A156610 -id:A156610 - cp's OEIS frontend

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.

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

A156608 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 = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, -2, 2, 2, -2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, -1, 2, 2, -1, 1, 1, 1, -2, 2, 2, -4, 2, 2, -2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, -1, 2, 2, -1, 2, 2, -1, 1, 1

Offset: 0

Roger L. Bagula, Feb 11 2009

The original definition of this sequence said it was based on the Cartan matrix of type D_n, so that matrix is somehow implicitly involved. - N. J. A. Sloane, Jun 25 2021

			Triangle begins:
  1;
  1,  1;
  1, -1,  1;
  1,  1,  1, 1;
  1,  1, -1, 1,  1;
  1, -2,  2, 2, -2,  1;
  1,  1,  2, 2,  2,  1, 1;
  1,  1, -1, 2,  2, -1, 1,  1;
  1, -2,  2, 2, -4,  2, 2, -2,  1;
  1,  1,  2, 2,  2,  2, 2,  2,  1, 1;
  1,  1, -1, 2,  2, -1, 2,  2, -1, 1, 1;

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

Cf. A129862, A007318 (m=0), this sequence (m=2), A156609 (m=3), A156610 (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, 2], {n,0,15}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 23 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, 2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 23 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,2) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jun 23 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 = 2.

Edited by G. C. Greubel, Jun 23 2021

A156609 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 = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -2, 1, 1, 4, 4, 1, 1, -4, 8, -4, 1, 1, 4, 8, 8, 4, 1, 1, -4, 8, -8, 8, -4, 1, 1, 4, 8, 8, 8, 8, 4, 1, 1, -4, 8, -8, 8, -8, 8, -4, 1, 1, 4, 8, 8, 8, 8, 8, 8, 4, 1, 1, -4, 8, -8, 8, -8, 8, -8, 8, -4, 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, -2, 1;
  1,  4, 4,  1;
  1, -4, 8, -4, 1;
  1,  4, 8,  8, 4,  1;
  1, -4, 8, -8, 8, -4, 1;
  1,  4, 8,  8, 8,  8, 4,  1;
  1, -4, 8, -8, 8, -8, 8, -4, 1;
  1,  4, 8,  8, 8,  8, 8,  8, 4,  1;
  1, -4, 8, -8, 8, -8, 8, -8, 8, -4, 1;

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

Cf. A129862, A007318 (m=0), A156608 (m=2), this sequence (m=3), A156610 (m=4), A156612.

function T(n,k)
  if k eq 0 or k eq n then return 1;
  elif n eq 2 and k eq 1 then return -2;
  elif k eq 1 or k eq n-1 then return 4*(-1)^(n+1);
  elif k eq 2 or k eq n-2 then return 8;
  elif (n mod 2) eq 0 then return 8*(-1)^k;
  else return 8;
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 24 2021

(* 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, 3], {n,0,15}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 23 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, 3], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 23 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,3) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jun 23 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 = 3.

T(n, k) defined by T(n, 0) = T(n, n) = 1, T(2, 1) = -2, T(n, 1) = T(n, n-1) = 4*(-1)^(n+1), T(n, 2) = T(n, n-2) = 8, T(n, k) = 8*(-1)^k if n mod 2 = 0, and T(n, k) = 8 otherwise. - G. C. Greubel, Jun 24 2021

Definition corrected and edited by G. C. Greubel, Jun 23 2021

A156612 Square array T(n, k) = Product_{j=1..n} A129862(k+1, j) with T(n, 0) = n!, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 0, 6, 1, 1, -1, 0, 24, 1, 1, -2, -1, 0, 120, 1, 1, -3, -8, -1, 0, 720, 1, 1, -4, -27, 32, 2, 0, 5040, 1, 1, -5, -64, 567, 128, 2, 0, 40320, 1, 1, -6, -125, 3584, 30618, -512, 2, 0, 362880, 1, 1, -7, -216, 14375, 745472, -4317138, -2048, -4, 0, 3628800

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

			Square array begins:
    1, 1,  1,   1,     1,      1 ...;
    1, 1,  1,   1,     1,      1 ...;
    2, 0, -1,  -2,    -3,     -4 ...;
    6, 0, -1,  -8,   -27,    -64 ...;
   24, 0, -1,  32,   567,   3584 ...;
  120, 0,  2, 128, 30618, 745472 ...;
Triangle begins as:
  1;
  1, 1;
  1, 1,  2;
  1, 1,  0,    6;
  1, 1, -1,    0,    24;
  1, 1, -2,   -1,     0,    120;
  1, 1, -3,   -8,    -1,      0,      720;
  1, 1, -4,  -27,    32,      2,        0,  5040;
  1, 1, -5,  -64,   567,    128,        2,     0, 40320;
  1, 1, -6, -125,  3584,  30618,     -512,     2,     0, 362880;
  1, 1, -7, -216, 14375, 745472, -4317138, -2048,    -4,      0, 3628800;

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Cf. A129862, A156608, A156609, A156610.

(* 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, 30}];
T[n_, k_]:= If[k==0, n!, Product[f[[j+1]], {j, n-1}]]/.x -> k+1;
Table[T[k, n - k], {n,0,15}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 25 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);
Table[t[k, n-k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 25 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 T(n,k): return factorial(n) if (k==0) else product( f(j, k+1) for j in (1..n-1) )
flatten([[T(k,n-k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jun 25 2021

T(n, k) = Product_{j=1..n} A129862(k+1, j) with T(n, 0) = n!.

Edited by G. C. Greubel, Jun 25 2021

cp's OEIS Frontend

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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A156608 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 = 2, read by rows.

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A156609 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 = 3, read by rows.

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Magma

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A156612 Square array T(n, k) = Product_{j=1..n} A129862(k+1, j) with T(n, 0) = n!, read by antidiagonals.

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