A112553 -id:A112553 - cp's OEIS frontend

A112552 A modified Chebyshev transform of the second kind.

1, 0, 1, -2, 0, 1, 0, -3, 0, 1, 3, 0, -4, 0, 1, 0, 6, 0, -5, 0, 1, -4, 0, 10, 0, -6, 0, 1, 0, -10, 0, 15, 0, -7, 0, 1, 5, 0, -20, 0, 21, 0, -8, 0, 1, 0, 15, 0, -35, 0, 28, 0, -9, 0, 1, -6, 0, 35, 0, -56, 0, 36, 0, -10, 0, 1, 0, -21, 0, 70, 0, -84, 0, 45, 0, -11, 0, 1, 7, 0, -56, 0, 126, 0, -120, 0, 55, 0, -12, 0, 1

Offset: 0

Paul Barry, Sep 13 2005

Row sums are A112553.
Inverse is A112554.
Riordan array product (1/(1+x^2), x)*(1/(1+x^2), x/(1+x^2)).

			Triangle begins as:
   1;
   0,   1;
  -2,   0,   1;
   0,  -3,   0,   1;
   3,   0,  -4,   0,   1;
   0,   6,   0,  -5,   0,   1;
  -4,   0,  10,   0,  -6,   0,  1;
   0, -10,   0,  15,   0,  -7,  0,  1;
   5,   0, -20,   0,  21,   0, -8,  0,   1;
   0,  15,   0, -35,   0,  28,  0, -9,   0,   1;
  -6,   0,  35,   0, -56,   0, 36,  0, -10,   0, 1;
   0, -21,   0,  70,   0, -84,  0, 45,   0, -11, 0, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A049310, A052952, A112553, A112554, A128174.

[(-1)^Floor((n-k)/2)*((1+(-1)^(n+k))/2)*Binomial(Floor((n+k+2)/2), k+1): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 13 2022

Table[(-1)^Floor[(n-k)/2]*((1+(-1)^(n+k))/2)*Binomial[(n+k+2)/2, k+1], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 13 2022 *)

flatten([[(-1)^floor((n-k)/2)*((1+(-1)^(n+k))/2)*binomial((n+k+2)/2, k+1) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jan 13 2022

Riordan array (1/(1+x^2)^2, x/(1+x^2)).
T(n, k) = (-1)^floor((n-k)/2)*Sum_{j=0..n} (1+(-1)^(n-j))*(1+(-1)^(j-k))*binomial((j+k)/2, k)/4.
Unsigned triangle = A128174 * A149310, as infinite lower triangular matrices, with row sums A052952: (1, 1, 3, 4, 8, 12, 21, 33, ...). - Gary W. Adamson, Oct 28 2007
T(n, k) = (-1)^floor((n-k)/2)*((1 + (-1)^(n+k))/2)*binomial((n+k+2)/2, k+1). - G. C. Greubel, Jan 13 2022
T(n,k) = A049310(n+1,k+1) . - R. J. Mathar, Feb 07 2024

A112554 Riordan array (c(x^2)^2, x*c(x^2)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 5, 0, 4, 0, 1, 0, 9, 0, 5, 0, 1, 14, 0, 14, 0, 6, 0, 1, 0, 28, 0, 20, 0, 7, 0, 1, 42, 0, 48, 0, 27, 0, 8, 0, 1, 0, 90, 0, 75, 0, 35, 0, 9, 0, 1, 132, 0, 165, 0, 110, 0, 44, 0, 10, 0, 1, 0, 297, 0, 275, 0, 154, 0, 54, 0, 11, 0, 1, 429, 0, 572, 0, 429, 0, 208, 0, 65, 0, 12, 0, 1

Offset: 0

Paul Barry, Sep 13 2005

Inverse of A112552.

The n-th row polynomial (in descending powers of x) is equal to the n-th degree Taylor polynomial of the polynomial function (1 - x^4)*(1 + x^2)^n about 0. For example, when n = 6, (1 - x^4)*(1 + x^2)^6 = 1 + 6*x^2 + 14*x^4 + 14*x^6 + O(x^8). - Peter Bala, Feb 19 2018

			Triangle begins
   1;
   0, 1;
   2, 0,  1;
   0, 3,  0, 1;
   5, 0,  4, 0, 1;
   0, 9,  0, 5, 0, 1;
  14, 0, 14, 0, 6, 0, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Peter Bala, A 4-parameter family of embedded Riordan arrays

Row sums A037952, matrix inverse A112552.

Cf. A000108, A037952 (row sums), A112552, A112553.

A112554:= func< n,k | ((1+(-1)^(n-k))/2)*(Binomial(n, Floor((n-k)/2)) - Binomial(n, Floor((n-k-4)/2))) >;
[A112554(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 13 2022

seq(seq((1 + (-1)^(n-k))/2*( binomial(n, floor((n - k)/2)) - binomial(n, floor((n - k - 4)/2 )) ), k = 0..n), n = 0..10); # Peter Bala, Feb 19 2018

T[n_, k_] := (1 + (-1)^(n-k))/2 (Binomial[n, Floor[(n-k)/2]] - Binomial[n, Floor[(n-k-4)/2]]);
Table[T[n, k], {n, 0, 12}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

# Algorithm of L. Seidel (1877)
# Prints the first n rows of a signed version of the triangle.
def Signed_A112554_triangle(n) :
    D = [0]*(n+4); D[1] = 1
    b = False; h = 2
    for i in range(2*n+2) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
            h += 1
        else :
            for k in range(1,h, 1) : D[k] -= D[k+1]
        b = not b
        if b and i > 0 : print([D[z] for z in (2..h-1)])
Signed_A112554_triangle(13) # Peter Luschny, May 01 2012

Sum_{k=0..n} T(n, k) = binomial(n+1, floor(n/2)) = A037952(n+1).

T(n, k) = ((1 + (-1)^(n-k))/2)*binomial(n, floor((n-k)/2)) - binomial(n, floor((n-k-4)/2 )). - Peter Bala, Feb 19 2018

A098554 G.f.: x(1-x^2)/((1+x^2)(1+x+x^2)).

Original entry on oeis.org

0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2

Offset: 0

N. J. A. Sloane, Oct 26 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
G. I. Lehrer and G. B. Segal, Homology stability for classical regular semisimple varieties, Math. Zeit., 236 (2001), 251-290; see Th. 7.12.
Index entries for linear recurrences with constant coefficients, signature (-1,-2,-1,-1).

I:=[0,1,-1,-2]; [n le 4 select I[n] else -Self(n-1) - 2*Self(n-2) -Self(n-3) - Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 17 2018

CoefficientList[Series[x*(1-x^2)/((1+x^2)*(1+x+x^2)),{x,0,110}],x] (* or *) LinearRecurrence[{-1,-2,-1,-1},{0,1,-1,-2},110] (* Harvey P. Dale, Jan 16 2016 *)

x='x+O('x^30); concat([0], Vec(x*(1-x^2)/((1+x^2)*(1+x+x^2)))) \\ G. C. Greubel, Jan 17 2018

Let b(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(0^(n-2k)-(-1)^(n-2k)). Then a(n) = b(n) - b(n-2), or a(n) = Sum_{j=0..n} b(n-j)*(binomial(1, j/2)*(-1)^(j/2)*(1+(-1)^j)/2). The g.f. is obtained from the g.f. x/(1+x) of 0^n-(-1)^n by applying the transformation G(x)->((1-x^2)/(1+x^2))G(x/(1+x^2)). - Paul Barry, Oct 26 2004
a(n) = (-1)^n*(A112553(n-1) - A112553(n-3)). - R. J. Mathar, Sep 27 2014
a(0)=0, a(1)=1, a(2)=-1, a(3)=-2, a(n) = a(n-1) - 2*a(n-2) - a(n-3) - a(n-4). - Harvey P. Dale, Jan 16 2016

A247918 Expansion of (1 + x) / ((1 - x^4) * (1 + x^4 - x^5)) in powers of x.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 2, 0, -1, 2, -1, 2, 1, -2, 4, -3, 1, 4, -5, 7, -4, -2, 10, -12, 11, -1, -11, 22, -23, 13, 11, -33, 45, -35, 3, 44, -78, 81, -37, -41, 122, -158, 119, 4, -163, 281, -276, 115, 167, -443, 558, -391, -52, 611, -1000, 949

Offset: 0

Michael Somos, Sep 26 2014

			G.f. = 1 + x + x^5 + x^6 + x^8 + x^11 + 2*x^13 - x^15 + 2*x^16 - x^17 + ...

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

Cf. A010892, A056594, A077905, A112553, A176971, A247907.

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!((1 + x)/((1-x^4)*(1+x^4-x^5)))); // G. C. Greubel, Aug 04 2018

CoefficientList[Series[(1+x)/((1-x^4)(1+x^4-x^5)), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 27 2014 *)

{a(n) = if( n<0, n=-8-n; polcoeff( -1/((1-x)*(1-x+x^2)*(1+x^2)*(1 - x^2 - x^3)) + x * O(x^n), n), polcoeff( 1/((1-x)*(1-x+x^2)*(1+x^2)*(1+x-x^3)) + x * O(x^n), n))};

def A247918_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)/((1-x^4)*(1+x^4-x^5)) ).list()
A247918_list(70) # G. C. Greubel, Aug 08 2022

G.f.: 1 / ((1 - x) * (1 - x + x^2) * (1 + x^2) * (1 + x - x^3)).
a(n) = a(n+1) + a(n+5) - mod(floor((n-1)/2), 2) for all n in Z.
a(n) = -A247907(-8-n) for all n in Z.
Convolution of A077905 and A112553.
a(n) = (35 + 7*(A056594(n) + 3*A056594(n-1)) + 10*(3*A010892(n) - A010892(n-1)) - 2*(A176971(n) + 4*A172971(n-1) + 12*A176971(n-2)))/70. - G. C. Greubel, Aug 08 2022

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A112552 A modified Chebyshev transform of the second kind.

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A112554 Riordan array (c(x^2)^2, x*c(x^2)), c(x) the g.f. of A000108.

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A098554 G.f.: x*(1-x^2)/((1+x^2)*(1+x+x^2)).

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A247918 Expansion of (1 + x) / ((1 - x^4) * (1 + x^4 - x^5)) in powers of x.

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Magma

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A098554 G.f.: x(1-x^2)/((1+x^2)(1+x+x^2)).