A115142 -id:A115142 - cp's OEIS frontend

A115141 Convolution of A115140 with itself.

1, -2, -1, -2, -5, -14, -42, -132, -429, -1430, -4862, -16796, -58786, -208012, -742900, -2674440, -9694845, -35357670, -129644790, -477638700, -1767263190, -6564120420, -24466267020, -91482563640, -343059613650, -1289904147324, -4861946401452, -18367353072152

Offset: 0

Wolfdieter Lang, Jan 13 2006

This is the so-called A-sequence for the Riordan triangles A053122, A110162, A129818, A158454 and signed A158909. For the notion of Z- and A-sequences for Riordan arrays see the W. Lang link under A006232 with details and references. Wolfdieter Lang, Dec 20 2010. [Revised, Nov 13 2012, Nov 22 2012 and Oct 22 2019]

a(n)*(-1)^n is the A-sequence for the Riordan triangle A111125. - Wolfdieter Lang, Jun 26 2011

			G.f. = 1 - 2*x - x^2 - 2*x^3 - 5*x^4 - 14*x^5 - 42*x^6 - 132*x^7 - 429*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000108, A115140, A115142 - A115149.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-2*x+Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019

a[n_] := -First[ ListConvolve[ cc = Array[ CatalanNumber, n-1, 0], cc]]; a[0] = 1; a[1] = -2; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 21 2011 *)
CoefficientList[Series[(1-2*x+Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

{a(n) = if( n<1, n==0, -(n==1) -binomial( 2*n-2, n-1) / n)} /* Michael Somos, Mar 28 2012 */

((1-2*x+sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^2 = (1-x) - x*c(x) with the o.g.f. c(x) = (1-sqrt(1-4*x) )/(2*x) of A000108 (Catalan numbers).
a(0)=1, a(1)=-2, a(n) = -C(n-1), n>=2, with C(n):=A000108(n) (Catalan). The start [1, -2] is row n=2 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
The convolution inverse is A000108(x)^2. - Michael Somos, Mar 28 2012
REVERT transform is A069271. - Michael Somos, Mar 28 2012
EULER transform of -A060165. - Michael Somos, Mar 28 2012
D-finite with recurrence: n*a(n) +2*(-2*n+3)*a(n-1)=0. - R. J. Mathar, Feb 21 2020

A115143 a(n) = -4binomial(2n-5, n-4)/n for n > 0 and a(0) = 1.

Original entry on oeis.org

1, -4, 2, 0, -1, -4, -14, -48, -165, -572, -2002, -7072, -25194, -90440, -326876, -1188640, -4345965, -15967980, -58929450, -218349120, -811985790, -3029594040, -11338026180, -42550029600, -160094486370, -603784920024, -2282138106804, -8643460269248, -32798844771700

Offset: 0

Wolfdieter Lang, Jan 13 2006

Previous name: Fourth convolution of A115140.
a(n+4) := - convolution ( A000108(n+1) ), n=0,1,... - Tilman Neumann, Jan 05 2009
Self-convolution of A115141. - R. J. Mathar, Sep 26 2012

Seiichi Manyama, Table of n, a(n) for n = 0..1669

Cf. A115139 - A115142, A115144 - A115149, A099376, A000108.

[1,-4,2] cat [-4*Binomial(2*n-5,n-4)/n: n in [3..30]]; // G. C. Greubel, Feb 12 2019

A115143 := n -> `if`(n=0, 1, -4*binomial(2*n-5,n-4)/n):
seq(A115143(n), n=0..28); # Peter Luschny, Feb 27 2017
A115143List := proc(m) local A, P, n; A := [1,-4,2,0]; P := [-1,0];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A115143List(27); # Peter Luschny, Mar 26 2022

Join[{1},Table[-4*Binomial[2n-5,n-4]/n,{n,30}]] (* Harvey P. Dale, Dec 01 2017 *)
CoefficientList[Series[(1-4*x+2*x^2+(1-2*x)*Sqrt[1-4*x])/2, {x,0,30}], x] (* G. C. Greubel, Feb 12 2019 *)

my(x='x+O('x^30)); Vec((1-4*x+2*x^2 +(1-2*x)*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019

[1,-4,2] + [-4*binomial(2*n-5,n-4)/n for n in (3..30)] # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^4 = P(5, x) - x*P(4, x)*c(x) with the o.g.f. c(x) := (1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(5, x) = 1-3*x+x^2 and P(4, x) = 1-2*x.
a(n) = -C4(n-4), n>=4, with C4(n) := A002057(n) (fourth convolution of Catalan numbers). a(0)=1, a(1)=-4, a(2)=2, a(3)=0. [1, -4, 2] is row n=4 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
E.g.f.: 1 - 3*x + 1/2*x^2 - x*Q(0), where Q(k)= 1 - 2*x/(k+2 - (k+2)*(2*k+1)/(2*k+1 - (k+2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 28 2013
D-finite with recurrence n*(n-4)*a(n) -2*(2*n-5)*(n-3)*a(n-1)=0. - R. J. Mathar, Sep 15 2024

Simpler name from Peter Luschny, Feb 27 2017

A115145 Sixth convolution of A115140.

Original entry on oeis.org

1, -6, 9, -2, 0, 0, -1, -6, -27, -110, -429, -1638, -6188, -23256, -87210, -326876, -1225785, -4601610, -17298645, -65132550, -245642760, -927983760, -3511574910, -13309856820, -50528160150, -192113383644, -731508653106, -2789279908316, -10649977831752, -40715807302800

Offset: 0

Wolfdieter Lang, Jan 13 2006

Seiichi Manyama, Table of n, a(n) for n = 0..1669

Cf. A115139, A115140, A115141, A115142, A115143.

Cf. A115144, A115146, A115147, A115148, A115149.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019

CoefficientList[Series[(1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

my(x='x+O('x^30)); Vec((1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2) *sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019

((1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^6 = P(7, x) - x*P(6, x)*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(7, x)=1-5*x+6*x^2-x^3 and P(6, x) = 1-4*x+3*x^2.
a(n) = -C6(n-6), n>=6, with C6(n) = A003517(n+2) (sixth convolution of Catalan numbers). a(0)=1, a(1)=-6, a(2)=9, a(3)=-2, a(4)=0=a(5). [1, -6, 9, -2] is row n=6 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
D-finite with recurrence +n*(n-6)*a(n) -2*(2*n-7)*(n-4)*a(n-1)=0. - R. J. Mathar, Sep 23 2021

A115146 Seventh convolution of A115140.

Original entry on oeis.org

1, -7, 14, -7, 0, 0, 0, -1, -7, -35, -154, -637, -2548, -9996, -38760, -149226, -572033, -2187185, -8351070, -31865925, -121580760, -463991880, -1771605360, -6768687870, -25880277150, -99035193894, -379300783092, -1453986335186, -5578559816632, -21422369201800

Offset: 0

Wolfdieter Lang, Jan 13 2006

Seiichi Manyama, Table of n, a(n) for n = 0..1670

Cf. A115139, A115140, A115141, A115142, A115143.

Cf. A115144, A115145, A115147, A115148, A115149.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019

CoefficientList[Series[(1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3) *Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

my(x='x+O('x^30)); Vec((1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3) *sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019

((1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3)*sqrt(1-4*x))/2 ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^7 = P(8, x) - x*P(7, x)*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(8, x)=1-6*x+10*x^2-4*x^3 and P(7, x)=1-5*x+6*x^2-x^3.
a(n) = -C7(n-7), n>=7, with C7(n):=A000588(n+3) (seventh convolution of Catalan numbers). a(0)=1, a(1)=-7, a(2)=14, a(3)=-7, a(4)=a(5)=a(6)=0. [1, -7, 14, -7] is row n=7 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
D-finite with recurrence n*(n-7)*a(n) -2*(n-4)*(2*n-9)*a(n-1)=0. - R. J. Mathar, Sep 15 2024

A144701 Hankel transform of expansion of 1/c(x)^3, c(x) the g.f. of A000108.

Original entry on oeis.org

1, -9, 26, -25, -36, 133, -132, -81, 375, -374, -144, 806, -805, -225, 1480, -1479, -324, 2451, -2450, -441, 3773, -3772, -576, 5500, -5499, -729, 7686, -7685, -900, 10385, -10384, -1089, 13651, -13650, -1296, 17538, -17537

Offset: 0

Paul Barry, Sep 19 2008

Hankel transform of A115142.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-4,-10,-16,-19,-16,-10,-4,-1).

Cf. A000108, A099504, A115142, A128504.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2)*(1-5*x+x^2)/(1+x+x^2)^4 )); // G. C. Greubel, Jun 16 2022

LinearRecurrence[{-4,-10,-16,-19,-16,-10,-4,-1}, {1,-9,26,-25,-36,133,-132,-81}, 40] (* G. C. Greubel, Jun 16 2022 *)

def A144701_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^2)*(1-5*x+x^2)/(1+x+x^2)^4 ).list()
A144701_list(40) # G. C. Greubel, Jun 16 2022

G.f.: (1+x)*(1-x)*(1-5*x+x^2)/(1+x+x^2)^4.

a(n) = (6 - 7*n - 9*n^2 - 2*n^3)*cos(2*Pi*n/3)/6 - sqrt(3)*(42 + 55*n + 21*n^2 + 2*n^3)*sin(2*Pi*n/3)/18.

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A115141 Convolution of A115140 with itself.

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A115143 a(n) = -4*binomial(2*n-5, n-4)/n for n > 0 and a(0) = 1.

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A115145 Sixth convolution of A115140.

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A115146 Seventh convolution of A115140.

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A144701 Hankel transform of expansion of 1/c(x)^3, c(x) the g.f. of A000108.

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Formula

A115143 a(n) = -4binomial(2n-5, n-4)/n for n > 0 and a(0) = 1.