A219233 -id:A219233 - cp's OEIS frontend

A110162 Riordan array ((1-x)/(1+x), x/(1+x)^2).

1, -2, 1, 2, -4, 1, -2, 9, -6, 1, 2, -16, 20, -8, 1, -2, 25, -50, 35, -10, 1, 2, -36, 105, -112, 54, -12, 1, -2, 49, -196, 294, -210, 77, -14, 1, 2, -64, 336, -672, 660, -352, 104, -16, 1, -2, 81, -540, 1386, -1782, 1287, -546, 135, -18, 1, 2, -100, 825, -2640, 4290, -4004, 2275, -800, 170, -20, 1

Offset: 0

Paul Barry, Jul 14 2005

Inverse of Riordan array A094527. Rows sums are A099837. Diagonal sums are A110164. Product of Riordan array A102587 and inverse binomial transform (1/(1+x), x/(1+x)).
Coefficients of polynomials related to Cartan matrices of types C_n and B_n: p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2), with p(x,0) = 1; p(x,1) = 2-x; p(x,2) = x^2-4*x-2. - Roger L. Bagula, Apr 12 2008
From Wolfdieter Lang, Nov 16 2012: (Start)
The alternating row sums are given in A219233.
For n >= 1 the row polynomials in the variable x^2 are R(2*n,x):=2*T(2*n,x/2) with Chebyshev's T-polynomials. See A127672 and also the triangle A127677.
(End)
From Peter Bala, Jun 29 2015: (Start)
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = x/(1 + x)^2 and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = (1 - 2*x + sqrt(1 - 4*x))/2. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

			Triangle T(n,k) begins:
m\k  0    1    2     3     4     5     6    7    8   9 10 ...
0:   1
1:  -2    1
2:   2   -4    1
3:  -2    9   -6     1
4:   2  -16   20    -8     1
5:  -2   25  -50    35   -10     1
6:   2  -36  105  -112    54   -12     1
7:  -2   49 -196   294  -210    77   -14    1
8:   2  -64  336  -672   660  -352   104  -16    1
9:  -2   81 -540  1386 -1782  1287  -546  135  -18   1
10:  2 -100  825 -2640  4290 -4004  2275 -800  170 -20  1
... Reformatted and extended by _Wolfdieter Lang_, Nov 16 2012
Row polynomial n=2: P(2,x) = 2 - 4*x + x^2. R(4,x):= 2*T(4,x/2) = 2 - 4*x^2 + x^4. For P and R see a comment above. - _Wolfdieter Lang_, Nov 16 2012.

G. C. Greubel, Rows n=0..100 of triangle, flattened
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
T. M. Richardson, The Reciprocal Pascal Matrix, arXiv:1405.6315 [math.CO], 2014.

Cf. A128411. See A127677 for an almost identical triangle.

Cf. A136674, A053122.

/* As triangle */ [[(-1)^(n-k)*(Binomial(n+k,n-k) + Binomial(n+k-1,n-k-1)): k in [0..n]]: n in [0.. 12]]; // Vincenzo Librandi, Jun 30 2015

Table[If[n==0 && k==0, 1, (-1)^(n-k)*(Binomial[n+k, n-k] + Binomial[n+k-1, n-k-1])], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 16 2018 *)

{T(n,k) = (-1)^(n-k)*(binomial(n+k,n-k) + binomial(n+k-1,n-k-1))};
for(n=0, 12, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 16 2018

[[(-1)^(n-k)*(binomial(n+k,n-k) + binomial(n+k-1,n-k-1)) for k in range(n+1)] for n in range(12)] # G. C. Greubel, Dec 16 2018

T(n,k) = (-1)^(n-k)*(C(n+k,n-k) + C(n+k-1,n-k-1)), with T(0,0) = 1. - Paul Barry, Mar 22 2007
From Wolfdieter Lang, Nov 16 2012: (Start)
O.g.f. row polynomials P(n,x) := Sum(T(n,k)*x^k, k=0..n): (1-z^2)/(1+(x-2)*z+z^2) (from the Riordan property).
O.g.f. column No. k: ((1-x)/(1+x))*(x/(1+x)^2)^k, k >= 0.
T(0,0) = 1, T(n,k) = (-1)^(n-k)*(2*n/(n+k))*binomial(n+k,n-k), n>=1, and T(n,k) = 0 if n < k. (From the Chebyshev T-polynomial formula due to Waring's formula.)
(End)
T(n,k) = -2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 29 2013

A129862 Triangle read by rows: T(n,k) is the coefficient [x^k] of (-1)^n times the characteristic polynomial of the Cartan matrix for the root system D_n.

Original entry on oeis.org

1, 2, -1, 4, -4, 1, 4, -10, 6, -1, 4, -20, 21, -8, 1, 4, -34, 56, -36, 10, -1, 4, -52, 125, -120, 55, -12, 1, 4, -74, 246, -329, 220, -78, 14, -1, 4, -100, 441, -784, 714, -364, 105, -16, 1, 4, -130, 736, -1680, 1992, -1364, 560, -136, 18, -1, 4, -164, 1161, -3312, 4950, -4356, 2379, -816, 171, -20, 1

Offset: 0

Roger L. Bagula, May 23 2007

Row sums of the absolute values are s(n) = 1, 3, 9, 21, 54, 141, 369, 966, 2529, 6621, 17334, ... (i.e., s(n) = 3*|A219233(n-1)| for n > 0). - R. J. Mathar, May 31 2014

			Triangle begins:
  1;
  2,   -1;
  4,   -4,    1;
  4,  -10,    6,    -1;
  4,  -20,   21,    -8,    1;
  4,  -34,   56,   -36,   10,    -1;
  4,  -52,  125,  -120,   55,   -12,    1;
  4,  -74,  246,  -329,  220,   -78,   14,   -1;
  4, -100,  441,  -784,  714,  -364,  105,  -16,   1;
  4, -130,  736, -1680, 1992, -1364,  560, -136,  18,  -1;
  4, -164, 1161, -3312, 4950, -4356, 2379, -816, 171, -20, 1;

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 60.
Sigurdur Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S. :ISBN 0-8218-2848-7, 1978, p. 464.

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A156608, A156609, A156610, A156612, A219233.

A129862 := proc(n,k)
    M := Matrix(n,n);
    for r from 1 to n do
    for c from 1 to n do
        if r = c then
            M[r,c] := 2;
        elif abs(r-c)= 1 then
            M[r,c] := -1;
        else
            M[r,c] := 0 ;
        end if;
    end do:
    end do:
    if n-2 >= 1 then
        M[n,n-2] := -1 ;
        M[n-2,n] := -1 ;
    end if;
    if n-1 >= 1 then
        M[n-1,n] := 0 ;
        M[n,n-1] := 0 ;
    end if;
    LinearAlgebra[CharacteristicPolynomial](M,x) ;
    (-1)^n*coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, May 31 2014

(* First program *)
t[n_, m_, d_]:= If[n==m, 2, If[(m==d && n==d-2) || (n==d && m==d-2), -1, If[(n==m- 1 || n==m+1) && n<=d-1 && m<=d-1, -1, 0]]];
M[d_]:= Table[t[n,m,d], {n,1,d}, {m,1,d}];
p[n_, x_]:= If[n==0, 1, CharacteristicPolynomial[M[n], x]];
T[n_, k_]:= SeriesCoefficient[p[n, x], {x, 0, k}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 21 2021 *)
(* Second program *)
Join[{{1}, {2, -1}}, CoefficientList[Table[(2-x)*LucasL[2(n-1), Sqrt[-x]], {n, 2, 10}], x]]//Flatten (* Eric W. Weisstein, Apr 04 2018 *)

def p(n,x): return 2*(2-x)*sum( ((n-1)/(2*n-k-2))*binomial(2*n-k-2, k)*(-x)^(n-k-1) for k in (0..n-1) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
[1,2,-1]+flatten([T(n) for n in (2..12)]) # G. C. Greubel, Jun 21 2021

T(n, k) = coefficients of ( (2-x)*Lucas(2*n-2, i*sqrt(x)) ) with T(0, 0) = 1, T(1, 0) = 2 and T(1, 1) = -1. - G. C. Greubel, Jun 21 2021

A156610 Triangle T(n, k, m) = round( t(n,m)/(t(k,m)*t(n-k,m)) ), with T(0, k, m) = 1, where t(n, k) = Product_{j=1..n} A129862(k+1, j), t(n, 0) = n!, and m = 4, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -3, 1, 1, 9, 9, 1, 1, -21, 63, -21, 1, 1, 54, 378, 378, 54, 1, 1, -141, 2538, -5922, 2538, -141, 1, 1, 369, 17343, 104058, 104058, 17343, 369, 1, 1, -966, 118818, -1861482, 4786668, -1861482, 118818, -966, 1, 1, 2529, 814338, 33387858, 224175618, 224175618, 33387858, 814338, 2529, 1

Offset: 0

Roger L. Bagula, Feb 11 2009

Cartan_Dn refers to a Cartan matrix of type D_n. - N. J. A. Sloane, Jun 25 2021

			Triangle begins:
  1;
  1,    1;
  1,   -3,      1;
  1,    9,      9,        1;
  1,  -21,     63,      -21,         1;
  1,   54,    378,      378,        54,         1;
  1, -141,   2538,    -5922,      2538,      -141,        1;
  1,  369,  17343,   104058,    104058,     17343,      369,      1;
  1, -966, 118818, -1861482,   4786668,  -1861482,   118818,   -966,    1;
  1, 2529, 814338, 33387858, 224175618, 224175618, 33387858, 814338, 2529, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A129862, A007318 (m=0), A156608 (m=2), A156609 (m=3), this sequence (m=4), A156612.

(* First program *)
b[n_, k_, d_]:= If[n==k, 2, If[(k==d && n==d-2) || (n==d && k==d-2), -1, If[(k==n- 1 || k==n+1) && n<=d-1 && k<=d-1, -1, 0]]];
M[d_]:= Table[b[n, k, d], {n, d}, {k, d}];
p[x_, n_]:= If[n==0, 1, CharacteristicPolynomial[M[n], x]];
f = Table[p[x, n], {n, 0, 20}];
t[n_, k_]:= If[k==0, n!, Product[f[[j+1]], {j, n-1}]]/.x -> k+1;
T[n_, k_, m_]:= Round[t[n, m]/(t[k, m]*t[n-k, m])];
Table[T[n, k, 4], {n,0,15}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 24 2021 *)
(* Second program *)
f[n_, x_]:= f[n,x]= If[n<2, (2-x)^n, (2-x)*LucasL[2*(n-1), Sqrt[-x]] ];
t[n_, k_]:= t[n,k]= If[k==0, n!, Product[f[j, x], {j, n-1}]]/.x -> (k+1);
T[n_, k_, m_]:= T[n,k,m]= Round[t[n,m]/(t[k,m]*t[n-k,m])];
Table[T[n, k, 4], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 24 2021 *)

@CachedFunction
def f(n,x): return (2-x)^n if (n<2) else 2*(2-x)*sum( ((n-1)/(2*n-j-2))*binomial(2*n-j-2, j)*(-x)^(n-j-1) for j in (0..n-1) )
def g(n,k): return factorial(n) if (k==0) else product( f(j, k+1) for j in (1..n-1) )
def T(n,k,m): return round( g(n,m)/(g(k,m)*g(n-k,m)) )
flatten([[T(n,k,4) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jun 24 2021

T(n, k, m) = round( t(n,m)/(t(k,m)*t(n-k,m)) ), with T(0, k, m) = 1, where t(n, k) = Product_{j=1..n} A129862(k+1, j), t(n, 0) = n!, and m = 4.

T(n, 1) = T(n, n-1) = [n==1] - 3*A219233(n-2)*[n >= 2]. - G. C. Greubel, Jun 24 2021

Definition corrected and edited by G. C. Greubel, Jun 24 2021

A075150 a(n) = L(n)*C(n), L(n)=Lucas numbers (A000032), C(n)=reflected Lucas numbers (see comment to A061084).

Original entry on oeis.org

4, -1, 9, -16, 49, -121, 324, -841, 2209, -5776, 15129, -39601, 103684, -271441, 710649, -1860496, 4870849, -12752041, 33385284, -87403801, 228826129, -599074576, 1568397609, -4106118241, 10749957124, -28143753121, 73681302249, -192900153616, 505019158609, -1322157322201

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 05 2002

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,2,1).

Cf. A000032, A001254, A061084, A219233.

A075150:= func< n | (-1)^n*Lucas(n)^2 >; // G. C. Greubel, Jun 14 2025

CoefficientList[Series[(4 + 7*x - x^2)/(1 + 2*x - 2*x^2 - x^3), {x, 0, 30}], x]
LinearRecurrence[{-2,2,1},{4,-1,9},50] (* Harvey P. Dale, Nov 08 2011 *)

a(n) = round((2+(1/2*(-3-sqrt(5)))^n+(1/2*(-3+sqrt(5)))^n)) \\ Colin Barker, Oct 01 2016

Vec((4+7*x-x^2)/(1+2*x-2*x^2-x^3) + O(x^30)) \\ Colin Barker, Oct 01 2016

def A075150(n): return (-1)**n*lucas_number2(n,1,-1)**2 # G. C. Greubel, Jun 14 2025

a(n) = (-1)^n*A000032(2*n) + 2.
a(n) = -2*a(n-1) + 2*a(n-2) + a(n-3) with a(0)=4, a(1)=-1, a(2)=9.
G.f.: (4 + 7*x - x^2)/(1 + 2*x - 2*x^2 - x^3).
a(n) = (-1)^n*A001254(n). - R. J. Mathar, Jan 11 2012
a(n) = 2 + (1/2*(-3-sqrt(5)))^n + (1/2*(-3+sqrt(5)))^n. - Colin Barker, Oct 01 2016
From G. C. Greubel, Jun 14 2025: (Start)
a(n) = A000032(n)*A000032(-n) = (-1)^n*A000032(n)^2.
a(n) = A219233(n) + 2 + [n=0].
E.g.f.: 2*exp(-3*x/2)*cosh(sqrt(5)*x/2) + 2*exp(x). (End)

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A110162 Riordan array ((1-x)/(1+x), x/(1+x)^2).

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A129862 Triangle read by rows: T(n,k) is the coefficient [x^k] of (-1)^n times the characteristic polynomial of the Cartan matrix for the root system D_n.

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A156610 Triangle T(n, k, m) = round( t(n,m)/(t(k,m)*t(n-k,m)) ), with T(0, k, m) = 1, where t(n, k) = Product_{j=1..n} A129862(k+1, j), t(n, 0) = n!, and m = 4, read by rows.

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A075150 a(n) = L(n)*C(n), L(n)=Lucas numbers (A000032), C(n)=reflected Lucas numbers (see comment to A061084).

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