A111125 -id:A111125 - cp's OEIS frontend

A219240 Coefficient array for the cube of Chebyshev's S polynomials.

1, 0, 0, 0, 1, -1, 0, 3, 0, -3, 0, 1, 0, 0, 0, -8, 0, 12, 0, -6, 0, 1, 1, 0, -9, 0, 30, 0, -45, 0, 30, 0, -9, 0, 1, 0, 0, 0, 27, 0, -108, 0, 171, 0, -136, 0, 57, 0, -12, 0, 1, -1, 0, 18, 0, -123, 0, 399, 0, -651, 0, 588, 0, -308, 0, 93, 0, -15, 0, 1, 0, 0, 0, -64, 0, 480, 0, -1488, 0, 2488, 0, -2472, 0, 1524, 0, -588, 0, 138, 0, -18, 0, 1

Offset: 0

Wolfdieter Lang, Dec 12 2012

The row lengths sequence is 3*n+1 = A016777(n).
For the coefficient triangle for Chebyshev's S polynomials see A049310.
The o.g.f. for S(n,x)^3, n >= 0, is GS(3;x,z) = (1+z^2+2*z*x)/ ((1+z^2-z*x)*(1+z^2-z*x*(x^2-3))). This is obtained from the de Moivre-Binet formula for S(n,x) and the binomial theorem.
In general the monic integer Chebyshev polynomial tau(n,x):= R(2*n+1,x)/x enters, where R(n,x) = 2*T(n,x/2) with Chebyshev's T polynomial (for R see A127672), and the coefficient triangle for tau is given in A111125 (here for the third power of S only tau(0,x) = 1 and tau(1,x) = x^2 - 3 enter).

			The array a(n,m) begins:
n\m   0  1  2  3  4    5   6    7  8    9 10  11 12  13 14 15
n=0:  1
n=1:  0  0  0  1
n=2: -1  0  3  0 -3    0   1
n=3:  0  0  0 -8  0   12   0  -6   0    1
n=4:  1  0 -9  0 30    0 -45   0  30    0 -9   0  1
n=5:  0  0  0 27  0 -108   0  171  0 -136  0  57  0 -12  0  1
...
Row n=6: [-1, 0, 18, 0, -123, 0, 399, 0, -651, 0, 588, 0, -308, 0, 93, 0, -15, 0, 1],
Row n=7: [0, 0, 0, -64, 0, 480, 0, -1488, 0, 2488, 0, -2472, 0, 1524, 0, -588, 0, 138, 0, -18, 0, 1],
Row n=8: [1, 0, -30, 0, 345, 0, -1921, 0, 5598, 0, -9540, 0, 10212, 0, -7137, 0, 3303, 0, -1003, 0, 192, 0, -21, 0, 1].
n=2: S(2,x)^3 = (x^2 - 1)^3 = -1 + 3*x^2 - 3*x^4 + x^6.
n=3: S(3,x)^3 = (x^3 - 2*x)^3 = -8*x^3 + 12*x^5 - 6*x^7 + x^9.

Cf. A049310, A127672, A158454 (square of S polynomials), A219234 (fourth power of S polynomials).

a(n,m) = [x^m] S(n, x)^3, n >= 0, 0 <= m <= 3*n, with Chebyshev's S polynomials (see A049310).
a(n,m) = [x^m]([z^n] GS(3;x,z)), with the o.g.f. GS(3;x,z) given above in a comment.
The row polynomials p(n, x) := Sum_{m=0..3*n} a(n,m)*x^m = S(n, x)^3 are (S(3*n+2, x) - 3*S(n, x))/(x^2 - 4). For the factorization of S polynomials see comments on A049310. - Wolfdieter Lang, Apr 09 2018

A220670 Coefficient triangle for powers of x^2 of polynomials appearing in a generalized Melham conjecture on alternating sums of third powers of Chebyshev's S polynomials with odd indices. Coefficients in powers of x^2 of 2 + (-1)^nS(2n,x).

Original entry on oeis.org

3, 3, -1, 3, -3, 1, 3, -6, 5, -1, 3, -10, 15, -7, 1, 3, -15, 35, -28, 9, -1, 3, -21, 70, -84, 45, -11, 1, 3, -28, 126, -210, 165, -66, 13, -1, 3, -36, 210, -462, 495, -286, 91, -15, 1, 3, -45, 330, -924, 1287, -1001, 455, -120, 17, -1, 3, -55, 495, -1716, 3003, -3003, 1820, -680, 153, -19, 1

Offset: 0

Wolfdieter Lang, Jan 07 2013

For the original Melham conjecture on sums of odd powers of even-indexed Fibonacci numbers see the references given in A217475. See especially the Wang and Zhang reference given there.
An analog conjecture stated for Chebyshev's S polynomials (see A049310) is product(tau(j,x),j=0..m)*sum(((-1)^k)*(S(2*k-1,x)/x)^(2*m+1),k=0..n)/(P(n,x^2)^2) = H(m,n,x^2), with P(n,x^2) := (1 - (-1)^n*S(2*n,x))/x^2 and certain integer polynomials H with degree (2*m-1)*n + binomial(m-1,2) in x^2. The coefficients of powers of x^2 of the monic integer polynomials tau(n,x):= 2*T(2*n+1,x/2)/x, with Chebyshev's T polynomials, are given by the signed A111125 triangle (see a comment there from Oct 23 2012). The coefficients of the powers of x^(2*j) of the polynomials P(n,x^2) are found in (-1)^(n-1)*A109954(n-1,j).
Here the conjecture is considered for m=1 (third powers): H(1,n,x^2) = sum(a(n,p)*x^(2*p),p=0..n), n >= 1. It is conjectured that in fact H(1,n,x^2) = 2 + (-1)^n*S(2*n,x). This has been checked by Maple for n=1..100.  Therefore we have added a(0,0) = 3 (in the conjecture above this would be the undetermined 0/0).
The original Melham conjecture for m=1 (third powers), appears by putting x = i (the imaginary unit): 1*4*sum(F(2*k)^3)/(1-F(2*n+1))^2 = sum(a(n,p)*(-1)^p) = 2 + F(2*n+1) (the unsigned row sums of the present triangle). This m=1 identity is, of course, proved.
The row sums of this triangle are given by 2 + (-1)^n*S(2*n,1) = 2 + (-1)^n*((2/sqrt(3))*sin((2*n+1)*Pi/3)) producing the period 6 sequence periodic (3, 2, 1, 1, 2, 3).

			The triangle a(n,p) begins:
n\p 0    1    2     3    4     5    6    7   8   9 10 ...
0:  3
1:  3   -1
2:  3   -3    1
3:  3   -6    5    -1
4:  3  -10   15    -7    1
5:  3  -15   35   -28    9    -1
6:  3  -21   70   -84   45   -11    1
7:  3  -28  126  -210  165   -66   13  -1
8:  3  -36  210  -462  495  -286   91  -15   1
9:  3  -45  330  -924 1287 -1001  455 -120  17  -1
10: 3  -55  495 -1716 3003 -3003 1820 -680 153 -19  1
...
Row n=2: H(1,2,x^2) := (-3+x^2)*(0 - (S(1,x)/x)^3 + (S(3,x)/x)^3)/((1 - S(4,x))/x^2)^2 = 3 - 3*x^2 + x^4 =
  2 + S(4,x).
Row n=3:  H(1,3,x^2) := (-3+x^2)*(0 - (S(1,x)/x)^3 + (S(3,x)/x)^3 - (S(5,x)/x)^3 )/((1 + S(6,x))/x^2)^2 =  3-6*x^2+5*x^4-x^6 = 2 - S(6,x).

R. S. Melham, Some conjectures concerning sums of odd powers of Fibonacci and Lucas numbers, The Fibonacci Quart. 46/47 (2008/2009), no. 4, 312-315.
K. Ozeki, On Melham's sum, The Fibonacci Quart. 46/47 (2008/2009), no. 2, 107-110.
H. Prodinger, On a sum of Melham and its variants, The Fibonacci Quart. 46/47 (2008/2009), no. 3, 207-215.
T. Wang and W. Zhang, Some identities involving Fibonacci, Lucas polynomials and their applications, Bull. Math. Soc. Sci. Math. Roumanie, Tome 55(103), No.1, (2012) 95-103.

Cf, A049310, A111125 (signed), A109954 (signed), A217475, A220671 (fifth powers).

a(n,p) = [x^(2*p)] H(1,n,x^2), with H(1,n,x^2) := (-3+x^2)*sum(((-1)^k)*(S(2*k-1,x)/x)^3,k=0..n)/((1 - (-1)^n*S(2*n,x))/x^2)^2, n >= 1, p = 0..n, and a(0,0):=3.

a(n,p) = [x^(2*p)] (2 + (-1)^n*S(2*n,x)), n >= 0, p = 0..n.

A220671 Coefficient array for powers of x^2 of polynomials appearing in a generalized Melham conjecture on alternating sums of fifth powers of Chebyshev S polynomials with odd indices.

Original entry on oeis.org

-14, 15, -20, 8, -1, 55, -170, 221, -153, 59, -12, 1, 115, -670, 1773, -2696, 2549, -1538, 589, -138, 18, -1, 195, -1850, 8215, -21530, 36330, -41110, 31865, -17080, 6314, -1579, 255, -24, 1, 295, -4150, 27735, -110795, 289540, -518290, 654595, -595805, 396316, -193906, 69641, -18129, 3327, -408, 30, -1

Offset: 1

Wolfdieter Lang, Jan 11 2013

The row lengths sequence is 3*n + 1 = A016777(n).
For the generalized Melham conjecture and links to the references concerned with the Melham conjecture on sums of fifth powers of even-indexed Fibonacci numbers see a comment under A220670.
Here the conjecture is considered for m=2 (fifth powers): H(2,n,x^2):= product(tau(j,x), j=0..2) * sum(((-1)^k)*(S(2*k-1,x)/x)^5, k=0..n) / (P(n,x^2)^2), with P(n,x^2):= (1 - (-1)^n*S(2*n,x))/x^2. For tau(j,x):= 2*T(2*j+1,x/2)/x, with Chebyshev's T polynomials see a Oct 23 2012 comment on A111125. For the polynomials P see signed A109954. The conjecture is that H(2,n,x^2) is an integer polynomial of degree 3*n:  H(2,n,x^2) = sum(a(n,p)*x^(2*p), p=0..3*n), n >= 1.
If one puts x = i (the imaginary unit) one obtains the original Melham conjecture for Fibonacci numbers F = A000045.
H(2,n,-1) = +44*sum(F(2*k)^5,k=0..n)/(1+F(2*n+1))^2, n>=1, which is conjectured to be -14 - 3*y(n) + 8*y(n)^2 + 4*y(n)^3, with y(n):=F(2*n+1) (see row m=2 of A217475).
It is conjectured that H(2,n,x^2) = h(2,n,x^2) - 3*z(n) + 8*z(n)^2 + 4*z(n)^3, with z(n):= ((-1)^n)*S(2*n,x), with h an integer polynomial of degree 3*n. See A220672 for the coefficients of h(2,n,x^2) for n = 1..5. Because h(2,n,-1) = -14 by the usual Melham conjecture, we put h(2,0,x^2) = -14.

			The array a(n,p) begins:
  n\p   0     1     2      3      4      5     6      7   8     9   10 11 12
  0:  -14
  1:   15   -20     8     -1
  2:   55  -170   221   -153     59    -12     1
  3:  115  -670  1773  -2696   2549  -1538   589   -138   18    -1
  4:  195 -1850  8215 -21530  36330 -41110 31865 -17080 6314 -1579 255 -24 1
...
Row n=5: [295, -4150, 27735, -110795, 289540, -518290, 654595, -595805, 396316, -193906, 69641, -18129, 3327, -408, 30, -1],
Row n=6: [415, -8120, 76118, -429531, 1599441, -4125672, 7621983, -10350335, 10539787, -8164410, 4853792, -2222153, 781514, -209172, 41823, -6047, 597, -36, 1].

Cf. A220670, A217475, A220672.

a(n,p) = [x^(2*p)] H(2,n,x^2), n>=1, with H(2,n,x^2) defined in a comment above. a(0,0) has been put to -14 ad hoc.

A231123 Array T(n,k) read by antidiagonals: T(n,k) = sum(i=0...n, (-1)^(n+i) * C(n+i,2i) * n/(2i+1) * k^(2i+1) ), n>0, k>1.

Original entry on oeis.org

2, 2, 18, 2, 123, 52, 2, 843, 724, 110, 2, 5778, 10084, 2525, 198, 2, 39603, 140452, 57965, 6726, 322, 2, 271443, 1956244, 1330670, 228486, 15127, 488, 2, 1860498, 27246964, 30547445, 7761798, 710647, 30248, 702, 2, 12752043, 379501252, 701260565, 263672646

Offset: 2

Ralf Stephan, Nov 04 2013

The polynomial x^(4n+2) - T(n,k)*x^(2n+1) + 1 is reducible. Example: x^10-123x^5+1=(x^2-3x+1)(x^8+3x^7+8x^6+21x^5+55x^4+21x^3+8x^2+3x+1). It is conjectured that for prime p=2n+1, these are the only values where this holds.

			Array starts
2, 18, 52, 110, 198, 322, 488, 702, 970,...
2, 123, 724, 2525, 6726, 15127, 30248, 55449, 95050,...
2, 843, 10084, 57965, 228486, 710647, 1874888, 4379769, 9313930,...
2, 5778, 140452, 1330670, 7761798, 33385282, 116212808, 345946302,...
2, 39603, 1956244, 30547445, 263672646, 1568397607, 7203319208,...

A. Schinzel, On reducible trinomials III. In: Selecta, Vol. I, European Mathematical Society 2007, pp. 625-626.

Ralf Stephan, On a class of reducible trinomials

T(i,k)=n=2*i+1;sum(m=0,(n-1)/2,(-1)^(m+(n-1)/2)*n*binomial((n+2*m+1)/2-1,2*m)/(2*m+1)*k^(2*m+1))

T(,2) = 2, T(1,n) = A121670(n), T(2,n) = A230586(n).

T(n,k) = sum(i=1..n, (-1)^i * A111125(n,i) * k^(2i+1) ).

A284966 Triangle read by rows: coefficients of the scaled Lucas polynomials x^(n/2)*L(n, sqrt(x)) for n >= 0, sorted by descending powers of x.

Original entry on oeis.org

2, 1, 0, 2, 1, 0, 0, 3, 1, 0, 0, 2, 4, 1, 0, 0, 0, 5, 5, 1, 0, 0, 0, 2, 9, 6, 1, 0, 0, 0, 0, 7, 14, 7, 1, 0, 0, 0, 0, 2, 16, 20, 8, 1, 0, 0, 0, 0, 0, 9, 30, 27, 9, 1, 0, 0, 0, 0, 0, 2, 25, 50, 35, 10, 1, 0, 0, 0, 0, 0, 0, 11, 55, 77, 44, 11, 1, 0, 0, 0, 0, 0, 0, 2, 36, 105, 112, 54, 12, 1

Offset: 0

Eric W. Weisstein, Apr 06 2017

For n >= 3, also the coefficients of the edge and vertex cover polynomials for the n-cycle graph C_n.

For more information on how this triangular array is related to the work of Charalambides (1991) and Moser and Abramson (1969), see the comments for triangular array A212634 (which contains additional formulas). The coefficients of these polynomials are given by formula (2.1), p. 291, in Charalambides (1991), where an obvious typo in the index of the summation must be corrected (floor(n/K) -> floor(n/K) - 1). - Petros Hadjicostas, Jan 27 2019

			First few polynomials are
  2;
  x;
  2*x + x^2;
  3*x^2 + x^3;
  2*x^2 + 4*x^3 + x^4;
  5*x^3 + 5*x^4 + x^5;
  ...
giving
  2;
  0, 1;
  0, 2, 1;
  0, 0, 3, 1;
  0, 0, 2, 4, 1;
  0, 0, 0, 5, 5, 1;
  ...

C. A. Charalambides, Lucas numbers and polynomials of order k and the length of the longest circular success run, The Fibonacci Quarterly, 29 (1991), 290-297.
W. O. J. Moser and M. Abramson, Enumeration of combinations with restricted differences and cospan, J. Combin. Theory, 7 (1969), 162-170.
Eric Weisstein's World of Mathematics, Cycle Graph
Eric Weisstein's World of Mathematics, Edge Cover Polynomial
Eric Weisstein's World of Mathematics, Lucas Polynomial
Eric Weisstein's World of Mathematics, Vertex Cover Polynomial

Cf. A034807 (Lucas polynomials x^(n/2)*L(n, 1/sqrt(x))).

Cf. A111125, A127677, A136481, A212634.

L := proc (n, K, x) -1 + sum((-1)^j*n*binomial(n - j*K, j)*x^j*(x+1)^(n - j*(K+1))/(n - j*K), j = 0 .. floor(n/(K + 1))) end proc; for i to 30 do expand(L(i, 2, x)) end do; # gives the g.f. of row n for 1 <= n <= 30. - Petros Hadjicostas, Jan 27 2019

CoefficientList[Table[x^(n/2) LucasL[n, Sqrt[x]], {n, 12}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[Table[2 x^n (-1/x)^(n/2) ChebyshevT[n, 1/(2 Sqrt[-1/x])], {n, 12}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[Table[FunctionExpand[2 (-(1/x))^(n/2) x^n Cos[n ArcSec[2 Sqrt[-(1/x)]]]], {n, 15}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[LinearRecurrence[{x, x}, {x, x (2 + x)}, 15], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)

First element T(n=0, k=0) and the example corrected by Petros Hadjicostas, Jan 27 2019

Name edited by Petros Hadjicostas, Jan 27 2019 and by Stefano Spezia, Mar 09 2025

A111126 Triangle read by rows: T(k,s) = binomial(k+s,2s+1)(2k-1)(2k+1)/(2s+3), k >= 1, 0 <= s <= k-1.

Original entry on oeis.org

1, 10, 3, 35, 28, 5, 84, 126, 54, 7, 165, 396, 297, 88, 9, 286, 1001, 1144, 572, 130, 11, 455, 2184, 3510, 2600, 975, 180, 13, 680, 4284, 9180, 9350, 5100, 1530, 238, 15, 969, 7752, 21318, 28424, 20995, 9044, 2261, 304, 17, 1330, 13167, 45144, 76076, 72618

Offset: 1

N. J. A. Sloane, Oct 16 2005

			Triangle starts:
1;
10,3;
35,28,5;
84,126,54,7;
165,396,297,88,9;

K. Dilcher and K. B. Stolarsky, A Pascal-type triangle characterizing twin primes, Amer. Math. Monthly, 112 (2005), 673-681.

Mirror image of A111127. Cf. A111125, A082985, A100218, A098599.

T:=(k,s)->binomial(k+s,2*s+1)*(2*k-1)*(2*k+1)/(2*s+3): for k from 1 to 10 do seq(T(k,s),s=0..k-1) od; # yields sequence in triangular form; Emeric Deutsch, Feb 01 2006

More terms from Emeric Deutsch, Feb 01 2006

A113214 Riordan array (1+2x,x(1+x)).

Original entry on oeis.org

1, 2, 1, 0, 3, 1, 0, 2, 4, 1, 0, 0, 5, 5, 1, 0, 0, 2, 9, 6, 1, 0, 0, 0, 7, 14, 7, 1, 0, 0, 0, 2, 16, 20, 8, 1, 0, 0, 0, 0, 9, 30, 27, 9, 1, 0, 0, 0, 0, 2, 25, 50, 35, 10, 1, 0, 0, 0, 0, 0, 11, 55, 77, 44, 11, 1, 0, 0, 0, 0, 0, 2, 36, 105, 112, 54, 12, 1, 0, 0, 0, 0, 0, 0, 13, 91, 182, 156, 65, 13, 1

Offset: 0

Paul Barry, Oct 18 2005

Row sums are Lucas numbers A000204. Diagonal sums are A007307(n+1). Inverse is (-1)^(n-k)A092392(n,k). Product with Pascal triangle (1/(1-x),x/(1-x)) is A111125.

			Triangle begins
  1;
  2,  1;
  0,  3,  1;
  0,  2,  4,  1;
  0,  0,  5,  5,  1;
  0,  0,  2,  9,  6,  1;
  0,  0,  0,  7, 14,  7,  1;
  0,  0,  0,  2, 16, 20,  8,  1;
Row 4: (1 + x*c(-x))^5 = 1 + 5*x + 5*x^2 + O(x^5). - _Peter Bala_, Sep 10 2021

R. Tauraso, Edge cover time for regular graphs, JIS 11 (2008) 08.4.4.

Cf. A000108, A000204, A007307, A092392, A111125.

T(n, k) = C(k, n-k) + 2*C(k, n-k-1).
T(n, k) = Sum_{j = 0..n} (-1)^(n-j)*C(n, j)*C(j+k, 2*k)*(2*j+1)/(2*k+1).
From Peter Bala, Sep 10 2021: (Start)
T(n,k) = T(n-1,k-1) + T(n-2,k-1) with boundary conditions T(n,n) = 1, T(1,0) = 2 and T(n,k) = 0 for k < 0 or k > n.
The entries in row n, read in reverse order, are the coefficients in the n-th degree Taylor polynomial of (1 + x*c(-x))^(n+1) at x = 0, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

A217479 Array of coefficients of polynomials providing the third term of the numerator of the generating function for odd powers (2*m+1) of Chebyshev S-polynomials. The present polynomials are called P(m;2,x^2), m >= 2.

Original entry on oeis.org

-8, 6, -27, 65, -56, 15, -61, 260, -469, 415, -176, 28, -114, 736, -2104, 3214, -2838, 1456, -400, 45, -190, 1714, -6988, 15699, -21461, 18760, -10614, 3768, -760, 66, -293, 3507, -19195, 58807, -112123, 141441, -122168, 73185, -30077, 8107, -1288, 91

Offset: 2

Wolfdieter Lang, Nov 14 2012

The row length of this irregular triangle is 2*(m-1), m >= 2.
For the o.g.f. of S(m,x)^(2*m+1), m>=0, with Chebyshev's S-polynomials (coefficient triangle A049310) see the comment on A217478. G(m;z,x) = Z(m;z,x)/N(m;z,x) with N(m;z,x) = product((1+z^2) - z*x*tau(k,x),k=0..m), and Z(m;z,x) = sum((1+z^2)^(m-l)*(-z*x)^l*P(m;l,x^2),l=0..m), where P(m,l,x^2) = sum(T(m,k)*S(2*k,x)*sigma(m;k,l,x^2), k=0..m)/(x^2-4)^m, with sigma(m;k,l,x^2) the elementary symmetric function of a product of l factors from tau(j,x), for j=0..m, with tau(k,x) missing. Here tau(j,x):= 2*T(2*j+1,x/2)/x = R(2*j+1,x)/x (see A127672 for the coefficients of R(n,x)).
The present array a(m,k) provides the P(m;2,x^2) coefficients, and m >= 2: P(m;2,x^2) = sum(a(m,k)*x^2,k=0..(2*m-3)).
Using inclusion-exclusion one can write (x^2-4)^m*P(m;2,x^2) =
sum(T(m,k)*S(2*k,x)*(sigma(m+1;2,x^2) - sum(tau(j,x),j=0..m)* tau(k,x) + tau(k,x)^2),k=0..m), with sigma(m+1;2,x^2) the elementary symmetric function of 2 factors from tau(j,x), for j=0,1,...,m. E.g.,m=2:  sigma(2+1;2,x^2) = tau(0,x)*tau(1,x) + tau(0,x)*tau(2,x) + tau(1,x)*tau(2,x). The identities Id(0;m,x^2) and Id(1;m,x^2) (given in the comment on A217478) together with the new identity Id(2;m,x^2) := sum(T(m,k)*S(2*k,x)*tau(k,x)^2,k=0..m) = (x^2-4)^m*((x^2-1)^(2*m+1) + 1)/x^2 are now used. The new identity is obtained from the de Moivre-Binet formula for S and tau using first twice the identity mentioned in a Nov 14 2012 comment on A113187, and then the identity q^3 - 1/q^3 = sqrt(x^2-4)*(x^2-1) (see the instance k=1 of the formula given in a Oct 18 2012 comment on A111125 with x -> q which is defined by  (x+sqrt(x^2-4))/2). This yields, after division by (x^2-4)^m, finally the polynomial P(2;m,x^2) = sigma(m+1;2,x^2) - sum(tau(j,x),j=0..m)*x^(2*m) + ((x^2-1)^(2*m+1) + 1)/x^2, for m >= 2.

			The array a(m,k) starts:
m\k   0    1     2     3      4     5      6    7    8  9 ...
2:   -8    6
3:  -27   65   -56    15
4:  -61  260  -469   415   -176    28
5: -114  736 -2104  3214  -2838  1456   -400   45
6: -190 1714 -6988 15699 -21461 18760 -10614 3768 -760 66
...
Row m=7:  -293, 3507, -19195, 58807, -112123, 141441, -122168, 73185, -30077, 8107, -1288, 91.
Row m=8: -427, 6536, -46102, 183762, -461654, 780716, -926345, 790773, -491397, 221760, -71139, 15405, -2016, 120.
Row 9: -596, 11346, -100077, 502036, -1600280, 3470116, -5352805, 6051236, -5110145, 3256825, -1568416, 564980, -148176, 26770, -2976, 153.
m=2: P(2;2,x^2) = tau(0,x)*tau(1,x) + tau(0,x)*tau(2,x) + tau(1,x)*tau(2,x) - (tau(0,x)+tau(1,x)+tau(2,x))*x^4 + (5 -10*x^2 + 10*x^4 - 5*x^6 + x^8)  = -8 + 6*x^2 = 2*(-4 + 3*x^2).
  The numerator of the o.g.f. for S(n,x)^5 is Z(2;z,x) = (1+z^2)^2 + (1+z^2)*(-x*z)*(3-4*x^2) + (-x*z)^2*2*(-4 + 3*x^2), where the last bracket in the second term comes from row m=2 of A217478. The denominator is N(2;z,x) = product((1+z^2)-z*x*tau(k,x), k=0..2). See the example of A217478.

Cf. A217478.

a(m,k) = [x^(2*k)] P(2;m,x^2), m >= 2, k = 0..(2*m-3), with P(2;m,x^2) given in the comment above.

A219235 Coefficient array for the third power of the monic integer Chebyshev polynomials 2T(2n+1,x/2)/x as a function of x^2.

Original entry on oeis.org

1, -27, 27, -9, 1, 125, -375, 450, -275, 90, -15, 1, -343, 2058, -5145, 7007, -5733, 2940, -952, 189, -21, 1, 729, -7290, 30861, -72927, 107406, -104652, 69768, -32319, 10395, -2277, 324, -27, 1, -1331, 19965, -127776, 461857, -1058145, 1641486, -1797818, 1427679, -834900, 361790, -115830, 27027, -4466, 495, -33, 1

Offset: 0

Wolfdieter Lang, Nov 27 2012

The length of row n of this array is 3*n+1; see A016777.
Define tau(n,x):= C(2*n+1,x)/x, with C the monic integer Chebyshev T-polynomials with their coefficients given in A127672 (C(n,x) = 2*T(n,x/2) is there called R(n,x)). The coefficients of the x^2-powers of tau(n) are found as signed A111125 (see the last of the comments from Oct 18 2012 there). The irregular triangle a(n,m) appears in tau(n,x)^3 = sum(a(n,m)*x(2*m),m=0..3*n), n>=0.
The o.g.f. of the row polynomials as function of x^2 is G(3;x,z) := sum(tau(n,x)^3*z^n, n=0..infinity) =
  (1 - (23-17*x^2+3*x^4)*z*(1-z) - z^3)/(((z+1)^2-x^2*z)*((z+1)^2-z*x^2*(x^2-3)^2)). From the odd part of the bisection of the o.g.f. for C(n,x)^3 divided by x^3.

			The array a(n,m) begins:
n\m   0    1     2    3     4     5     6   7   8  9 ...
0:    1
1:  -27   27    -9    1
2:  125 -375   450 -275    90   -15     1
3: -343 2058 -5145 7007 -5733  2940  -952 189 -21  1
...
Row n=4: [729, -7290, 30861, -72927, 107406, -104652, 69768, -32319, 10395, -2277, 324, -27, 1].
Row n=5: [-1331, 19965, -127776, 461857, -1058145, 1641486, -1797818, 1427679, -834900, 361790, -115830, 27027, -4466, 495, -33, 1].
Row n=1 polynomial p(1,x) := -27 + 27*x - 9*x^2 + 1*x^3 with p(1,x^2) = tau(1,x)^3 = (-3 + x^2)^3 = -27+27*x^2-9*x^4+x^6.

Cf. A111125 (tau(n,x) coefficients if signed).

a(n,m) = [x^(2*m)] tau(n,x)^3, n>=0, m=0,1,...,3*n, with the monic integer polynomials tau(n,x) defined above in a comment.

A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 7, 5, 1, 4, 14, 16, 7, 1, 5, 25, 41, 29, 9, 1, 6, 41, 91, 92, 46, 11, 1, 7, 63, 182, 246, 175, 67, 13, 1, 8, 92, 336, 582, 550, 298, 92, 15, 1, 9, 129, 582, 1254, 1507, 1079, 469, 121, 17, 1, 10, 175, 957, 2508, 3718, 3367, 1925, 696, 154

Offset: 0

Philippe Deléham, Jan 24 2014

Triangle T(n,k), read by rows, given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Row sums are A111282(n+1) = A025169(n-1).
Diagonal sums are A122391(n+1) = A003945(n-1).

			Triangle begins:
  1;
  1,  1;
  2,  3,   1;
  3,  7,   5,   1;
  4, 14,  16,   7,   1;
  5, 25,  41,  29,   9,  1;
  6, 41,  91,  92,  46, 11,  1;
  7, 63, 182, 246, 175, 67, 13, 1;

Cf. Columns: A028310, A004006.
Cf. Diagonals: A000012, A005408, A130883.
Cf. Similar sequences: A078812, A085478, A111125, A128908, A165253, A207606.
Cf. A321620.

# The function RiordanSquare is defined in A321620.
RiordanSquare(1+x/(1-x)^2, 8); # Peter Luschny, Mar 06 2022

CoefficientList[#, y] & /@
CoefficientList[
Series[(1 - x + x^2)/(1 - 2*x - x*y + x^2), {x, 0, 12}], x] (* Wouter Meeussen, Jan 25 2014 *)

G.f.: (1 - x + x^2)/(1 - 2*x - x*y + x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = 2, T(2,1) = 3, T(2,2) = 1, T(n,k) = 0 if k < 0 or k > n.
The Riordan square (see A321620) of 1 + x/(1 - x)^2. - Peter Luschny, Mar 06 2022

cp's OEIS Frontend

A219240 Coefficient array for the cube of Chebyshev's S polynomials.

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A220670 Coefficient triangle for powers of x^2 of polynomials appearing in a generalized Melham conjecture on alternating sums of third powers of Chebyshev's S polynomials with odd indices. Coefficients in powers of x^2 of 2 + (-1)^n*S(2*n,x).

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A220671 Coefficient array for powers of x^2 of polynomials appearing in a generalized Melham conjecture on alternating sums of fifth powers of Chebyshev S polynomials with odd indices.

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A231123 Array T(n,k) read by antidiagonals: T(n,k) = sum(i=0...n, (-1)^(n+i) * C(n+i,2i) * n/(2i+1) * k^(2i+1) ), n>0, k>1.

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A284966 Triangle read by rows: coefficients of the scaled Lucas polynomials x^(n/2)*L(n, sqrt(x)) for n >= 0, sorted by descending powers of x.

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A111126 Triangle read by rows: T(k,s) = binomial(k+s,2s+1)*(2k-1)*(2k+1)/(2s+3), k >= 1, 0 <= s <= k-1.

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A113214 Riordan array (1+2x,x(1+x)).

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A217479 Array of coefficients of polynomials providing the third term of the numerator of the generating function for odd powers (2*m+1) of Chebyshev S-polynomials. The present polynomials are called P(m;2,x^2), m >= 2.

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A219235 Coefficient array for the third power of the monic integer Chebyshev polynomials 2*T(2*n+1,x/2)/x as a function of x^2.

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A220670 Coefficient triangle for powers of x^2 of polynomials appearing in a generalized Melham conjecture on alternating sums of third powers of Chebyshev's S polynomials with odd indices. Coefficients in powers of x^2 of 2 + (-1)^nS(2n,x).

A111126 Triangle read by rows: T(k,s) = binomial(k+s,2s+1)(2k-1)(2k+1)/(2s+3), k >= 1, 0 <= s <= k-1.

A219235 Coefficient array for the third power of the monic integer Chebyshev polynomials 2T(2n+1,x/2)/x as a function of x^2.