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

0,4

Comments

Row sums are A112553.

Inverse is A112554.

Riordan array product (1/(1+x^2), x)*(1/(1+x^2), x/(1+x^2)).

Links

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

Formula

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

Example

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;

Mathematica

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 *)

Prog

(Magma) [(-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
(Sage) 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

Crossrefs

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

Sequence in context: A298972 A143728 A127368 * A048154 A320602 A134511

Adjacent sequences: A112549 A112550 A112551 * A112553 A112554 A112555

Keyword

easy,sign,tabl

Author

Paul Barry, Sep 13 2005

Status

approved