A049310 -id:A049310 - cp's OEIS frontend

A101950 Product of A049310 and A007318 as lower triangular matrices.

1, 1, 1, 0, 2, 1, -1, 1, 3, 1, -1, -2, 3, 4, 1, 0, -4, -2, 6, 5, 1, 1, -2, -9, 0, 10, 6, 1, 1, 3, -9, -15, 5, 15, 7, 1, 0, 6, 3, -24, -20, 14, 21, 8, 1, -1, 3, 18, -6, -49, -21, 28, 28, 9, 1, -1, -4, 18, 36, -35, -84, -14, 48, 36, 10, 1, 0, -8, -4, 60, 50, -98, -126, 6, 75, 45, 11, 1, 1, -4, -30, 20, 145, 36, -210

Offset: 0

Paul Barry, Dec 22 2004

A Chebyshev and Pascal product.
Row sums are n+1, diagonal sums the constant sequence 1 resp. A023434(n+1). Riordan array (1/(1-x+x^2),x/(1-x+x^2)).
Apart from signs, identical with A104562.
Subtriangle of the triangle given by [0,1,-1,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 27 2010
The Fi1 and Fi2 sums lead to A004525 and the Gi1 sums lead to A077889, see A180662 for the definitions of these triangle sums. - Johannes W. Meijer, Aug 06 2011
Also the convolution triangle of the inverse of 6th cyclotomic polynomial A010892. - Peter Luschny, Oct 08 2022

			Triangle begins:
   1,
   1, 1,
   0, 2, 1,
  -1, 1, 3, 1,
  -1,-2, 3, 4, 1,
  ...
Triangle [0,1,-1,1,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] begins : 1 ; 0,1 ; 0,1,1 ; 0,0,2,1 ; 0,-1,1,3,1 ; 0,-1,-2,3,4,1 ; ... - _Philippe Deléham_, Jan 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1325
Jerry Ray Dias, Properties and relationships of conjugated polyenes having a reciprocal eigenvalue spectrum - dendralene and radialene hydrocarbons, Croatica Chem. Acta, 77 (2004), 325-330. [p. 328].
Jonathan L. Gross, Toufik Mansour, Thomas W. Tucker, and David G. L. Wang, Root geometry of polynomial sequences. II: Type (1,0), J. Math. Anal. Appl. 441, No. 2, 499-528 (2016).

Cf. A049310, A007318, A104562, A010892, A084938.

A101950 := proc(n,k) local j,k1: add((-1)^((n-j)/2)*binomial((n+j)/2,j)*(1+(-1)^(n+j))* binomial(j,k)/2, j=0..n) end: seq(seq(A101950(n,k),k=0..n), n=0..11); # Johannes W. Meijer, Aug 06 2011
# Uses function PMatrix from A357368. Adds a row on top and a column to the left.
PMatrix(10, n -> [0, 1, 1, 0, -1,-1][irem(n, 6) + 1]); # Peter Luschny, Oct 08 2022

T[0, 0] = 1; T[n_, k_] /; k>n || k<0 = 0; T[n_, k_] := T[n, k] = T[n-1, k-1]+T[n-1, k]-T[n-2, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 07 2014, after Philippe Deléham *)

T(n, k) = Sum_{j=0..n} (-1)^((n-j)/2)*C((n+j)/2,j)*(1+(-1)^(n+j))*C(j,k)/2.
T(0,0) = 1, T(n,k) = 0,if k>n or if k<0, T(n,k) = T(n-1,k-1) + T(n-1,k) - T(n-2,k). - Philippe Deléham, Jan 26 2010
p(n,x) = (x+1)*p(n-1,x)-p(n-2,x) with p(0,x) = 1 and p(1,x) = x+1 [Dias].
G.f.: 1/(1-x-x^2-y*x). - Philippe Deléham, Feb 10 2012
T(n,0) = A010892(n), T(n+1,1) = A099254(n), T(n+2,2) = A128504(n). - Philippe Deléham, Mar 07 2014
T(n,k) = C(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], 4) for n>=1. - Peter Luschny, Apr 25 2016

Typo in formula corrected and information added by Johannes W. Meijer, Aug 06 2011

A168561 Riordan array (1/(1-x^2), x/(1-x^2)). Unsigned version of A049310.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 1, 0, 3, 0, 4, 0, 1, 1, 0, 6, 0, 5, 0, 1, 0, 4, 0, 10, 0, 6, 0, 1, 1, 0, 10, 0, 15, 0, 7, 0, 1, 0, 5, 0, 20, 0, 21, 0, 8, 0, 1, 1, 0, 15, 0, 35, 0, 28, 0, 9, 0, 1, 0, 6, 0, 35, 0, 56, 0, 36, 0, 10, 0, 1, 1, 0, 21, 0, 70, 0, 84, 0, 45, 0, 11, 0, 1

Offset: 0

Philippe Deléham, Nov 29 2009

Row sums: A000045(n+1), Fibonacci numbers.
A168561*A007318 = A037027, as lower triangular matrices. Diagonal sums : A077957. - Philippe Deléham, Dec 02 2009
T(n,k) is the number of compositions of n+1 into k+1 odd parts. Example: T(4,2)=3 because we have 5 = 1+1+3 = 1+3+1 = 3+1+1.
Coefficients of monic Fibonacci polynomials (rising powers of x). Ftilde(n, x) = x*Ftilde(n-1, x) + Ftilde(n-2, x), n >=0, Ftilde(-1,x) = 0, Ftilde(0, x) = 1. G.f.: 1/(1 - x*z - z^2). Compare with Chebyshev S-polynomials (A049310). - Wolfdieter Lang, Jul 29 2014

			The triangle T(n,k) begins:
n\k 0  1   2   3   4    5    6    7    8    9  10  11  12  13 14 15 ...
0:  1
1:  0  1
2:  1  0   1
3:  0  2   0   1
4:  1  0   3   0   1
5:  0  3   0   4   0    1
6:  1  0   6   0   5    0    1
7:  0  4   0  10   0    6    0    1
8:  1  0  10   0  15    0    7    0    1
9:  0  5   0  20   0   21    0    8    0    1
10: 1  0  15   0  35    0   28    0    9    0   1
11: 0  6   0  35   0   56    0   36    0   10   0   1
12: 1  0  21   0  70    0   84    0   45    0  11   0   1
13: 0  7   0  56   0  126    0  120    0   55   0  12   0   1
14: 1  0  28   0 126    0  210    0  165    0  66   0  13   0  1
15: 0  8   0  84   0  252    0  330    0  220   0  78   0  14  0  1
... reformatted by _Wolfdieter Lang_, Jul 29 2014.
------------------------------------------------------------------------

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - From _N. J. A. Sloane_, May 10 2012
Tom Copeland, Addendum to Elliptic Lie Triad
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A162515 (rows reversed), A112552, A102426 (deflated).

A168561:=proc(n,k) if n-k mod 2 = 0 then binomial((n+k)/2,k) else 0 fi end proc:
seq(seq(A168561(n,k),k=0..n),n=0..12) ; # yields sequence in triangular form

Table[If[EvenQ[n + k], Binomial[(n + k)/2, k], 0], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 16 2017 *)

T(n,k) = if ((n+k) % 2, 0, binomial((n+k)/2,k));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print();); \\ Michel Marcus, Oct 09 2016

Sum_{k=0..n} T(n,k)*x^k = A059841(n), A000045(n+1), A000129(n+1), A006190(n+1), A001076(n+1), A052918(n), A005668(n+1), A054413(n), A041025(n), A099371(n+1), A041041(n), A049666(n+1), A041061(n), A140455(n+1), A041085(n), A154597(n+1), A041113(n) for x = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 respectively. - Philippe Deléham, Dec 02 2009
T(2n,2k) = A085478(n,k). T(2n+1,2k+1) = A078812(n,k). Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000045(n+1), A006131(n), A015445(n), A168579(n), A122999(n) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Dec 02 2009
T(n,k) = binomial((n+k)/2,k) if (n+k) is even; otherwise T(n,k)=0.
G.f.: (1-z^2)/(1-t*z-z^2) if offset is 1.
T(n,k) = T(n-1,k-1) + T(n-2,k), T(0,0) = 1, T(0,1) = 0. - Philippe Deléham, Feb 09 2012
Sum_{k=0..n} T(n,k)^2 = A051286(n). - Philippe Deléham, Feb 09 2012
From R. J. Mathar, Feb 04 2022: (Start)
Sum_{k=0..n} T(n,k)*k = A001629(n+1).
Sum_{k=0..n} T(n,k)*k^2 = 0,1,4,11,... = 2*A055243(n)-A099920(n+1).
Sum_{k=0..n} T(n,k)*k^3 = 0,1,8,29,88,236,... = 12*A055243(n) -6*A001629(n+2) +A001629(n+1)-6*(A001872(n)-2*A001872(n-1)). (End)

Typo in name corrected (1(1-x^2) changed to 1/(1-x^2)) by Wolfdieter Lang, Nov 20 2010

A127670 Discriminants of Chebyshev S-polynomials A049310.

Original entry on oeis.org

1, 4, 32, 400, 6912, 153664, 4194304, 136048896, 5120000000, 219503494144, 10567230160896, 564668382613504, 33174037869887488, 2125764000000000000, 147573952589676412928, 11034809241396899282944, 884295678882933431599104, 75613185918270483380568064

Offset: 1

Wolfdieter Lang, Jan 23 2007

a(n-1) is the number of fixed n-cell polycubes that are proper in n-1 dimensions (Barequet et al., 2010).
From Rigoberto Florez, Sep 02 2018: (Start)
a(n-1) is the discriminant of the Morgan-Voyce Fibonacci-type polynomial B(n).
Morgan-Voyce Fibonacci-type polynomials are defined as B(0) = 0, B(1) = 1 and B(n) = (x+2)*B(n-1) - B(n-2) for n > 1.
The absolute value of the discriminant of Fibonacci polynomial F(n) is a(n-1).
Fibonacci polynomials are defined as F(0) = 0, F(1) = 1 and F(n) = x*F(n-1) + F(n-2) for n > 1. (End)
The first 6 values are the dimensions of the polynomial ring in 3n variables xi, yi, zi for 1 <= i <= n modulo the ideal generated by x1^a y1^b z1^c + ... + xn^a yn^b zn^c for 0 < a+b+c <= n (see Fact 2.8.1 in Haiman's paper). - Mike Zabrocki, Dec 31 2019

			n=3: The zeros are [sqrt(2),0,-sqrt(2)]. The Vn(xn[1],...,xn[n]) matrix is [[1,1,1],[sqrt(2),0,-sqrt(2)],[2,0,2]]. The squared determinant is 32 = a(3). - _Wolfdieter Lang_, Aug 07 2011

Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016, http://www.csun.edu/~ctoth/Handbook/chap14.pdf.
G. Barequet and M. Shalah, Automatic Proofs for Formulae Enumerating Proper Polycubes, 31st International Symposium on Computational Geometry (SoCG'15). Editors: Lars Arge and János Pach; pp. 19-22, 2015.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219 for T and U polynomials.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Andrei Asinowski, Gill Barequet, Ronnie Barequet, and Gunter Rote, Proper n-Cell Polycubes in n - 3 Dimensions, Journal of Integer Sequences, Vol. 15 (2012), #12.8.4.
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012. See Th. 1. [From _N. J. A. Sloane_, Oct 16 2010]
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica 30 (2010), pp. 257-275.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, preprint, 1993.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), no. 1, 17--76.
O. Khorunzhiy, Enumeration of tree-type diagrams assembled from oriented chains of edges, arXiv:2207.00766 [math.CO], 2022.
Andrew Snowden, Measures for the colored circle, arXiv:2302.08699 [math.CO], 2023.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Cf. A007701 (T-polynomials), A086804 (U-polynomials), A171860 and A191092 (fixed n-cell polycubes proper in n-2 and n-3 dimensions, resp.).
Cf. A243953, A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.
A317403 is essentially the same sequence.
Diagonal 1 of A195739.

[((n+1)^n/(n+1)^2)*2^n: n in [1..20]]; // Vincenzo Librandi, Jun 23 2014

Table[((n + 1)^n)/(n + 1)^2 2^n, {n, 1, 30}] (* Vincenzo Librandi, Jun 23 2014 *)

a(n) = ((n+1)^(n-2))*2^n, n >= 1.
a(n) = (Det(Vn(xn[1],...,xn[n])))^2 with the determinant of the Vandermonde matrix Vn with elements (Vn)i,j:= xn[i]^j, i=1..n, j=0..n-1 and xn[i]:=2*cos(Pi*i/(n+1)), i=1..n, are the zeros of S(n,x):=U(n,x/2).
a(n) = ((-1)^(n*(n-1)/2))*Product_{j=1..n} ((d/dx)S(n,x)|_{x=xn[j]}), n >= 1, with the zeros xn[j], j=1..n, given above.
a(n) = A007830(n-2)*A000079(n), n >= 2. - Omar E. Pol, Aug 27 2011
E.g.f.: -LambertW(-2*x)*(2+LambertW(-2*x))/(4*x). - Vaclav Kotesovec, Jun 22 2014

Slightly edited by Gill Barequet, May 24 2011

A054450 Triangle of partial row sums of unsigned triangle A049310(n,m), n >= m >= 0 (Chebyshev S-polynomials).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 4, 4, 1, 1, 8, 8, 5, 5, 1, 1, 13, 12, 12, 6, 6, 1, 1, 21, 21, 17, 17, 7, 7, 1, 1, 34, 33, 33, 23, 23, 8, 8, 1, 1, 55, 55, 50, 50, 30, 30, 9, 9, 1, 1, 89, 88, 88, 73, 73, 38, 38, 10, 10, 1, 1, 144, 144, 138, 138, 103, 103, 47, 47, 11, 11, 1, 1

Offset: 0

Wolfdieter Lang, Apr 27 2000 and May 08 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is Fib(z)/(1-x*z/(1-z^2)) where Fib(x)=1/(1-x-x^2) = g.f. for A000045(n+1) (Fibonacci numbers without 0).

This is the first member of the family of Riordan-type matrices obtained from the unsigned convolution matrix A049310 by repeated application of the partial row sums procedure.

			Triangle begins as:
   1;
   1,  1;
   2,  1,  1;
   3,  3,  1,  1;
   5,  4,  4,  1,  1;
   8,  8,  5,  5,  1,  1;
  13, 12, 12,  6,  6,  1,  1;
  21, 21, 17, 17,  7,  7,  1,  1;
  34, 33, 33, 23, 23,  8,  8,  1,  1;
  55, 55, 50, 50, 30, 30,  9,  9,  1, 1;
  89, 88, 88, 73, 73, 38, 38, 10, 10, 1, 1;
  ...
Fourth row polynomial (n=3): p(3,x) = 3 + 3*x + x^2 + x^3.

Cf. A000027, A000045, A029907 (row sums), A038793, A038794, A038795, A038796.

Cf. A049310, A052952, A053121, A054451, A054453, A099571, A099572, A152948.

A049310:= func< n,k | ((n+k) mod 2) eq 0 select (-1)^(Floor((n+k)/2)+k)*Binomial(Floor((n+k)/2), k) else 0 >;
A054450:= func< n,k | (&+[Abs(A049310(n,j)): j in [k..n]]) >;
[A054450(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 25 2022

A049310[n_, k_]:= A049310[n, k]= If[n<0, 0, If[k==n, 1, A049310[n-1, k-1] - A049310[n-2, k] ]];
A054450[n_, k_]:= A054450[n, k]= Sum[Abs[A049310[n,j]], {j,k,n}];
Table[A054450[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 25 2022 *)

@CachedFunction
def A049310(n, k):
    if (n<0): return 0
    elif (k==n): return 1
    else: return A049310(n-1, k-1) - A049310(n-2, k)
def A054450(n,k): return sum( abs(A049310(n,j)) for j in (k..n) )
flatten([[A054450(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 25 2022

T(n, m) = Sum_{k=m..n} |A049310(n, k)| (sequence of partial row sums in column m).

Column m recursion: T(n, m) = Sum_{j=m..n} T(j-1, m)*|A049310(n-j, 0)| + |A049310(n, m)|, n >= m >= 0, a(n, m) := 0 if n

T(n, 0) = A000045(n+1).

T(n, 1) = A052952(n-1).

T(n, 2) = A054451(n-2).

Sum_{k=0..n} T(n, k) = A029907(n) = A054453(n, 0).

G.f. for column m: Fib(x)*(x/(1-x^2))^m, m >= 0, with Fib(x) = g.f. A000045(n+1).

The corresponding square array has T(n, k) = Sum_{j=0..floor(k/2)} binomial(n+k-j, j). - Paul Barry, Oct 23 2004

From G. C. Greubel, Jul 25 2022: (Start)

T(n, 3) = A099571(n-3).

T(n, 4) = A099572(n-4).

T(n, n) = T(n, n-1) = A000012(n).

T(n, n-2) = A000027(n), n >= 2.

T(n, n-3) = A000027(n), n >= 3.

T(n, n-4) = A152948(n), n >= 4.

T(n, n-5) = A152948(n), n >= 5.

T(n, n-6) = A038793(n), n >= 6.

T(n, n-8) = A038794(n), n >= 8.

T(n, n-10) = A038795(n), n >= 10.

T(n, n-12) = A038796(n), n >= 12. (End)

A054451 Third column of triangle A054450 (partial row sums of unsigned Chebyshev triangle A049310).

Original entry on oeis.org

1, 1, 4, 5, 12, 17, 33, 50, 88, 138, 232, 370, 609, 979, 1596, 2575, 4180, 6755, 10945, 17700, 28656, 46356, 75024, 121380, 196417, 317797, 514228, 832025, 1346268, 2178293, 3524577, 5702870, 9227464, 14930334, 24157816, 39088150, 63245985, 102334135

Offset: 0

Wolfdieter Lang, Apr 27 2000

Equals triangle A173284 * [1, 2, 3, ...]. - Gary W. Adamson, Mar 03 2010

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-3,1,1).

Cf. A054450, A049310, A000045, A052952.
Cf. A007382.
Cf. A173284.

BB:=1/(1-k^2)^2/(1-k-k^2): seq(coeff(series(BB, k, n+1), k, n), n=0..50); # Zerinvary Lajos, May 16 2007

LinearRecurrence[{1,3,-2,-3,1,1},{1,1,4,5,12,17},40] (* Harvey P. Dale, Oct 06 2024 *)

Vec(-1/((x-1)^2*(x+1)^2*(x^2+x-1)) + O(x^100)) \\ Colin Barker, Jun 14 2015

a(n) = A054450(n+2, 2).
G.f.: Fib(x)/(1-x^2)^2, with Fib(x)=1/(1-x-x^2) = g.f. A000045 (Fibonacci numbers without 0).
a(2*k) = A027941(k)= F(2*k+3)-1; a(2*k+1)= F(2*(k+2))-(k+2)= A054452(k), k >= 0.
a(n-2) = Fibonacci(n+1) - binomial(n-floor(n/2), floor(n/2)), or a(n-2) = Sum_{i=0..floor(n/2)-1} binomial(n-i, i). - Jon Perry, Mar 18 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+2, k). - Paul Barry, Oct 23 2004

More terms from James Sellers, Apr 28 2000

A134511 abs(A049310) * A128174 provided both arrays are read with offset (n,k) = (0,0).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 5, 0, 4, 0, 1, 0, 8, 0, 5, 0, 1, 13, 0, 12, 0, 6, 0, 1, 0, 21, 0, 17, 0, 7, 0, 1, 34, 0, 33, 0, 23, 0, 8, 0, 1, 0, 55, 0, 50, 0, 30, 0, 9, 0, 1, 89, 0, 88, 0, 73, 0, 38, 0, 10, 0, 1, 0, 144, 0, 138, 0, 103, 0, 47, 0, 11, 0, 1, 233, 0, 232, 0, 211, 0, 141, 0, 57, 0, 12, 0, 1

Offset: 0

Gary W. Adamson, Oct 28 2007

A112552(unsigned) = A128174 * A049310.
Row sums = A134512: (1, 1, 3, 4, 10, 14, 32, 46, 99, 145, ...).
From Petros Hadjicostas, Sep 03 2019: (Start)
To prove Alois P. Heinz's claim (see the Formula section and his Maple program below) we note that, for n >= 0 and 0 <= k <= n, T(n, n-k) = Sum_{r = 0 .. infinity} abs(A049310(n,r)) * A128174(r,n-k) = Sum_{r = n-k..n} abs(A049310(n,r)) * A128174(r,n-k). But A049310(n,r) = 0 when n + r is odd and A128174(r,n-k) = 1 iff r + n - k is even. Thus, when k is odd, T(n, n-k) = 0.
Assume now k is even. Then T(n, n-k) = Sum_{r = n-k..n and n+r even} abs(A049310(n,r)) =  Sum_{r = n-k..n and n+r even} binomial((n+r)/2, r). Letting m = n-r (which is even), we see that the summation ranges from m = 0 to k over even numbers. Thus, let s = m/2, and so T(n, n-k) = Sum_{s = 0 .. k/2} binomial(n-s, n-2*s) = Sum_{s = 0 .. k/2} binomial(n-s, s) = F(n+1, k/2), where F(.,.) is the incomplete Fibonacci number from the references (see also the Formula section below).
(End)

			First few rows of the triangle T(n,k):
   1;
   0,  1;
   2,  0,  1;
   0,  3,  0,  1;
   5,  0,  4,  0,  1;
   0,  8,  0,  5,  0,  1;
  13,  0, 12,  0,  6,  0,  1;
   0, 21,  0, 17,  0,  7,  0,  1;
  34,  0, 33,  0, 23,  0,  8,  0,  1;
   0, 55,  0, 50,  0, 30,  0,  9,  0,  1;
  ...

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 141, flattened)
H. Belbachir and A. Belkhir, Combinatorial Expressions Involving Fibonacci, Hyperfibonacci, and Incomplete Fibonacci Numbers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.4.3.
A. Dil and I. Mezo, A symmetric algorithm for hyperharmonic and Fibonacci numbers, Appl. Math. Comp. 206 (2008), 942-951; in Eqs. (11), see the incomplete Fibonacci numbers.
Piero Filipponi, Incomplete Fibonacci and Lucas numbers, P. Rend. Circ. Mat. Palermo (Serie II) 45(1) (1996), 37-56; see Table 1 (p. 39) that contains the incomplete Fibonacci numbers.
A. Pintér and H.M. Srivastava, Generating functions of the incomplete Fibonacci and Lucas numbers, Rend. Circ. Mat. Palermo (Serie II) 48(3) (1999), 591-596.

Cf. A038730, A038792, A049310, A128174, A134512.

A(4n,2n) gives: A038736.

N:= 20: # for the first N rows
T128174:= Matrix(N,N,(i,j) -> `if`(j<=i, (i-j+1) mod 2, 0)):
T049310:= Matrix(N,N):
for i from 1 to N do
     P:= orthopoly[U](i-1,x/2);
     for j from 1 to i do
       T049310[i,j]:= abs(coeff(P,x,j-1))
     od
od:
A:= T049310 . T128174:
for i from 1 to N do
convert(A[i,1..i],list)
od;  # Robert Israel, Mar 02 2018
# second Maple program:
T:= (n, k)-> `if`((n+k)::odd, 0, add(binomial(n-s, s), s=0..(n-k)/2)):
seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Sep 02 2019

T[n_, k_] := If[OddQ[n+k], 0, Sum[Binomial[n-s, s], {s, 0, (n-k)/2}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 31 2021, after Alois P. Heinz *)

abs(A049310) * A128174 as infinite lower triangular matrices assuming both of them have offset (n,k) = (0,0).
From Petros Hadjicostas, Sep 03 2019: (Start)
Let F(m,r) = Sum_{j = 0..r} binomial(m-1-j, j) be the incomplete Fibonacci numbers from the references (defined for m >= 1 and 0 <= r <= floor((m-1)/2)).
As Alois P. Heinz observed, for n >= 0 and 0 <= k <= n, T(n, n-k) = F(n+1, k/2) when k is even, and = 0 otherwise (see his Maple program below).
(End)

Edited by Robert Israel, Mar 02 2018

A131774 2*A065941 - A049310.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 0, 4, 1, 1, 2, 3, 4, 1, 2, -1, 8, 2, 6, 1, 1, 2, 4, 8, 7, 6, 1, 2, -2, 12, 0, 20, 6, 8, 1, 1, 2, 4, 12, 15, 20, 13, 8, 1, 2, -3, 16, -6, 42, 9, 40, 12, 10, 1

Offset: 1

Gary W. Adamson, Jul 14 2007

Row sums = the Lucas numbers, A000032, starting (1, 3, 4, 7, 11, ...). Generally, N*A065941 - (N-1)*A049310 = triangles with row sums = Fibonacci-like sequences starting (1, (N+1), (N+1+1), ...). With N = 2, row sums of the triangle A131774 = (1, 3, 4, 7, ...).

			First few rows of the triangle:
  1;
  2,  1;
  1,  2,  1;
  2,  0,  4,  1;
  1,  2,  3,  4,  1;
  2, -1,  8,  2,  6,  1;
  1,  2,  4,  8,  7,  6,  1;
  ...

Cf. A065941, A049310, A000032, A131775, A131776, A131777, A131778.

2*A065941 - A049310 as infinite lower triangular matrices.

A131325 Triangle |3*|A049310(n,k)| - 2| read by rows, 0 <= k <= n.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 4, 2, 1, 1, 2, 7, 2, 1, 2, 7, 2, 10, 2, 1, 1, 2, 16, 2, 13, 2, 1, 2, 10, 2, 28, 2, 16, 2, 1, 1, 2, 28, 2, 43, 2, 19, 2, 1, 2, 13, 2, 58, 2, 61, 2, 22, 2, 1, 1, 2, 43, 2, 103, 2, 82, 2, 25, 2, 1, 2, 16, 2, 103, 2, 166, 2, 106, 2, 28, 2, 1, 1, 2, 61, 2, 208, 2, 250, 2

Offset: 0

Gary W. Adamson, Jun 28 2007

			First few rows of the triangle:
  1;
  2,  1;
  1,  2,  1;
  2,  4,  2,  1;
  1,  2,  7,  2,  1;
  2,  7,  2, 10,  2,  1;
  1,  2, 16,  2, 13,  2,  1;
  ...

Cf. A049310, A131324, A131326 (row sums), A131327.

A131325 := proc(n,k)
    abs(3*abs(A049310(n,k))-2) ;
end proc:
seq(seq(A131325(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Aug 13 2012

Definition corrected by David Scambler, Aug 12 2012

A131775 3A065941 - 2A049310.

Original entry on oeis.org

1, 3, 1, 1, 3, 1, 3, -1, 6, 1, 1, 3, 3, 6, 1, 3, -3, 12, 1, 9, 1, 1, 3, 3, 12, 8, 9, 1, 3, -5, 18, -5, 30, 6, 12, 1, 1, 3, 1, 18, 15, 30, 16, 12, 1, 3, -7, 24, -19, 63, 3, 60, 14, 15, 1

Offset: 1

Gary W. Adamson, Jul 14 2007

Row sums = A000285, (a Fibonacci-like sequence) starting (1, 4, 5, 9, 14, 23, ...).

			Table begins:
  1;
  3,  1;
  1,  3,  1;
  3, -1,  6,  1;
  1,  3,  3,  6,  1;
  3, -3, 12,  1,  9,  1;
  1,  3,  3, 12,  8,  9,  1;
  3, -5, 18, -5, 30,  6, 12,  1;
  1,  3,  1, 18, 15, 30, 16, 12,  1;
  ...

Cf. A065941, A049310, A131774, A131776, A131777, A131778.

3*A065941 - 2*A049310.

A131776 4A065941 - 3A049310.

Original entry on oeis.org

1, 4, 1, 1, 4, 1, 4, -2, 8, 1, 1, 4, 3, 8, 1, 4, -5, 16, 0, 12, 1, 1, 4, 2, 16, 9, 12, 1, 4, -8, 24, -10, 40, 6, 16, 1, 1, 4, -2, 24, 15, 40, 19, 16, 1, 4, -11, 32, -32, 84, -3, 80, 16, 20, 1

Offset: 1

Gary W. Adamson, Jul 14 2007

Row sums = A022095, a Fibonacci-like sequence starting (1, 5, 6, 11, 17, 28, ...).

			First few rows of the triangle:
  1;
  4,  1;
  1,  4,  1;
  4, -2,  8,  1;
  1,  4,  3,  8,  1;
  4, -5, 16,  0, 12,  1;
  1,  4,  2, 16,  9, 12,  1;
  ...

Cf. A065941, A049310, A022095, A131774, A131775, A131777, A131778.

4*A065941 - 3*A049310 as infinite lower triangular matrices.

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A101950 Product of A049310 and A007318 as lower triangular matrices.

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A168561 Riordan array (1/(1-x^2), x/(1-x^2)). Unsigned version of A049310.

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A127670 Discriminants of Chebyshev S-polynomials A049310.

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A054450 Triangle of partial row sums of unsigned triangle A049310(n,m), n >= m >= 0 (Chebyshev S-polynomials).

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A054451 Third column of triangle A054450 (partial row sums of unsigned Chebyshev triangle A049310).

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A134511 abs(A049310) * A128174 provided both arrays are read with offset (n,k) = (0,0).

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A131775 3A065941 - 2A049310.

A131776 4A065941 - 3A049310.