A129862 -id:A129862 - cp's OEIS frontend

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.

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

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

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

A135185 Triangle read by rows: E. F. Cornelius Jr. and Phill Schultz-based polynomials for the D_n Cartan Matrices in sequence A129862 that give a triangular sequence.

Original entry on oeis.org

1, 2, -4, 1, -4, 4, 2, -11, 19, -4, 12, -82, 142, -84, 4, 288, -2208, 4142, -2262, 422, -4, 34560, -277056, 566448, -327528, 49938, -2504, 4, 24883200, -202141440, 427397472, -267303264, 44330580, -1580666, 17368, -4, 125411328000, -1021977907200, 2179398320640, -1397599077312

Offset: 1

Roger L. Bagula, Feb 04 2008

These polynomials are suggested by Corollary 2.4 in the E. F. Cornelius Jr. and Phill Schultz paper.
It appears that many other fundamental matrix sequences can be related to these Cornelius-Schultz polynomials by a similar treatment.
Here a factor of 4 was necessary to convert them all to integer form.

			{1},
{2, -4},
{1, -4, 4},
{2, -11, 19, -4},
{12, -82, 142, -84, 4},
{288, -2208, 4142, -2262, 422, -4},
{34560, -277056, 566448, -327528, 49938, -2504, 4},
{24883200, -202141440, 427397472, -267303264, 44330580, -1580666, 17368, -4},

E. F. Cornelius Jr. and Phill Schultz, Sequences Generated by Polynomials, The American Mathematical Monthly, Vol. 115, No. 2 (Feb., 2008), pp. 154-158.

Cf. A129862.

cp's OEIS Frontend

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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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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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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A135185 Triangle read by rows: E. F. Cornelius Jr. and Phill Schultz-based polynomials for the D_n Cartan Matrices in sequence A129862 that give a triangular sequence.

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