K-Theory, p. 288 (called Newton polynomials).
M. Eiermann and R. Varga, Zeros and local extreme points of Faber polynomials associated with hypocycloidal domains, Elect. Trans. on Numer. Analysis, Vol. 1, p. 49-71, 1993.
H. Figueroa and J. Gracia-Bondia, Combinatorial Hopf algebras in quantum field theory I, arXiv:0408145 [hep-th], 2005, (normalized versions on pp. 42 and 78, denoted as Schur polynomials).
R. Friedrich and J. McKay, Formal groups, Witt vectors and free probability, arXiv:1204.6522 [math.OA], 2012.
A. Hatcher, Vector Bundles and K-Theory, Version 2.2, 2017, p. 63.
M. Hazewinkel, Three lectures on formal group laws, Canadian Mathematical Society Conference Proceedings, Vol. 5, p. 51-67, 1986.
Y. He, The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning, arXiv:1812.02893 [hep-th], 2018, p. 146.
O. Iena, On symbolic computations with Chern classes: Remarks on the library CHERN.LIB for SINGULAR, p. 6.
B. Konopelchenko, Quantum deformations of associative algebras and integrable systems, arXiv:0802.3022 [nlin.SI], 2008.
R. Lu, Regularized equivariant Euler classes and gamma functions, Ph.D. thesis, Dept. of Pure Mathematics, University of Adelaide, 2008, p. 86.
K. Maslanka, Effective method of computing Li’S coefficients and their propertie, arXiv:0402168v5 [math.NT], 2004, p. 6.
MathOverflow, Canonical reference for Chern characteristic classes, a question posed by Tom Copeland, 2019.
J. McKay and A. Sebbar, On replicable functions: an introduction, Frontiers in Number Theory, Physics, and Geometry II, pp. 373-386.
L. Nicolaescu, Lectures on the Geometry of Manifolds, p.337, 2018.
J. Novak and M. LaCroix, Three lectures on free probability, arXiv:1205.2097 [math.CO], 2012.
K. Ono and L. Rolen, On Witten's extremal partition functions , arXiv:1807.00444 [math.NT], 2019.
T. Takebe, Lee-Peng Teo, and A. Zabrodin, Löwner equations and dispersionless hierarchies, arXiv:math/0605161 [math.CV], p. 24, 2006.
Lee-Peng Teo, Analytic functions and integrable hierarchies-characterization of tau functions, Letters in Mathematical Physics, Vol. 64, Issue 1, Apr 2003, pp. 75-92 (also arXiv:hep-th/0305005, 2003).
Wikipedia, Newton identities.
D. Zagier, Traces of singular moduli, 2011.

Cf. Row sums of absolute values: A000225.
Cf. A034807, A036039, A036040, A094587, A110162, A111125, A113279, A115131.
Cf. A127671, A127672, A127677, A130561, A133314, A156308, A208513, A210258.
Cf. A217476, A220668.
Cf. A263634.

F[0] = 1; F[1] = -b[1]; F[2] = b[1]^2 - 2 b[2]; F[n_] := F[n] = -b[1] F[n - 1] - Sum[b[n - k] F[k], {k, 1, n - 2}] - n b[n] // Expand;
row[n_] := (List @@ F[n]) /. b[_] -> 1 // Reverse;
Table[row[n], {n, 1, 8}] // Flatten // Rest (* Jean-François Alcover, Jun 12 2017 *)

-log(1 + b(1) x + b(2) x^2 + ...) = Sum_{n>=1} F(n,b(1),...,b(n)) * x^n/n.
-d(1 + b(1) x + b(2) x^2 + ...)/dx / (1 + b(1) x + b(2) x^2 + ...) = Sum_{n>=1} F(n,b(1),...,b(n)) x^(n-1).
F(n,b(1),...,b(n)) = -n*b(n) - Sum_{k=1..n-1} b(n-k)*F(k,b(1),...,b(k)).
Umbrally, with B(x) = 1 + b(1) x + b(2) x^2 + ..., B(x) = exp[log(1-F.x)] and 1/B(x) = exp[-log(1-F.x)], establishing a connection to the e.g.f. of A036039 and the symmetric polynomials.
The Stirling partition polynomials of the first kind St1(n,b1,b2,...,bn;-1) = IF(n,b1,b2,...,bn) (cf. the Copeland link Lagrange a la Lah, signed A036039, and p. 184 of Airault and Bouali), i.e., the cyclic partition polynomials for the symmetric groups, and the Faber polynomials form an inverse pair for isolating the indeterminates in their definition, for example, F(3,IF(1,b1),IF(2,b1,b2)/2!,IF(3,b1,b2,b3)/3!)= b3, with bk = b(k), and IF(3,F(1,b1),F(2,b1,b2),F(3,b1,b2,b3))/3!= b3.
The polynomials specialize to F(n,t,t,...) = (1-t)^n - 1.
See Newton Identities on Wikipedia on relation between the power sum symmetric polynomials and the complete homogeneous and elementary symmetric polynomials for an expression in multinomials for the coefficients of the Faber polynomials.
(n-1)! F(n,x[1],x[2]/2!,...,x[n]/n!) = - p_n(x[1],...,x[n]), where p_n are the cumulants of A127671 expressed in terms of the moments x[n]. - Tom Copeland, Nov 17 2015
-(n-1)! F(n,B(1,x[1]),B(2,x[1],x[2])/2!,...,B(n,x[1],...,x[n])/n!) = x[n] provides an extraction of the indeterminates of the complete Bell partition polynomials B(n,x[1],...,x[n]) of A036040. Conversely, IF(n,-x[1],-x[2],-x[3]/2!,...,-x[n]/(n-1)!) = B(n,x[1],...,x[n]). - Tom Copeland, Nov 29 2015
For a square matrix M, determinant(I - x M) = exp[-Sum_{k>0} (trace(M^k) x^k / k)] = Sum_{n>0} [ P_n(-trace(M),-trace(M^2),...,-trace(M^n)) x^n/n! ] = 1 + Sum_{n>0} (d[n] x^n), where P_n(x[1],...,x[n]) are the cycle index partition polynomials of A036039 and d[n] = P_n(-trace(M),-trace(M^2),...,-trace(M^n)) / n!. Umbrally, det(I - x M)= exp[log(1 - b. x)] = exp[P.(-b_1,..,-b_n)x] = 1 / (1-d.x), where b_k = tr(M^k). Then F(n,d[1],...,d[n]) = tr[M^n]. - Tom Copeland, Dec 04 2015
Given f(x) = -log(g(x)) = -log(1 + b(1) x + b(2) x^2 + ...) = Sum_{n>=1} F(n,b(1),...,b(n)) * x^n/n, action on u_n = F(n,b(1),...,b(n)) with A133932 gives the compositional inverse finv(x) of f(x), with F(1,b(1)) not equal to zero, and f(g(finv(x))) = f(e^(-x)). Note also that exp(f(x)) = 1 / g(x) = exp[Sum_{n>=1} F(n,b(1),...,b(n)) * x^n/n] implies relations among A036040, A133314, A036039, and the Faber polynomials. - Tom Copeland, Dec 16 2015
The Dress and Siebeneicher paper gives combinatorial interpretations and various relations that the Faber polynomials must satisfy for integral values of its arguments. E.g., Eqn. (1.2) p. 2 implies [2 * F(1,-1) + F(2,-1,b2) + F(4,-1,b2,b3,b4)]  mod(4) = 0. This equation implies that [F(n,b1,b2,...,bn)-(-b1)^n] mod(n) = 0 for n prime. - Tom Copeland, Feb 01 2016
With the elementary Schur polynomials S(n,a_1,a_2,...,a_n) = Lah(n,a_1,a_2,...,a_n) / n!, where Lah(n,...) are the refined Lah polynomials of A130561, F(n,S(1,a_1),S(2,a_1,a_2),...,S(n,a_1,...,a_n)) = -n * a_n since sum_{n > 0} a_n x^n = log[sum{n >= 0} S(n,a_1,...,a_n) x^n]. Conversely, S(n,-F(1,a_1),-F(2,a_1,a_2)/2,...,-F(n,a_1,...,a_n)/n) = a_n. - Tom Copeland, Sep 07 2016
See Corollary 3.1.3 on p. 38 of Ardila and Copeland's two MathOverflow links to relate the Faber polynomials, with arguments being the signed elementary symmetric polynomials, to the logarithm of determinants, traces of powers of an adjacency matrix, and number of walks on graphs. - Tom Copeland, Jan 02 2017
The umbral inverse polynomials IF appear on p. 19 of Konopelchenko as partial differential operators. - Tom Copeland, Nov 19 2018

More terms from Jean-François Alcover, Jun 12 2017

A087960 a(n) = (-1)^binomial(n+1,2).

Original entry on oeis.org

1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1

Offset: 0

W. Edwin Clark, Sep 17 2003

Period 4: repeat [1, -1, -1, 1]. - Joerg Arndt, Feb 14 2016

Also equal to the sign of product(j-i, 1<=j

Hankel transform of A097331, A097332. [Paul Barry, Aug 10 2009]

The Kn22 sums, see A180662, of triangle A108299 equal the terms of this sequence. [Johannes W. Meijer, Aug 14 2011]

			a(1) = -1 since (-1)^binomial(2,2) = (-1)^1 = -1.
G.f. = 1 - x - x^2 + x^3 + x^4 - x^5 - x^6 + x^7 + x^8 - x^9 - x^10 + ...

I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.

Gary Detlefs and Wolfdieter Lang, Improved Formula for the Multi-Section of the Linear Three-Term Recurrence Sequence, arXiv:2304.12937 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A000217, A021913, A057077, A097331, A097332, A108299, A111125, A180662.

a087960 n = (-1) ^ (n * (n + 1) `div` 2)
a087960_list = cycle [1,-1,-1,1]  -- Reinhard Zumkeller, Nov 15 2015

[(-1)^Binomial(n+1,2) : n in [0..100]]; // Wesley Ivan Hurt, Jul 07 2016

A087960:=n->(-1)^binomial(n+1,2): seq(A087960(n), n=0..100); # Wesley Ivan Hurt, Jul 07 2016

(-1)^Binomial[Range[0,110],2] (* or *) LinearRecurrence[{0,-1},{1,1},110] (* Harvey P. Dale, Jul 07 2014 *)
a[ n_] := (-1)^(n (n + 1) / 2); (* Michael Somos, Jul 20 2015 *)
a[ n_] := (-1)^Quotient[ n + 1, 2]; (* Michael Somos, Jul 20 2015 *)

{a(n) = (-1)^((n + 1)\2)}; /* Michael Somos, Jul 20 2015 */

def A087960(n): return -1 if n+1&2 else 1 # Chai Wah Wu, Jan 31 2023

a(n) = (-1)^A000217(n).
a(n) = (-1)^floor((n+1)/2). - Benoit Cloitre and Ray Chandler, Sep 19 2003
G.f.: (1-x)/(1+x^2). - Paul Barry, Aug 10 2009
a(n) = I^(n*(n+1)). - Bruno Berselli, Oct 17 2011
a(n) = Product_{k=1..n} 2*cos(2*k*Pi/(2*n+1)) for n>=0 (for n=0 the empty product is put to 1). See the Gradstein-Ryshik reference, p. 63, 1.396 2. with x = sqrt(-1). - Wolfdieter Lang, Oct 22 2013
a(n) + a(n-2) = 0 for n>1, a(n) = a(n-4) for n>3. - Wesley Ivan Hurt, Jul 07 2016
E.g.f.: cos(x) - sin(x). - Ilya Gutkovskiy, Jul 07 2016
a(n) = Sum_{s=0..n} (-1)^(n-s)*A111125(n, s)*2^s (row polynomials of signed A111125 evaluated at 2). - Wolfdieter Lang, May 02 2021

More terms from Benoit Cloitre and Ray Chandler, Sep 19 2003

Offset and Vandermonde formula corrected by R. J. Mathar, Sep 25 2009

A050486 a(n) = binomial(n+6,6)*(2n+7)/7.

Original entry on oeis.org

1, 9, 44, 156, 450, 1122, 2508, 5148, 9867, 17875, 30888, 51272, 82212, 127908, 193800, 286824, 415701, 591261, 826804, 1138500, 1545830, 2072070, 2744820, 3596580, 4665375, 5995431, 7637904, 9651664, 12104136, 15072200, 18643152, 22915728, 28001193

Offset: 0

Barry E. Williams, Dec 26 1999

If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-8) is the number of 8-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
7-dimensional square numbers, sixth partial sums of binomial transform of [1,2,0,0,0,...]. a(n) = Sum_{i=0..n} C(n+6,i+6)*b(i), where b(i) = [1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
2*a(n) is number of ways to place 6 queens on an (n+6) X (n+6) chessboard so that they diagonally attack each other exactly 15 times. The maximal possible attack number, p=binomial(k,2)=15 for k=6 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form a corresponding complete graph. - Antal Pinter, Dec 27 2015
Coefficients in the terminating series identity 1 - 9*n/(n + 8) + 44*n*(n - 1)/((n + 8)*(n + 9)) - 156*n*(n - 1)*(n - 2)/((n + 8)*(n + 9)*(n + 10)) + ... = 0 for n = 1,2,3,.... Cf. A005585 and A053347. - Peter Bala, Feb 18 2019

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 15, 17.
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.

Partial sums of A040977, A005585.
Fourth column (s=3, without leading zeros) of A111125. - Wolfdieter Lang, Oct 18 2012
Cf. A084960 (unsigned fourth column divided by 64). - Wolfdieter Lang, Aug 04 2014
Cf. A053120, A084930.

[Binomial(n+6, 6) + 2*Binomial(n+6, 7): n in [0..35]]; // Vincenzo Librandi, Jun 09 2013

A050486:=n->binomial(n+6,6)*(2*n+7)/7: seq(A050486(n), n=0..50); # Wesley Ivan Hurt, Jan 01 2016

CoefficientList[Series[(1 + x) / (1 - x)^8, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)
Table[SeriesCoefficient[(1 + x)/(1 - x)^8, {x, 0, n}], {n, 0, 28}] (* or *)
Table[Binomial[n + 6, 6] (2 n + 7)/7, {n, 0, 30}] (* Michael De Vlieger, Dec 31 2015 *)

a(n)=binomial(n+6,6)*(2*n+7)/7 \\ Charles R Greathouse IV, Sep 24 2015

A050486_list, m = [], [2]+[1]*7
for _ in range(10**2):
    A050486_list.append(m[-1])
    for i in range(7):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

a(n) = (-1)^n*A053120(2*n+7, 7)/64 (1/64 of eighth unsigned column of Chebyshev T-triangle, zeros omitted).
G.f.: (1+x)/(1-x)^8.
a(n) = 2*C(n+7, 7)-C(n+6, 6). - Paul Barry, Mar 04 2003
a(n) = C(n+6,6)+2*C(n+6,7). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
a(n) = (-1)^n*A084930(n+3, 3)/64. Compare with the first line above. - Wolfdieter Lang, Aug 04 2014
a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8) for n>7. - Wesley Ivan Hurt, Jan 01 2016
From Amiram Eldar, Jan 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 24871/25 - 7168*log(2)/5.
Sum_{n>=0} (-1)^n/a(n) = 1792*Pi/5 - 28126/25. (End)

A127677 Scaled coefficient table for Chebyshev polynomials 2T(2n, sqrt(x)/2) (increasing even scaled powers, without zero entries).

Original entry on oeis.org

2, -2, 1, 2, -4, 1, -2, 9, -6, 1, 2, -16, 20, -8, 1, -2, 25, -50, 35, -10, 1, 2, -36, 105, -112, 54, -12, 1, -2, 49, -196, 294, -210, 77, -14, 1, 2, -64, 336, -672, 660, -352, 104, -16, 1, -2, 81, -540, 1386, -1782, 1287, -546, 135, -18, 1, 2, -100, 825, -2640, 4290, -4004, 2275, -800, 170, -20, 1

Offset: 0

Wolfdieter Lang, Mar 07 2007

2*T(2*n,x) = Sum_{m=0..n} a(n,m)*(2*x)^(2*m).
Closely related to A284982, which has opposite signs and rows begin with 0 of alternating signs instead of +/2. - Eric W. Weisstein, Apr 07 2017
Bisection triangle of A127672 (without zero entries, even part). The odd part is ((-1)^(n-m))*A111125(n,m).
If the leading 2 is replaced by a 1 we get the essentially identical sequence A110162. - N. J. A. Sloane, Jun 09 2007
Also row n gives coefficients of characteristic polynomial of the Cartan matrix for the root system B_n (or, equally, C_n). - Roger L. Bagula, May 23 2007
From Wolfdieter Lang, Oct 04 2013: (Start)
This triangle a(n,m) is used to express the length ratio side/R given by s(4*n+2) = 2*sin(Pi/(4*n+2)) = 2*cos(2*n*Pi/(4*n+2)) in a regular (4*n+2)-gon, inscribed in a circle with radius R, in terms of rho(4*n+2) = 2*cos(Pi/4*n+2), the length ratio of (the smallest diagonal)/side (for n=2 there is no such diagonal).
s(4*n+2) = Sum_{m=0..n}a(n,m)*rho(4*n+2)^(2*m). This formula is needed to show that the total sum of all length ratios in a (4*n+2)-gon is an integer in the algebraic number field Q(rho(4*n+2)). Note that rho(4*n+2) has degree delta(4*n+2) = A055034(4*n+2). Therefore one has to take s(4*n+2) modulo C(4*n+2, x=rho(4*n+2)), the minimal polynomial of rho(4*n+2) (see A187360). Thanks go to Seppo Mustonen for asking me to look into this problem. See ((-1)^(n-m))*A111125(n,m) for the (4*n)-gon situation. (End)

			The triangle a(n,m) starts:
n\m  0    1    2     3     4     5     6     7    8   9  10 ...
0:   2
1:  -2    1
2:   2   -4    1
3:  -2    9   -6     1
4:   2  -16   20    -8     1
5:  -2   25  -50    35   -10     1
6:   2  -36  105  -112    54   -12     1
7:  -2   49 -196   294  -210    77   -14     1
8:   2  -64  336  -672   660  -352   104   -16    1
9:  -2   81 -540  1386 -1782  1287  -546   135  -18   1
10:  2 -100  825 -2640  4290 -4004  2275  -800  170 -20  1
... Reformatted and extended by _Wolfdieter Lang_, Nov 21 2012.
n=3: [-2,9,-6,1] stands for -2*1 + 9*(2*x)^2 -6*(2*x)^4 +1*(2*x)^6 = 2*(1+18*x^2-48*x^4+32*x^6) = 2*T(6,x).
(4*n+2)-gon side/radius s(4*n+2) as polynomial in rho(4*n+2) = smallest diagonal/side: n=0: s(2) = 2 (rho(2)=0); n=1: s(6) = -2 + rho(6)^2 = -2 + 3 = 1, (C(6,x) = x^2 - 3); n=2: s(10) = 2 - 4*rho(10)^2 + 1*rho(10)^4 = 2 - 4*rho(10)^2 + (5*rho(10)^2 - 5) = -3 + rho(10)^2, (C(10,x) = x^4 - 5*x^2 + 5). - _Wolfdieter Lang_, Oct 04 2013

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 62
Sigurdur Helgasson,Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S. :ISBN 0-8218-2848-7, 1978,p. 463.

Wolfdieter Lang, First 10 rows and more.
P. Damianou , On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
P. Damianou and C. Evripidou, Characteristic and Coxeter polynomials for affine Lie algebras, arXiv preprint arXiv:1409.3956 [math.RT], 2014.
Yidong Sun, Numerical triangles and several classical sequences, Fib. Quart., Nov. 2005, pp. 359-370. See Table 1.6 (an unsigned version).
Eric Weisstein's World of Mathematics, Cartan Matrix
Eric Weisstein's World of Mathematics, Dynkin Diagram

Cf. A284982 (opposite signs and rows begin with 0).
Row sums (signed): -A061347(n+3) for n>=0.
Row sums (unsigned): A005248(n) = L(2*n), where L=Lucas.
Cf. A005248, A053122.
Cf. A263916, A111125, A208513, A034807, A127672.

T[n_, m_, d_] := If[ n == m, 2, If[n == d && m == d - 1, -2, If[(n == m - 1 || n == m + 1), -1, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] a = Join[M[1], Table[CoefficientList[CharacteristicPolynomial[M[d], x], x], {d, 1, 10} ]] (* Roger L. Bagula, May 23 2007 *)
CoefficientList[2 ChebyshevT[2 Range[0, 10], Sqrt[x]/2], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[Table[(-1)^n LucasL[2 n, Sqrt[-x]], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)

a(n,m) = {if(n>=2, -2*a(n-1,m)+a(n-1,m-1)-a(n-2,m), if(n==0, if(m!=0,0,2), if(m==0,-2, if(m==1,1,0))))};
for(n=0,10,for(m=0,n,print1(a(n,m),", "))) \\ Hugo Pfoertner, Jul 19 2020

a(n,m) = 0 if n < m; a(n,0) = 2*(-1)^n; a(n,m) = ((-1)^(n+m))*n*binomial(n+m-1, 2*m-1)/m.
a(n,m) = 0 if n < m, a(0,0) = 2, a(n,m) = (-1)^(n-m)*(2*n/(n+m))*binomial(n+m, n-m), n >= 1. From Waring's formula applied to Chebyshev's T-polynomials. See also A110162. - Wolfdieter Lang, Nov 21 2012
The o.g.f. Sum_{n>=0} p(n,x)*z^n, n>=0, for the row polynomials p(n,x) := Sum_{m=0..n} a(n,m)*x^m is (2 + z*(2-x))/((z+1)^2 - z*x). Here p(n,x) = R(2*n,sqrt(x)) := 2*T(2*n,sqrt(x)/2) with Chebyshev's T-polynomials. For the R-polynomials see A127672. - Wolfdieter Lang, Nov 28 2012
From Tom Copeland, Nov 07 2015: (Start)
A logarithmic generator is 2*(1-log(1+x))-log(1-t*x/(1+x)^2) = 2 - log(1+(2-t)*x+x^2) = 2 + (-2 + t)*x + (2 - 4*t + t^2) x^2/2 + (-2 + 9*t - 6*t^2 + t^3) x^3/3 + ..., so a number of relations to the Faber polynomials of A263916 hold with p(0,x) = 2:
1) p(n,x) = F(n,(2-x),1,0,0,..)
2) p(n,x) = (-1)^n 2 + F(n,-x,2x,-3x,...,(-1)^n n*x)
3) p(n,x) = (-1)^n [2 + F(n,x,2x,3x,...,n*x)].
The unsigned array contains the partial sums of A111125 modified by appending a column of zeros, except for an initial two, to A111125. Then the difference of consecutive rows of unsigned A127677, further modified by appending an initial rows of zeros, generates the modified A111125. Cf. A208513 and A034807.
For relations among the characteristic polynomials of Cartan matrices of the Coxeter root groups, Chebyshev polynomials, cyclotomic polynomials, and the polynomials of this entry, see Damianou (p. 12, 20, and 21) and Damianou and Evripidou (p. 7).
See A111125 for a relation to the squares of the odd row polynomials here with the constant removed.
p(n,x)^2 = 2 + p(2*n,x). See also A127672. (End)
a(n,m) = -2*a(n-1,m) + a(n-1,m-1) - a(n-2,m) for n >= 2 with initial conditions a(0,0) = 2, a(1,0) = -2, a(1,1) = 1, a(0,m) = 0 for m != 0, a(1,m) = 0 for m != 0,1. - William P. Orrick, Jun 09 2020
p(n,x) = (x-2)*p(n-1,x) - p(n-2,x) for n >= 2. - William P. Orrick, Jun 09 2020

Definition corrected by Eric W. Weisstein, Apr 06 2017

A054333 1/256 of tenth unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).

Original entry on oeis.org

1, 11, 65, 275, 935, 2717, 7007, 16445, 35750, 72930, 140998, 260338, 461890, 791350, 1314610, 2124694, 3350479, 5167525, 7811375, 11593725, 16921905, 24322155, 34467225, 48208875, 66615900, 91018356, 123058716, 164750740

Offset: 0

Barry E. Williams, Wolfdieter Lang, Mar 15 2000

If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-10) is the number of 10-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
9-dimensional square numbers, eighth partial sums of binomial transform of [1,2,0,0,0,...]. a(n)=sum{i=0,n,C(n+8,i+8)*b(i)}, where b(i)=[1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
2*a(n) is number of ways to place 8 queens on an (n+8) X (n+8) chessboard so that they diagonally attack each other exactly 28 times. The maximal possible attack number, p=binomial(k,2) =28 for k=8 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form the corresponding complete graph. - Antal Pinter, Dec 27 2015

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.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.
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..5000
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.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
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.

Partial sums of A053347. Cf. A053120, A000581.
Cf. A005585, A040977, A050486, A053347. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009
Cf. A111125, fifth column (s=4, without leading zeros). - Wolfdieter Lang, Oct 18 2012

List([0..30],n->(2*n+9)*Binomial(n+8,8)/9); # Muniru A Asiru, Dec 06 2018

[Binomial(n+8,8)+2*Binomial(n+8,9): n in [0..40]]; // Vincenzo Librandi, Feb 14 2016

LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {1, 11, 65, 275, 935, 2717, 7007, 16445, 35750, 72930}, 30] (* Vincenzo Librandi, Feb 14 2016 *)

vector(40, n, n--; (2*n+9)*binomial(n+8, 8)/9) \\ G. C. Greubel, Dec 02 2018

[(2*n+9)*binomial(n+8, 8)/9 for n in range(40)] # G. C. Greubel, Dec 02 2018

a(n) = (2*n+9)*binomial(n+8, 8)/9 = ((-1)^n)*A053120(2*n+9, 9)/2^8.
G.f.: (1+x)/(1-x)^10.
a(n) = 2*C(n+9, 9) - C(n+8, 8). - Paul Barry, Mar 04 2003
a(n) = C(n+8,8) + 2*C(n+8,9). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
E.g.f.: (1/362880)*exp(x)*(362880 + 3628800*x + 7983360*x^2 + 6773760*x^3 + 2751840*x^4 + 592704*x^5 + 70560*x^6 + 4608*x^7 + 153*x^8 + 2*x^9). - Stefano Spezia, Dec 03 2018
From Amiram Eldar, Jan 26 2022: (Start)
Sum_{n>=0} 1/a(n) = 294912*log(2)/35 - 7153248/1225.
Sum_{n>=0} (-1)^n/a(n) = 73728*Pi/35 - 8105688/1225. (End)

A115141 Convolution of A115140 with itself.

Original entry on oeis.org

1, -2, -1, -2, -5, -14, -42, -132, -429, -1430, -4862, -16796, -58786, -208012, -742900, -2674440, -9694845, -35357670, -129644790, -477638700, -1767263190, -6564120420, -24466267020, -91482563640, -343059613650, -1289904147324, -4861946401452, -18367353072152

Offset: 0

Wolfdieter Lang, Jan 13 2006

This is the so-called A-sequence for the Riordan triangles A053122, A110162, A129818, A158454 and signed A158909. For the notion of Z- and A-sequences for Riordan arrays see the W. Lang link under A006232 with details and references. Wolfdieter Lang, Dec 20 2010. [Revised, Nov 13 2012, Nov 22 2012 and Oct 22 2019]

a(n)*(-1)^n is the A-sequence for the Riordan triangle A111125. - Wolfdieter Lang, Jun 26 2011

			G.f. = 1 - 2*x - x^2 - 2*x^3 - 5*x^4 - 14*x^5 - 42*x^6 - 132*x^7 - 429*x^8 + ...

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

Cf. A000108, A115140, A115142 - A115149.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-2*x+Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019

a[n_] := -First[ ListConvolve[ cc = Array[ CatalanNumber, n-1, 0], cc]]; a[0] = 1; a[1] = -2; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 21 2011 *)
CoefficientList[Series[(1-2*x+Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

{a(n) = if( n<1, n==0, -(n==1) -binomial( 2*n-2, n-1) / n)} /* Michael Somos, Mar 28 2012 */

((1-2*x+sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^2 = (1-x) - x*c(x) with the o.g.f. c(x) = (1-sqrt(1-4*x) )/(2*x) of A000108 (Catalan numbers).
a(0)=1, a(1)=-2, a(n) = -C(n-1), n>=2, with C(n):=A000108(n) (Catalan). The start [1, -2] is row n=2 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
The convolution inverse is A000108(x)^2. - Michael Somos, Mar 28 2012
REVERT transform is A069271. - Michael Somos, Mar 28 2012
EULER transform of -A060165. - Michael Somos, Mar 28 2012
D-finite with recurrence: n*a(n) +2*(-2*n+3)*a(n-1)=0. - R. J. Mathar, Feb 21 2020

A057788 Expansion of (1+x)/(1-x)^12.

Original entry on oeis.org

1, 13, 90, 442, 1729, 5733, 16744, 44200, 107406, 243542, 520676, 1058148, 2057510, 3848222, 6953544, 12183560, 20764055, 34512075, 56071470, 89224590, 139299615, 213696795, 322561200, 479634480, 703323660, 1018031196, 1455797448, 2058314440, 2879378332

Offset: 0

N. J. A. Sloane, Nov 04 2000

1/2^10 of twelfth unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).
If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-12) is the number of 12-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
11-dimensional square numbers, tenth partial sums of binomial transform of [1,2,0,0,0,...]. a(n) = sum_{i=0..n} C(n+10,i+10)*b(i), where b(i)=[1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
2*a(n) is number of ways to place 10 queens on an (n+10) X (n+10) chessboard so that they diagonally attack each other exactly 45 times. The maximal possible attack number, p=binomial(k,2) =45 for k=10 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form the corresponding complete graph. - Antal Pinter, Dec 27 2015

T. D. Noe, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
Index entries for linear recurrences with constant coefficients, signature (12,-66, 220,-495,792,-924,792,-495,220,-66,12,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A053120, A054334, A054333, A053347, A002415, A005585, A040977, A050486.
Partial sums of A054334.
Sixth column of A111125 (s=5, without leading zeros). - Wolfdieter Lang, Oct 18 2012

List([0..30], n -> (2*n+11)*Binomial(n+10, 10)/11); # G. C. Greubel, Dec 02 2018

[Binomial(n+10,10)*(2*n+11)/11: n in [0..40]]; // Vincenzo Librandi, Feb 14 2016

A057788 := proc(n)
        1/39916800*(2*n+11) *(n+10) *(n+9) *(n+8) *(n+7) *(n+6) *(n+5) *(n+4) *(n+3) *(n+2) *(n+ 1) ; end proc: # R. J. Mathar, Mar 22 2011

Table[(2*n+11)*Binomial[n+10, 10]/11, {n,0,40}] (* G. C. Greubel, Dec 02 2018 *)
CoefficientList[Series[(1 + x) / (1 - x)^12, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 14 2016 *)
LinearRecurrence[{12,-66,220,-495,792,-924,792,-495,220,-66,12,-1},{1,13,90,442,1729,5733,16744,44200,107406,243542,520676,1058148},30] (* Harvey P. Dale, Sep 07 2022 *)

Vec((1+x)/(1-x)^12+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

[(2*n+11)*binomial(n+10, 10)/11 for n in range(40)] # G. C. Greubel, Dec 02 2018

a(n) = 2*C(n+11, 11) - C(n+10, 10). - Paul Barry, Mar 04 2003
a(n) = C(n+10,10) + 2*C(n+10,11). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
a(n) = C(n+10,10)*(2n+11)/11. - Antal Pinter, Dec 27 2015
a(n) = 12*a(n-1)-66*a(n-2)+220*a(n-3)-495*a(n-4)+792*a(n-5)-924*a(n-6)+792*a(n-7)-495*a(n-8)+220*a(n-9)-66*a(n-10)+12*a(n-11)-a(n-12) for n >11. - Vincenzo Librandi, Feb 14 2016
a(n) = (2*n+11)*binomial(n+10, 10)/11. - G. C. Greubel, Dec 02 2018
From Amiram Eldar, Jan 26 2022: (Start)
Sum_{n>=0} 1/a(n) = 419751541/13230 - 2883584*log(2)/63.
Sum_{n>=0} (-1)^n/a(n) = 720896*Pi/63 - 237793798/6615. (End)

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cp's OEIS Frontend

A053122 Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).

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A049666 a(n) = Fibonacci(5*n)/5.

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A111418 Right-hand side of odd-numbered rows of Pascal's triangle.

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A263916 Coefficients of the Faber partition polynomials.

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A087960 a(n) = (-1)^binomial(n+1,2).

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A050486 a(n) = binomial(n+6,6)*(2n+7)/7.

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A127677 Scaled coefficient table for Chebyshev polynomials 2T(2n, sqrt(x)/2) (increasing even scaled powers, without zero entries).