A034807 -id:A034807 - cp's OEIS frontend

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

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.

A117938 Triangle, columns generated from Lucas Polynomials.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 4, 1, 4, 11, 14, 7, 1, 5, 18, 36, 34, 11, 1, 6, 27, 76, 119, 82, 18, 1, 7, 38, 140, 322, 393, 198, 29, 1, 8, 51, 234, 727, 1364, 1298, 478, 47, 1, 9, 66, 364, 1442, 3775, 5778, 4287, 1154, 76, 1, 10, 83, 536, 2599, 8886, 19602, 24476, 14159, 2786, 123

Offset: 1

Gary W. Adamson, Apr 03 2006

Companion triangle using Fibonacci polynomial generators = A073133. Inverse binomial transforms of the columns defines rows of A117937 (with some adjustments of offset).

A309220 is another version of the same triangle (except it omits the last diagonal), and perhaps has a clearer definition. - N. J. A. Sloane, Aug 13 2019

			First few rows of the triangle are:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  6,   4;
  1, 4, 11,  14,   7;
  1, 5, 18,  36,  34,  11;
  1, 6, 27,  76, 119,  82,  18;
  1, 7, 38, 140, 322, 393, 198, 29;
  ...
For example, T(7,4) = 76 = f(4), x^3 + 3*x = 64 + 12 = 76.

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A104509, A114525, A117936, A117937, A118980, A118981, A309220.

Cf. A000204 (diagonal), A059100 (column 3), A061989 (column 4).

Lucas := proc(n,x) # see A114525
    option remember;
    if  n=0 then
        2;
    elif n =1 then
        x ;
    else
        x*procname(n-1,x)+procname(n-2,x) ;
    end if;
    expand(%) ;
end proc:
A117938 := proc(n::integer,k::integer)
    if k = 1 then
        1;
    else
        subs(x=n-k+1,Lucas(k-1,x)) ;
    end if;
end proc:
seq(seq(A117938(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Aug 16 2019

T[n_, k_]:= LucasL[k-1, n-k+1] - Boole[k==1];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Oct 28 2021 *)

def A117938(n,k): return 1 if (k==1) else round(2^(1-k)*( (n-k+1 + sqrt((n-k)*(n-k+2) + 5))^(k-1) + (n-k+1 - sqrt((n-k)*(n-k+2) + 5))^(k-1) ))
flatten([[A117938(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 28 2021

Columns are f(x), x = 1, 2, 3, ..., of the Lucas Polynomials: (1, defined different from A034807 and A114525); (x); (x^2 + 2); (x^3 + 3*x); (x^4 + 4*x^2 + 2); (x^5 + 5*x^3 + 5*x); (x^6 + 6*x^4 + 9*x^2 + 2); (x^7 + 7*x^5 + 14*x^3 + 7*x); ...

Terms a(51) and a(52) corrected by G. C. Greubel, Oct 28 2021

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

A338838 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] where adjacent values cannot be consecutive modulo n.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 0, 0, 1, 4, 4, 0, 0, 1, 5, 10, 10, 10, 10, 1, 6, 18, 36, 60, 84, 60, 1, 7, 28, 84, 210, 434, 630, 462, 1, 8, 40, 160, 544, 1552, 3440, 5168, 3920, 1, 9, 54, 270, 1170, 4338, 13158, 30366, 47178, 36954, 1, 10, 70, 420, 2220, 10220, 39780, 125220, 298060, 476220, 382740

Offset: 0

Xiangyu Chen, Nov 11 2020

In a convex n-gon, the number of paths using k-1 diagonals and k non-repeated vertices.

			n\k  0    1    2    3    4    5    6    7    8
0    1
1    1    1
2    1    2    0
3    1    3    0    0
4    1    4    4    0    0
5    1    5    10   10   10   10
6    1    6    18   36   60   84   60
7    1    7    28   84   210  434  630  462
8    1    8    40   160  544  1552 3440 5168 3920

Right diagonal is A002493.

Cf. A011973, A034807, A338526, A338849, A000166.

isokd(d, n) = my(x=abs(d)); (x==1) || (x==(n-1));
isok(s, p, n) = {my(w = vector(#s, k, s[p[k]])); for (i=1, #s-1, if (isokd(w[i+1] - w[i], n) == 1, return (0))); return (1);}
T(n, k) = {my(nb = 0); forsubset([n, k], s, for(i=1, k!, if (isok(s, numtoperm(k, i), n), nb++););); nb;} \\ Michel Marcus, Nov 21 2020

T(n,k) = n*(A338526(n-1,k-1)-S(n-1,k-1)) for k>0 except T(2,2)=0, T(n,0)=1, where S(n,k) = 2*A338526(n-1,k-1)-S(n-1,k-1) for k>0, S(n,0)=0.

A355342 G.f.: A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) C(x)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

0, 1, -2, -1, 3, 0, 1, -4, 2, 0, -1, 5, -5, 0, 0, 1, -6, 9, -2, 0, 0, -1, 7, -14, 7, 0, 0, 0, 1, -8, 20, -16, 2, 0, 0, 0, -1, 9, -27, 30, -9, 0, 0, 0, 0, 1, -10, 35, -50, 25, -2, 0, 0, 0, 0, -1, 11, -44, 77, -55, 11, 0, 0, 0, 0, 0, 1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0, -1, 13, -65, 156, -182, 91, -13, 0, 0, 0, 0, 0, 0, 1, -14, 77, -210, 294, -196, 49, -2, 0, 0, 0, 0, 0, 0, -1, 15, -90, 275, -450, 378, -140, 15, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, Jul 22 2022

sign

			G.f.: A(x) = x - 2*x^2 - x^3 + 3*x^4 + x^6 - 4*x^7 + 2*x^8 - x^10 + 5*x^11 - 5*x^12 + x^15 - 6*x^16 + 9*x^17 - 2*x^18 - x^21 + 7*x^22 - 14*x^23 + 7*x^24 + x^28 - 8*x^29 + 20*x^30 - 16*x^31 + 2*x^32 - x^36 + 9*x^37 - 27*x^38 + 30*x^39 - 9*x^40 + x^45 - 10*x^46 + 35*x^47 - 50*x^48 + 25*x^49 - 2*x^50 + ...
such that
A(x) = ... + x^6/C(x)^4 - x^3/C(x)^3 + x/C(x)^2 - 1/C(x) + 1 - x*C(x) + x^3*C(x)^2 - x^6*C(x)^3 + x^10*C(x)^4 +- ...
where
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + 16796*x^10 + ... + A000108(n)*x^n + ...
The coefficients of x^k in (-1)^n * x^(n*(n+1)/2) * (C(x)^n - 1/C(x)^(n+1)) begin:
n = 0: [0, 1,  1,  2,  5,  14,  42,  132,  429,  1430,  4862,  16796, ...];
n = 1: [0, 0, -3, -3, -7, -19, -56, -174, -561, -1859, -6292, -21658, ...];
n = 2: [0, 0,  0,  0,  5,   5,  15,   45,  141,   457,  1520,   5159, ...];
n = 3: [0, 0,  0,  0,  0,   0,   0,   -7,   -7,   -28,   -91,   -301,  ...];
n = 4: [0, 0,  0,  0,  0,   0,   0,    0,    0,     0,     0,      9, ...]; ...
forming a table the column sums of which yield this sequence.
The g.f. may also be written as
A(x) = 0 + (-2*x + 1)*x - (-3*x + 1)*x^3 + (2*x^2 - 4*x + 1)*x^6 - (5*x^2 - 5*x + 1)*x^10 + (-2*x^3 + 9*x^2 - 6*x + 1)*x^15 - (-7*x^3 + 14*x^2 - 7*x + 1)*x^21 + (2*x^4 - 16*x^3 + 20*x^2 - 8*x + 1)*x^28 - (9*x^4 - 30*x^3 + 27*x^2 - 9*x + 1)*x^36 + (-2*x^5 + 25*x^4 - 50*x^3 + 35*x^2 - 10*x + 1)*x^45 + ...
compare to
(1 + 2*y*x)/(1+x + y*x^2) = 1 - (-2*y + 1)*x + (-3*y + 1)*x^2 - (2*y^2 - 4*y + 1)*x^3 + (5*y^2 - 5*y + 1)*x^4 - (-2*y^3 + 9*y^2 - 6*y + 1)*x^5 + (-7*y^3 + 14*y^2 - 7*y + 1)*x^6 - (2*y^4 - 16*y^3 + 20*y^2 - 8*y + 1)*x^7 + (9*y^4 - 30*y^3 + 27*y^2 - 9*y + 1)*x^8 - (-2*y^5 + 25*y^4 - 50*y^3 + 35*y^2 - 10*y + 1)*x^9 + ...
The terms of this sequence may be written as a triangle:
0,
1, -2,
-1, 3, 0,
1, -4, 2, 0,
-1, 5, -5, 0, 0,
1, -6, 9, -2, 0, 0,
-1, 7, -14, 7, 0, 0, 0,
1, -8, 20, -16, 2, 0, 0, 0,
-1, 9, -27, 30, -9, 0, 0, 0, 0,
1, -10, 35, -50, 25, -2, 0, 0, 0, 0,
-1, 11, -44, 77, -55, 11, 0, 0, 0, 0, 0,
1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0,
-1, 13, -65, 156, -182, 91, -13, 0, 0, 0, 0, 0, 0,
1, -14, 77, -210, 294, -196, 49, -2, 0, 0, 0, 0, 0, 0,
-1, 15, -90, 275, -450, 378, -140, 15, 0, 0, 0, 0, 0, 0, 0,
1, -16, 104, -352, 660, -672, 336, -64, 2, 0, 0, 0, 0, 0, 0, 0,
...

Paul D. Hanna, Table of n, a(n) for n = 0..1275

Cf. A244422, A355341, A355343.

Cf. A034807.

{a(n) = my(A,C = serreverse(x-x^2 +x^2*O(x^n))/x);
A = sum(m=-n-1,n+1, (-1)^m * x^(m*(m+1)/2) * C^m); polcoeff(A,n)}
for(n=0,70,print1(a(n),", "))

{a(n) = my(A,C = serreverse(x-x^2 +x^2*O(x^n))/x, M = sqrtint(2*n+9));
A = sum(m=0,M, (-1)^m * x^(m*(m+1)/2) * (C^m - 1/C^(m+1))); polcoeff(A,n)}
for(n=0,70,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n) * x^n is equal to the following expressions; here, C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(1) A(x) = -1/C(x) * Product_{n>=1} (1 - x^n/C(x)) * (1 - x^(n-1)*C(x)) * (1-x^n), by the Jacobi triple product identity.
(2) A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * C(x)^n.
(3) A(x) = Sum_{n>=0} (-1)^n * x^(n*(n+1)/2) * (C(x)^n - 1/C(x)^(n+1)).
(4) A(x) = 1 - Sum_{n>=0} x^(n*(n+1)/2) * ( [y^n] (1 + 2*y*x)/(1+x + y*x^2) ).
(5) A(x) = 1 - Sum_{n>=1} (-1)^n * x^(n*(n-1)/2) * Sum_{k=0..n} A244422(n,k) * x^k.

A157000 Triangle T(n,k) = (n/k)*binomial(n-k-1, k-1) read by rows.

Original entry on oeis.org

2, 3, 4, 2, 5, 5, 6, 9, 2, 7, 14, 7, 8, 20, 16, 2, 9, 27, 30, 9, 10, 35, 50, 25, 2, 11, 44, 77, 55, 11, 12, 54, 112, 105, 36, 2, 13, 65, 156, 182, 91, 13, 14, 77, 210, 294, 196, 49, 2, 15, 90, 275, 450, 378, 140, 15, 16, 104, 352, 660, 672, 336, 64, 2, 17, 119, 442, 935, 1122, 714, 204, 17

Offset: 2

Roger L. Bagula, Feb 20 2009

Row sums are A001610(n-1).
Triangle A034807 (coefficients of Lucas polynomials) with the first column omitted. - Philippe Deléham, Mar 17 2013
T(n,k) is the number of ways to select k knights from a round table of n knights, no two adjacent. - Bert Seghers, Mar 02 2014

			The table starts in row n=2, column k=1 as:
   2;
   3;
   4,  2;
   5,  5;
   6,  9,   2;
   7, 14,   7;
   8, 20,  16,   2;
   9, 27,  30,   9;
  10, 35,  50,  25,  2;
  11, 44,  77,  55, 11;
  12, 54, 112, 105, 36, 2;

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 199

G. C. Greubel, Rows n = 2..100 of triangle, flattened

Cf. A132460, A113279, A082985, A029635.

[[n*Binomial(n-k-1,k-1)/k: k in [1..Floor(n/2)]]: n in [2..20]]; // G. C. Greubel, Apr 25 2019

Table[(n/k)*Binomial[n-k-1, k-1], {n,2,20}, {k,1,Floor[n/2]}]//Flatten (* modified by G. C. Greubel, Apr 25 2019 *)

a(n,k)=n*binomial(n-k-1,k-1)/k; \\ Charles R Greathouse IV, Jul 10 2011

[[n*binomial(n-k-1,k-1)/k for k in (1..floor(n/2))] for n in (2..20)] # G. C. Greubel, Apr 25 2019

T(n,k) = binomial(n-k,k) + binomial(n-k-1,k-1). - Bert Seghers, Mar 02 2014

Offset 2, keyword:tabf, more terms by the Assoc. Eds. of the OEIS, Nov 01 2010

cp's OEIS Frontend

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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A117938 Triangle, columns generated from Lucas Polynomials.

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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.

A355342 G.f.: A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) C(x)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).