A115139 -id:A115139 - cp's OEIS frontend

A115142 Third convolution of A115140.

1, -3, 0, -1, -3, -9, -28, -90, -297, -1001, -3432, -11934, -41990, -149226, -534888, -1931540, -7020405, -25662825, -94287120, -347993910, -1289624490, -4796857230, -17902146600, -67016296620, -251577050010, -946844533674, -3572042254128, -13505406670700

Offset: 0

Wolfdieter Lang, Jan 13 2006

Seiichi Manyama, Table of n, a(n) for n = 0..1668

Cf. A000108, A000245, A071724.

Cf. A115139, A115140, A115141, A115143 - A115149.

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

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

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

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

O.g.f.: 1/c(x)^3 = P(4, x) - x*P(3, 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(4, x)=1-2*x and P(3, x)=1-x.
a(n) = -C3(n-3), n >= 3, with C3(n):= A000245(n+1) (third convolution of Catalan numbers). a(0)=1, a(1)=-3, a(2)=0. [1, -3] is the row n=3 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
D-finite with recurrence +n*(n-3)*a(n) -2*(n-2)*(2*n-5)*a(n-1)=0. - R. J. Mathar, Sep 23 2021
O.g.f.: (1/8)*(1 + sqrt(1 - 4*x))^3 = ( hypergeom([-1/4, -3/4], [-1/2], 4*x) )^2. - Peter Bala, Mar 04 2022

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

A115144 Fifth convolution of A115140.

Original entry on oeis.org

1, -5, 5, 0, 0, -1, -5, -20, -75, -275, -1001, -3640, -13260, -48450, -177650, -653752, -2414425, -8947575, -33266625, -124062000, -463991880, -1739969550, -6541168950, -24647883000, -93078189750, -352207870014, -1335293573130, -5071418015120, -19293438101000

Offset: 0

Wolfdieter Lang, Jan 13 2006

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

Cf. A000344, A115139 - A115143, A115145 - A115149.

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

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

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

((1-5*x+5*x^2 +(1-3*x+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)^5 = P(6, x) - x*P(5, 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(6, x)=1-4*x+3*x^2 and P(5, x)=1-3*x+x^2.
a(n) = -C5(n-5), n>=5, with C5(n) = A000344(n+2) (fifth convolution of Catalan numbers). a(0)=1, a(1)=-5, a(2)=5, a(3)=0=a(4). [1, -5, 5] is row n=5 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
D-finite with recurrence +n*(n-5)*a(n) -2*(n-3)*(2*n-7)*a(n-1)=0. - R. J. Mathar, Sep 23 2021
From Peter Bala, Mar 05 2023: (Start)
a(n) = binomial(2*n - 6, n) - binomial(2*n - 6, n + 1).
a(n) = = -5/(n - 5)*binomial(2*n - 6, n) for n != 5.
a(n) = -A000344(n-3) for n >= 5. (End)

A115147 Eighth convolution of A115140.

Original entry on oeis.org

1, -8, 20, -16, 2, 0, 0, 0, -1, -8, -44, -208, -910, -3808, -15504, -62016, -245157, -961400, -3749460, -14567280, -56448210, -218349120, -843621600, -3257112960, -12570420330, -48507033744, -187187399448, -722477682080, -2789279908316, -10772391370048

Offset: 0

Wolfdieter Lang, Jan 13 2006

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

Cf. A115139 - A115146, A115148, A115149.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2-4*x^3)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019

CoefficientList[Series[(1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2-4*x^3) *Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

my(x='x+O('x^30)); Vec((1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2 -4*x^3)*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019

((1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2-4*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)^8 = P(9, x) - x*P(8, 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(9, x)= 1-7*x+15*x^2-10*x^3+x^4 and P(8, x)=1-6*x+10*x^2-4*x^3.

a(n) = -C8(n-8), n>=8, with C8(n) = A003518(n+3) (eighth convolution of Catalan numbers). a(0)=1, a(1)=-8, a(2)=30, a(3)=-16, a(4)=2, a(5)=a(6)=a(7)=0. [1, -8, 20, -16, 2] is row n=8 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.

A115148 Ninth convolution of A115140.

Original entry on oeis.org

1, -9, 27, -30, 9, 0, 0, 0, 0, -1, -9, -54, -273, -1260, -5508, -23256, -95931, -389367, -1562275, -6216210, -24582285, -96768360, -379629720, -1485507600, -5801732460, -22626756594, -88152205554, -343176898988, -1335293573130, -5193831553416

Offset: 0

Wolfdieter Lang, Jan 13 2006

Seiichi Manyama, Table of n, a(n) for n = 0..1671

Cf. A115139 - A115147, A115149.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2-10*x^3+x^4)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019

CoefficientList[Series[(1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2-10*x^3 +x^4)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

my(x='x+O('x^30)); Vec((1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2 -10*x^3+x^4)*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019

((1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2-10*x^3+x^4) *sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^9 = P(10, x) - x*P(9, 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(10, x)=1-8*x+21*x^2-20*x^3+5*x^4 and P(9, x)=1-7*x+15*x^2-10*x^3+x^4.

a(n) = -C9(n-9), n>=9, with C9(n) = A001392(n+4) (eighth convolution of Catalan numbers). a(0)=1, a(1)=-9, a(2)=27, a(3)=-30, a(4)=9, a(5)=a(6)=a(7)=a(8)=0. [1, -9, 27, -30, 9] is row n=9 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.

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

A128502 Convolution array for Chebyshev's S(n,x)=U(n,x/2) polynomials.

Original entry on oeis.org

1, 2, 3, -2, 4, -6, 5, -12, 3, 6, -20, 12, 7, -30, 30, -4, 8, -42, 60, -20, 9, -56, 105, -60, 5, 10, -72, 168, -140, 30, 11, -90, 252, -280, 105, -6, 12, -110, 360, -504, 280, -42, 13, -132, 495, -840, 630, -168, 7, 14, -156, 660, -1320, 1260, -504, 56, 15, -182, 858, -1980, 2310, -1260, 252, -8, 16, -210, 1092

Offset: 0

Wolfdieter Lang Apr 04 2007

S1(n,x):=sum(S(n-k,x)*S(k,x),k=0..n)= sum(a(n,m)*x^(n-2*m),m=0..floor(n/2)).
The unsigned column sequences, m>=0, divided by (m+1) give Pascal triangle column sequences for m+1.
G.f. for column m sequence: ((-1)^m)*(m+1)*(x^(2*m))/(1-x)^(m+2), m>=0.
Row polynomials P1(n,x):= sum(a(n,m)*x^m,m=0..floor(n/2)) (increasing powers of x).
Written as a triangle with increasing powers of x this is A294519. - Wolfdieter Lang, Nov 12 2017

			[1];[2];[3,-2],[4,-6];[5,-12,3];[6,-20,12];[7,-30,30,-4];[8,-42,60,-20];...
n=4: [5,-12,3] stands for the polynomial S1(4,x) = 5*x^4-12*x^2+3 = 2*(S(4,x)*1+S(3,x)*S(1,x))+S(2,x)*S(2,x).
n=4: [5,-12,3] stands also for the row polynomial P1(4,x) = 5-12*x+3*x^2.

W. Lang, First 15 rows and more.

Row sums (signed array) give A099254. Unsigned row sums are A001629(n+2).
Cf. A115139 (with offset n>=0 is S(n, x) array, decreasing powers of x).
Cf. A294519 (as triangle).

a(n,m)=binomial(n-m,m)*(n+1-m)*(-1)^m, m=0..floor(n/2), n>=0.
a(n,m)=binomial(n+1-m,m+1)*(m+1)*(-1)^m, m=0..floor(n/2), n>=0.
G.f. for S1(n,x): 1/(1-x*z+z^2)^2.
G.f. for P1(n,x): 1/(1-z+x*z^2)^2.

A135936 Irregular triangle read by rows: row n gives coefficients of Boubaker polynomial B_n(x) in order of decreasing exponents (another version).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 0, -2, 1, -1, -3, 1, -2, -3, 2, 1, -3, -2, 5, 1, -4, 0, 8, -2, 1, -5, 3, 10, -7, 1, -6, 7, 10, -15, 2, 1, -7, 12, 7, -25, 9, 1, -8, 18, 0, -35, 24, -2, 1, -9, 25, -12, -42, 49, -11, 1, -10, 33, -30, -42, 84, -35, 2, 1, -11, 42, -55, -30, 126, -84, 13, 1, -12, 52, -88, 0, 168, -168, 48, -2, 1, -13, 63, -130, 55, 198, -294

Offset: 0

N. J. A. Sloane, Mar 09 2008

See A135929 and A138034 for further information.

			The Boubaker polynomials B_0(x), B_1(x), B_2(x), ... are:
  1
  x
  x^2    + 2
  x^3    + x
  x^4             - 2
  x^5    - x^3  - 3*x
  x^6  - 2*x^4  - 3*x^2    + 2
  x^7  - 3*x^5  - 2*x^3  + 5*x
  x^8  - 4*x^6           + 8*x^2    - 2
  x^9  - 5*x^7  + 3*x^5 + 10*x^3  - 7*x
  ...

R. J. Mathar, Mar 11 2008, Table of n, a(n) for n = 0..160

Cf. A138034.

A135936 := proc(n,m) coeftayl( coeftayl( (1+3*t^2)/(1-x*t+t^2),t=0,n), x=0,m) ; end: for n from 0 to 25 do for m from n to 0 by -2 do printf("%d, ",A135936(n,m)) ; od; od; # R. J. Mathar, Mar 11 2008

T[n_, m_] := SeriesCoefficient[SeriesCoefficient[
   (1+3*t^2)/(1-x*t+t^2), {t, 0, n}], {x, 0, m}];
Table[T[n, m], {n, 0, 25}, {m, n, 0, -2}] // Flatten (* Jean-François Alcover, Mar 11 2023, after R. J. Mathar *)

Conjectures from Thomas Baruchel, Jun 03 2018: (Start)
T(n,m) = 4*A115139(n+1,m) - 3*A132460(n,m).
T(n,m) = (-1)^m * (binomial(n-m, m) - 3*binomial(n-m-1, m-1)). (End)

More terms from R. J. Mathar, Mar 11 2008

A205573 Array M read by antidiagonals in which successive rows evidently converge to A001405 (central binomial coefficients).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 3, 5, 1, 1, 1, 2, 3, 6, 8, 1, 1, 1, 2, 3, 6, 10, 13, 1, 1, 1, 2, 3, 6, 10, 19, 21, 1, 1, 1, 2, 3, 6, 10, 20, 33, 34, 1, 1, 1, 2, 3, 6, 10, 20, 35, 61, 55, 1, 1, 1, 2, 3, 6, 10, 20, 35, 69, 108, 89, 1

Offset: 0

L. Edson Jeffery, Jan 29 2012

CONJECTURE 1. Let M(n,k) (n,k >= 0) denote the entry in row n and column k of the array. For all n, M(n,j) = A001405(j), j=0,...,2*n+1; hence row n of M -> A001405 as n -> infinity.

Taking finite differences of even numbered columns from the top -> down yields triangle A205946 with row sums A000984, central binomial coefficients; while odd numbered columns yield triangle A205945 with row sums A001700. A205946 and A205945 represent the bisection of A191314. - Gary W. Adamson, Feb 01 2012

			Array begins:
  1, 1, 1, 1, 1,  1,  1,  1,  1,   1,   1,...
  1, 1, 2, 3, 5,  8, 13, 21, 34,  55,  89,...
  1, 1, 2, 3, 6, 10, 19, 33, 61, 108, 197,...
  1, 1, 2, 3, 6, 10, 20, 35, 69, 124, 241,...
  1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 251,...
  1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252,...
  ...
According to Conjecture 2, row n = 3 has g.f. F_3(x) = (1-2*x^2)/(1-x-3*x^2+2*x^3+x^4).

L. E. Jeffery, Unit-primitive matrices

Cf. A001405, A108299, A115139, A191314.

Cf. also A205945, A205946, A001700, A000984.

Let N=2*n+3. For each n>0, define the (n+1) X (n+1) tridiagonal unit-primitive matrix (see [Jeffery]) B_n = A_{N,1} = [0,1,0,...,0; 1,0,1,0,...,0; 0,1,0,1,0,...,0; ...; 0,...,0,1,0,1; 0,...,0,1,1], and put B_0 = [1]. Then, for all n, M(n,k)=[(B_n)^k]{n+1,n+1}, k=0,1,..., where X{n+1,n+1} denotes the lower right corner entry of X.
CONJECTURE 2 (Rows of M). Let S(n,i) denote term i in row n of A115139, i=0,...,floor(n/2), and let T(n,j) denote term j in row n of A108299, j=0,...,n.  The generating function for row n of M is of the form F_n(x) =sum[i=0,...,floor(n/2) S(n,i)*x^(2*i)]/sum[j=0,...,n T(n,j)*x^j].
CONJECTURE 3 (Columns of M). Let D(m,k) denote term m in column k of A191314, m=0,...,floor(k/2). The generating function for column k of M is of the form G_k(x)=sum[m=0,...,floor(k/2) D(m,k)*x^m]/(1-x).

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A115142 Third convolution of A115140.

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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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A115144 Fifth convolution of A115140.

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A115147 Eighth convolution of A115140.

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A115148 Ninth convolution of A115140.

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A115145 Sixth convolution of A115140.

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A115146 Seventh convolution of A115140.

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