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 1-7 of 7 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

A156599 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 = 5, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -4, 1, 1, 15, 15, 1, 1, -56, 210, -56, 1, 1, 209, 2926, 2926, 209, 1, 1, -780, 40755, -152152, 40755, -780, 1, 1, 2911, 567645, 7909187, 7909187, 567645, 2911, 1, 1, -10864, 7906276, -411126352, 1534382278, -411126352, 7906276, -10864, 1, 1, 40545, 110120220, 21370664028, 297662820390, 297662820390, 21370664028, 110120220, 40545, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 11 2009

Keywords

Examples

			Triangle begins:
  1;
  1,    1;
  1,   -4,      1;
  1,   15,     15,       1;
  1,  -56,    210,     -56,       1;
  1,  209,   2926,    2926,     209,      1;
  1, -780,  40755, -152152,   40755,   -780,    1;
  1, 2911, 567645, 7909187, 7909187, 567645, 2911, 1;

Links

Crossrefs

Cf. A007318 (m=0), A034801 (m=4), this sequence (m=5), A156600 (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, 5], {n,0,12}, {k,0,n}]//TableForm (* modified by G. C. Greubel, May 23 2019; 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, 5], {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, 5) 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 = 5.

Extensions

Edited by G. C. Greubel, May 23 2019; 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

A173005 A product triangle sequence based on recursion:a=4; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a).

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 80, 80, 1, 1, 711, 6320, 711, 1, 1, 6319, 499201, 499201, 6319, 1, 1, 56160, 39430560, 350439102, 39430560, 56160, 1, 1, 499121, 3114515040, 246007756722, 246007756722, 3114515040, 499121, 1, 1, 4435929, 246007257601
Offset: 0

Views

Author

Roger L. Bagula, Feb 07 2010

Keywords

Comments

Row sums are:
{1, 2, 11, 162, 7744, 1011042, 429412544, 498245541768, 1880728607247424,
19394268001029953928, 650631110504313946320896,...}.
a = 1; A034801.
a = 2; A156600.
a = 3; A156602.
This result seems to connect these new recursions directly to q-forms.

Examples

			{1},
{1, 1},
{1, 9, 1},
{1, 80, 80, 1},
{1, 711, 6320, 711, 1},
{1, 6319, 499201, 499201, 6319, 1},
{1, 56160, 39430560, 350439102, 39430560, 56160, 1},
{1, 499121, 3114515040, 246007756722, 246007756722, 3114515040, 499121, 1},
{1, 4435929, 246007257601, 172697094835902, 1534842394188558, 172697094835902, 246007257601, 4435929, 1},
{1, 39424240, 19431458835440, 121233114567545603, 9575881454449171680, 9575881454449171680, 121233114567545603, 19431458835440, 39424240, 1},
{1, 350382231, 1534839240742160, 85105473729326613333, 59743922859711995180563, 530973050767752120484320, 59743922859711995180563, 85105473729326613333, 1534839240742160, 350382231, 1}

Crossrefs

A034801, A156600., A156602.

Programs

  • Mathematica
    Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = (2*a + 1)*f[n - 1, a] - f[n - 2, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]

Formula

a=4; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a);
c(n)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
t(n,m)=c(n)/(c(m)*c(n-m)

A173006 A product triangle sequence based on recursion:a=5; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a).

Original entry on oeis.org

1, 1, 1, 1, 11, 1, 1, 120, 120, 1, 1, 1309, 14280, 1309, 1, 1, 14279, 1699201, 1699201, 14279, 1, 1, 155760, 202190640, 2205562898, 202190640, 155760, 1, 1, 1699081, 24058986960, 2862818956682, 2862818956682, 24058986960, 1699081, 1, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 07 2010

Keywords

Comments

Row sums are:
{1, 2, 13, 242, 16900, 3426962, 2610255700, 5773759285448, 47972252879976100,
1157507562695117906888, 104909162208463229766370000,...}.
a = 1; A034801.
a = 2; A156600.
a = 3; A156602.
This result seems to connect these new recursions directly to q-forms.

Examples

			{1},
{1, 1},
{1, 11, 1},
{1, 120, 120, 1},
{1, 1309, 14280, 1309, 1},
{1, 14279, 1699201, 1699201, 14279, 1},
{1, 155760, 202190640, 2205562898, 202190640, 155760, 1},
{1, 1699081, 24058986960, 2862818956682, 2862818956682, 24058986960, 1699081, 1},
{1, 18534131, 2862817257601, 3715936800366098, 40534653607660438, 3715936800366098, 2862817257601, 18534131, 1},
{1, 202176360, 340651194667560, 4823283104057937603, 573930157592104171920, 573930157592104171920, 4823283104057937603, 340651194667560, 202176360, 1},
{1, 2205405829, 40534629348182040, 6260617753130421176727, 8126277060814812179812443, 88644086770258081457215920, 8126277060814812179812443, 6260617753130421176727, 40534629348182040, 2205405829, 1}

Crossrefs

A034801, A156600., A156602.

Programs

  • Mathematica
    Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = (2*a + 1)*f[n - 1, a] - f[n - 2, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]

Formula

a=5; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a);
c(n)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
t(n,m)=c(n)/(c(m)*c(n-m)
Showing 1-7 of 7 results.

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