cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A156600 Triangle T(n, k, m) = t(n, m)/(t(k, m)*t(n-k, m)), where t(n, k) = Product_{j=1..n} p(j, k+1), p(n, x) = Sum_{j=0..n} (-1)^j*A053122(n, j)*x^j, and m = 6, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -5, 1, 1, 24, 24, 1, 1, -115, 552, -115, 1, 1, 551, 12673, 12673, 551, 1, 1, -2640, 290928, -1394030, 290928, -2640, 1, 1, 12649, 6678672, 153331178, 153331178, 6678672, 12649, 1, 1, -60605, 153318529, -16865038190, 80805530806, -16865038190, 153318529, -60605, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 11 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,     1;
  1,    -5,       1;
  1,    24,      24,         1;
  1,  -115,     552,      -115,         1;
  1,   551,   12673,     12673,       551,       1;
  1, -2640,  290928,  -1394030,    290928,   -2640,     1;
  1, 12649, 6678672, 153331178, 153331178, 6678672, 12649, 1;

Links

Crossrefs

Cf. A007318 (m=0), A034801 (m=4), A156599 (m=5), this sequence (m=6), A156601 (m=7), A156602 (m=8), A156603.
Cf. A053122.

Programs

  • Mathematica
    (* First program *)
b[n_, k_]:= If[k==n, 2, If[k==n-1 || k==n+1, -1, 0]];
M[d_]:= Table[b[n, k], {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]], {j, n}]/.x->(k+1)];
T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];
Table[T[n, k, 6], {n,0,12}, {k,0,n}]//TableForm (* modified by G. C. Greubel, Jun 25 2021 *)
(* Second program *)
t[n_, k_]:= t[n, k]= If[n==0, 1, If[k==0, (n-1)!, Product[(-1)^j*Simplify[ChebyshevU[j, x/2 - 1]], {j,0,n-1}]/.x->(k+1)]];
T[n_, k_, m_]:= T[n, k, m]= t[n, m]/(t[k, m]*t[n-k, m]);
Table[T[n, k, 6], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 25 2021 *)
  • Sage
    @CachedFunction
def t(n, k):
    if (n==0): return 1
    elif (k==0): return factorial(n-1)
    else: return product( (-1)^j*chebyshev_U(j, (k-1)/2) for j in (0..n-1) )
def T(n,k,m): return t(n,m)/(t(k,m)*t(n-k,m))
flatten([[T(n, k, 6) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 25 2021

Formula

T(n, k, m) = t(n, m)/(t(k, m)*t(n-k, m)), where t(n, k) = Product_{j=1..n} p(j, k+1), p(n, x) = Sum_{j=0..n} (-1)^j*A053122(n, j)*x^j, and m = 6.

Extensions

Edited by G. C. Greubel, Jun 25 2021

A156602 Triangle T(n, k, m) = t(n, m)/(t(k, m)*t(n-k, m)), where t(n, k) = Product_{j=1..n} p(j, k+1), p(n, x) = Sum_{j=0..n} (-1)^j*A053122(n, j)*x^j, and m = 8, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -7, 1, 1, 48, 48, 1, 1, -329, 2256, -329, 1, 1, 2255, 105985, 105985, 2255, 1, 1, -15456, 4979040, -34127170, 4979040, -15456, 1, 1, 105937, 233908896, 10988845010, 10988845010, 233908896, 105937, 1, 1, -726103, 10988739073, -3538373981506, 24252380937070, -3538373981506, 10988739073, -726103, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 11 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,      1;
  1,     -7,         1;
  1,     48,        48,           1;
  1,   -329,      2256,        -329,           1;
  1,   2255,    105985,      105985,        2255,         1;
  1, -15456,   4979040,   -34127170,     4979040,    -15456,      1;
  1, 105937, 233908896, 10988845010, 10988845010, 233908896, 105937, 1;

Links

Crossrefs

Cf. A007318 (m=0), A034801 (m=4), A156599 (m=5), A156600 (m=6), A156601 (m=7), this sequence (m=8), A156603.
Cf. A053122.

Programs

  • Mathematica
    (* First program *)
b[n_, k_]:= If[k==n, 2, If[k==n-1 || k==n+1, -1, 0]];
M[d_]:= Table[b[n, k], {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]], {j, n}]/.x->(k+1)];
T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];
Table[T[n, k, 8], {n,0,12}, {k,0,n}]//TableForm (* modified by G. C. Greubel, Jun 25 2021 *)
(* Second program *)
t[n_, k_]:= t[n, k]= If[n==0, 1, If[k==0, (n-1)!, Product[(-1)^j*Simplify[ChebyshevU[j, x/2 - 1]], {j,0,n-1}]/.x->(k+1)]];
T[n_, k_, m_]:= T[n, k, m]= t[n, m]/(t[k, m]*t[n-k, m]);
Table[T[n, k, 8], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 25 2021 *)
  • Sage
    @CachedFunction
def t(n, k):
    if (n==0): return 1
    elif (k==0): return factorial(n-1)
    else: return product( (-1)^j*chebyshev_U(j, (k-1)/2) for j in (0..n-1) )
def T(n,k,m): return t(n,m)/(t(k,m)*t(n-k,m))
flatten([[T(n, k, 8) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 25 2021

Formula

T(n, k, m) = t(n, m)/(t(k, m)*t(n-k, m)), where t(n, k) = Product_{j=1..n} p(j, k+1), p(n, x) = Sum_{j=0..n} (-1)^j*A053122(n, j)*x^j, and m = 8.

Extensions

Edited by G. C. Greubel, Jun 25 2021

A156601 Triangle T(n, k, m) = t(n, m)/(t(k, m)*t(n-k, m)), where t(n, k) = Product_{j=1..n} p(j, k+1), p(n, x) = Sum_{j=0..n} (-1)^j*A053122(n, j)*x^j, and m = 7, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -6, 1, 1, 35, 35, 1, 1, -204, 1190, -204, 1, 1, 1189, 40426, 40426, 1189, 1, 1, -6930, 1373295, -8004348, 1373295, -6930, 1, 1, 40391, 46651605, 1584821667, 1584821667, 46651605, 40391, 1, 1, -235416, 1584781276, -313786692648, 1828884203718, -313786692648, 1584781276, -235416, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 11 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,     1;
  1,    -6,        1;
  1,    35,       35,          1;
  1,  -204,     1190,       -204,          1;
  1,  1189,    40426,      40426,       1189,        1;
  1, -6930,  1373295,   -8004348,    1373295,    -6930,     1;
  1, 40391, 46651605, 1584821667, 1584821667, 46651605, 40391, 1;

Links

Crossrefs

Cf. A007318 (m=0), A034801 (m=4), A156599 (m=5), A156600 (m=6), this sequence (m=7), A156602 (m=8), A156603.
Cf. A053122.

Programs

  • Mathematica
    (* First program *)
b[n_, k_]:= If[k==n, 2, If[k==n-1 || k==n+1, -1, 0]];
M[d_]:= Table[b[n, k], {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]], {j, n}]/.x->(k+1)];
T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];
Table[T[n, k, 7], {n,0,12}, {k,0,n}]//TableForm (* modified by G. C. Greubel, Jun 25 2021 *)
(* Second program *)
t[n_, k_]:= t[n, k]= If[n==0, 1, If[k==0, (n-1)!, Product[(-1)^j*Simplify[ChebyshevU[j, x/2 - 1]], {j,0,n-1}]/.x->(k+1)]];
T[n_, k_, m_]:= T[n, k, m]= t[n, m]/(t[k, m]*t[n-k, m]);
Table[T[n, k, 7], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 25 2021 *)
  • Sage
    @CachedFunction
def t(n, k):
    if (n==0): return 1
    elif (k==0): return factorial(n-1)
    else: return product( (-1)^j*chebyshev_U(j, (k-1)/2) for j in (0..n-1) )
def T(n,k,m): return t(n,m)/(t(k,m)*t(n-k,m))
flatten([[T(n, k, 7) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 25 2021

Formula

T(n, k, m) = t(n, m)/(t(k, m)*t(n-k, m)), where t(n, k) = Product_{j=1..n} p(j, k+1), p(n, x) = Sum_{j=0..n} (-1)^j*A053122(n, j)*x^j, and m = 7.

Extensions

Edited by G. C. Greubel, Jun 25 2021

A156603 Square array T(n, k) = Product_{j=1..n} p(j, k+1), p(n, x) = Sum_{j=0..n} (-1)^j*A053122(n, j)*x^j, and 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, 0, 0, 120, 1, 1, -3, -6, 0, 0, 720, 1, 1, -4, -24, 24, 0, 0, 5040, 1, 1, -5, -60, 504, 120, 0, 0, 40320, 1, 1, -6, -120, 3360, 27720, -720, 0, 0, 362880, 1, 1, -7, -210, 13800, 702240, -3991680, -5040, 0, 0, 3628800
Offset: 0

Views

Author

Roger L. Bagula, Feb 11 2009

Keywords

Examples

			Square array begins as:
    1, 1,  1,    1,        1,          1,            1 ...;
    1, 1,  1,    1,        1,          1,            1 ...;
    2, 0, -1,  -2,        -3,         -4,           -5 ...;
    6, 0,  0,   -6,      -24,        -60,         -120 ...;
   24, 0,  0,   24,      504,       3360,        13800 ...;
  120, 0,  0,  120,    27720,     702240,      7603800 ...;
  720, 0,  0, -720, -3991680, -547747200, -20074032000 ...;
Antidiagonal rows begin as:
  1;
  1, 1;
  1, 1,  2;
  1, 1,  0,    6;
  1, 1, -1,    0,   24;
  1, 1, -2,    0,     0,    120;
  1, 1, -3,   -6,     0,      0,      720;
  1, 1, -4,  -24,    24,      0,        0,  5040;
  1, 1, -5,  -60,   504,    120,        0,     0, 40320;
  1, 1, -6, -120,  3360,  27720,     -720,     0,     0, 362880;
  1, 1, -7, -210, 13800, 702240, -3991680, -5040,     0,      0, 3628800;

Links

Crossrefs

Cf. A007318, A034801, A053122, A156599, A156600, A156601, A156602.

Programs

  • Mathematica
    (* First program *)
b[n_, k_]:= If[k==n, 2, If[k==n-1 || k==n+1, -1, 0]];
M[d_]:= Table[b[n, k], {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]], {j, n}]/.x->(k+1)];
Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 25 2021 *)
(* Second program *)
T[n_, k_]:= If[n==0, 1, If[k==0, n!, Product[(-1)^j*Simplify[ChebyshevU[j, x/2-1]], {j, 0, n-1}]/.x->(k+1)]];
Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 25 2021 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n==0): return 1
    elif (k==0): return factorial(n)
    else: return product( (-1)^j*chebyshev_U(j, (k-1)/2) for j in (0..n-1) )
flatten([[T(k, n-k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jun 25 2021

Formula

T(n, k) = Product_{j=1..n} p(j, k+1), p(n, x) = Sum_{j=0..n} (-1)^j*A053122(n, j)*x^j, and T(n, 0) = n! (square array).
T(n, k) = Product_{j=0..n-1} (-1)^j*ChebyshevU(j, (k-1)/2) with T(n, 0) = n! for n >= 1, and T(0, k) = 1 (square array). - G. C. Greubel, Jun 25 2021

Extensions

Edited by G. C. Greubel, Jun 25 2021
Showing 1-4 of 4 results.

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