A106181 -id:A106181 - cp's OEIS frontend

A106180 Matrix inverse of number triangle A046854.

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 0, 2, -2, -1, 1, -2, 2, 3, -3, -1, 1, 0, -5, 5, 4, -4, -1, 1, 5, -5, -9, 9, 5, -5, -1, 1, 0, 14, -14, -14, 14, 6, -6, -1, 1, -14, 14, 28, -28, -20, 20, 7, -7, -1, 1, 0, -42, 42, 48, -48, -27, 27

Offset: 0

Paul Barry, Apr 24 2005

First column is A105523; second column is A106181.
Triangle T(n,k), 0 <= k <= n, read by rows given by [ -1, 1, -1, 1, -1, 1, -1, 1, -1, 1,...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 29 2006
A124448*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 16 2007

			Triangle begins
   1;
  -1,  1;
   0, -1,  1;
   1, -1, -1,  1;
   0,  2, -2, -1,  1;
  -2,  2,  3, -3, -1,  1;
   0, -5,  5,  4, -4, -1,  1;

Cf. A000108.

Riordan array (1-y, y) where y=-(1-sqrt(1+4x^2))/(2x).
Sum_{k=0..n} abs(T(n,k)) = A063886(n). - Philippe Deléham, Oct 06 2006
T(0,0)=1; T(n,k)=0 if k < 0 or if k > n; T(n,0) = -T(n-1,0) - T(n-1,1); T(n,k) = T(n,k-1) - T(n-1,k+1) for k >= 1. - Philippe Deléham, Oct 27 2007
T(2n,0) = A000007(n); T(2n+2,2k+2) = -T(2n+2,2k+1) = (-1)^(n-k)*A039598(n,k); T(2n+1,2k+1) = -T(2n+1,2k) = (-1)^(n-k)*A039599(n,k). - Philippe Deléham, Oct 29 2007
Sum_{k>=0} T(m,k)*T(n,k)*(-1)^k = T(m+n,0) = A105523(m+n). - Philippe Deléham, Jan 24 2010

A099363 An inverse Chebyshev transform of 1-x.

Original entry on oeis.org

1, -1, 1, -2, 2, -5, 5, -14, 14, -42, 42, -132, 132, -429, 429, -1430, 1430, -4862, 4862, -16796, 16796, -58786, 58786, -208012, 208012, -742900, 742900, -2674440, 2674440, -9694845, 9694845, -35357670, 35357670, -129644790, 129644790, -477638700, 477638700, -1767263190, 1767263190

Offset: 0

Paul Barry, Oct 13 2004

Second binomial transform of the expansion of c(-x)^3. The g.f. is transformed to 1-x under the Chebyshev transformation A(x)->(1/(1+x^2))A(x/(1+x^2)).

A208355(n) = abs(a(n)). - Reinhard Zumkeller, Mar 03 2012

Cf. A000245.

Cf. A106181, A129996 and A208355, which also consist of duplicated Catalan numbers A000108. - M. F. Hasler, Aug 25 2012

Table[(-1)^n CatalanNumber[Floor[(n+1)/2]], {n, 0, 38}] (* Jean-François Alcover, Jun 11 2019 *)

A099363(n)=(-1)^n*A000108((n+1)\2) \\ M. F. Hasler, Aug 25 2012

def A099363_list(n) :
    D = [0]*(n+2); D[1] = 1
    b = True; h = 2; R = []
    for i in range(2*n-1) :
        if b :
            for k in range(h,0,-1) : D[k] -= D[k-1]
            h += 1; R.append((-1)^(h//2)*D[2])
        else :
            for k in range(1,h, 1) : D[k] += D[k+1]
        b = not b
    return R
A099363_list(39) # Peter Luschny, Jun 03 2012

G.f.: (1-(1-x)c(x^2))/x where c(x) is the g.f. of the Catalan numbers A000108.
a(n) = sum{k=0..n, (k+1)C(n, (n-k)/2)(0^k-sum{j=0..k, C(k, j)(-1)^(k-j)*j})(1+(-1)^(n-k))/(n+k+2)}.
a(n) = (-1)^n A208355(n) = (-1)^n A000108([(n+1)/2]): Repeated Catalan numbers with alternating sign. - M. F. Hasler, Aug 25 2012
Conjecture: (n+3)*a(n) +(-n-1)*a(n-1) -4*n*a(n-2) +4*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 26 2012

A129996 a(n) = (-1)^[(n+1)/2] A000108([n/2]+1).

Original entry on oeis.org

1, -1, -2, 2, 5, -5, -14, 14, 42, -42, -132, 132, 429, -429, -1430, 1430, 4862, -4862, -16796, 16796, 58786, -58786, -208012, 208012, 742900, -742900, -2674440, 2674440, 9694845, -9694845, -35357670, 35357670, 129644790, -129644790, -477638700, 477638700

Offset: 0

Paul Curtz, Jun 15 2007

sign

Terms of A000108(1,...) repeated, changing sign between the two repeated terms of same magnitude, but keeping the same sign for the next (larger) term. - M. F. Hasler, Aug 25 2012

Cf. A099363, A106181 and A208355, which also consist of duplicated terms of A000108. - M. F. Hasler, Aug 25 2012

A129996(n)=(-1)^((n+1)\2)*A000108(n\2+1) \\ - M. F. Hasler, Aug 25 2012

First differences of A129110: a(n)=A129110(n+1)-A129110(n).

Definition (formula) corrected by M. F. Hasler, Aug 25 2012

A129110 A transformation of the Catalan sequence.

Original entry on oeis.org

1, 2, 1, -1, 1, 6, 1, -13, 1, 43, 1, -131, 1, 430, 1, -1429, 1, 4863, 1, -16795, 1, 58787, 1, -208011, 1, 742901, 1, -2674439, 1, 9694846, 1, -35357669, 1, 129644791, 1, -477638699, 1, 1767263191, 1, -6564120419, 1, 24466267021, 1, -91482563639, 1, 343059613651

Offset: 0

Paul Curtz, May 24 2007

sign

a(2n)=1. a(2n+1)= 1+(-1)^n*C(n+1), where C=A000108.

a(n+1)-a(n) = A129996(n) (similar to A099363 and A106181).

Edited and extended by R. J. Mathar, Jul 04 2008

cp's OEIS Frontend

A106180 Matrix inverse of number triangle A046854.

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A099363 An inverse Chebyshev transform of 1-x.

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A129996 a(n) = (-1)^[(n+1)/2] A000108([n/2]+1).

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A129110 A transformation of the Catalan sequence.

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