A178112 -id:A178112 - cp's OEIS frontend

A178113 Transform of C(n+1,floor((n+1)/2)) by A178112.

1, 2, 2, 4, 5, 10, 13, 26, 35, 70, 96, 192, 267, 534, 750, 1500, 2123, 4246, 6046, 12092, 17303, 34606, 49721, 99442, 143365, 286730, 414584, 829168, 1201917, 2403834, 3492117, 6984234, 10165779, 20331558, 29643870, 59287740, 86574831, 173149662

Offset: 0

Paul Barry, May 20 2010

Hankel transform is A178115.

Cf. A178112, A178115, A005773.

a(n) = Sum_{k=0..n} C(floor(n/2),floor(k/2))*((1+(-1)^(n-k))/2)*C(k+1,floor((k+1)/2)). [The formula seems to generate A026392, not these terms. - R. J. Mathar, Feb 10 2015]

Conjecture: a(2*n) = A005773(n+1), a(2*n+1) = 2*a(2*n). - Jason Yuen, Feb 09 2025

A178111 Number triangle T(n,k)=(-1)^((n-k)/2)C(floor(n/2),floor(k/2))(1+(-1)^(n-k))/2.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -1, 0, 1, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, -1, 0, 3, 0, -3, 0, 1, 0, -1, 0, 3, 0, -3, 0, 1, 1, 0, -4, 0, 6, 0, -4, 0, 1, 0, 1, 0, -4, 0, 6, 0, -4, 0, 1, -1, 0, 5, 0, -10, 0, 10, 0, -5, 0, 1, 0, -1, 0, 5, 0, -10, 0, 10, 0, -5, 0, 1, 1, 0, -6, 0, 15, 0, -20, 0, 15, 0, -6, 0, 1

Offset: 0

Paul Barry, May 20 2010

Coefficient array of orthogonal polynomials P(n,x)=xP(n-1,x)-((1+(-1)^n)/2)*P(n-2,x), P(0,x)=1,P(1,x)=x.

Inverse is A178112.

			Triangle begins
1,
0, 1,
-1, 0, 1,
0, -1, 0, 1,
1, 0, -2, 0, 1,
0, 1, 0, -2, 0, 1,
-1, 0, 3, 0, -3, 0, 1,
0, -1, 0, 3, 0, -3, 0, 1,
1, 0, -4, 0, 6, 0, -4, 0, 1,
0, 1, 0, -4, 0, 6, 0, -4, 0, 1,
-1, 0, 5, 0, -10, 0, 10, 0, -5, 0, 1
Production matrix is
0, 1,
-1, 0, 1,
0, 0, 0, 1,
0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 1,
0, 0, 0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Production matrix of inverse is
0, 1,
1, 0, 1,
0, 0, 0, 1,
0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1

P := (n,x) -> `if`(n < 2, x^n, x*P(n-1,x) - ((1+(-1)^n)/2)*P(n-2,x)):
ListTools:-Flatten([seq(PolynomialTools:-CoefficientList(P(n,x), x),n=0..12)]);
# Peter Luschny, Aug 10 2019

cp's OEIS Frontend

A178113 Transform of C(n+1,floor((n+1)/2)) by A178112.

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Formula

A178111 Number triangle T(n,k)=(-1)^((n-k)/2)C(floor(n/2),floor(k/2))(1+(-1)^(n-k))/2.

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Maple

cp's OEIS Frontend

A178113 Transform of C(n+1,floor((n+1)/2)) by A178112.

Views

Author

Keywords

Comments

Crossrefs

Formula

A178111 Number triangle T(n,k)=(-1)^((n-k)/2)*C(floor(n/2),floor(k/2))*(1+(-1)^(n-k))/2.

Views

Author

Keywords

Comments

Examples

Programs

Maple

A178111 Number triangle T(n,k)=(-1)^((n-k)/2)C(floor(n/2),floor(k/2))(1+(-1)^(n-k))/2.