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 1-5 of 5 results.

A166343 Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+12*x+x^2)/(1-x)^4, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 27, 27, 1, 1, 58, 162, 58, 1, 1, 121, 718, 718, 121, 1, 1, 248, 2759, 5744, 2759, 248, 1, 1, 503, 9765, 36771, 36771, 9765, 503, 1, 1, 1014, 32816, 205674, 367710, 205674, 32816, 1014, 1, 1, 2037, 106560, 1052408, 3072594, 3072594, 1052408, 106560, 2037, 1
Offset: 1

Views

Author

Roger L. Bagula, Oct 12 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,    1;
  1,   12,      1;
  1,   27,     27,       1;
  1,   58,    162,      58,       1;
  1,  121,    718,     718,     121,       1;
  1,  248,   2759,    5744,    2759,     248,       1;
  1,  503,   9765,   36771,   36771,    9765,     503,      1;
  1, 1014,  32816,  205674,  367710,  205674,   32816,   1014,    1;
  1, 2037, 106560, 1052408, 3072594, 3072594, 1052408, 106560, 2037, 1;

References

  • Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91

Links

Crossrefs

Cf. A166340, A166341, A166344, A166345, A166346, A166349.
Cf. A063521, A123125.

Programs

  • Mathematica
    (* First program *)
p[x_, 1]:= x/(1-x)^2;
p[x_, 2]:= x*(1+x)/(1-x)^3;
p[x_, 3]:= x*(1+12*x+x^2)/(1-x)^4;
p[x_, n_]:= p[x, n]= x*D[p[x, n-1], x]
Table[CoefficientList[(1-x)^(n+1)*p[x, n]/x, x], {n,12}]//Flatten
(* Second program *)
b[n_, k_, m_]:= If[n<2, 1, If[k==0, 0, k^(n-1)*((m+3)*k^2 - m)/3]];
t[n_, k_, m_]:= t[n,k,m]= Sum[(-1)^(k-j)*Binomial[n+1, k-j]*b[n,j,m], {j,0,k}];
T[n_, k_, m_]:= T[n,k,m]= If[k==1, 1, t[n-1,k,m] - t[n-1,k-1,m]];
Table[T[n,k,4], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 11 2022 *)
  • Sage
    def b(n,k,m):
    if (n<2): return 1
    elif (k==0): return 0
    else: return k^(n-1)*((m+3)*k^2 - m)/3
@CachedFunction
def t(n,k,m): return sum( (-1)^(k-j)*binomial(n+1, k-j)*b(n,j,m) for j in (0..k) )
def A166343(n,k): return 1 if (k==1) else t(n-1,k,4) - t(n-1,k-1,4)
flatten([[A166343(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 11 2022

Formula

T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+12*x+x^2)/(1-x)^4.
From G. C. Greubel, Mar 11 2022: (Start)
T(n, k) = t(n-1, k) - t(n-1, k-1), T(n,1) = 1, where t(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n+1, k-j)*b(n, j), b(n, k) = k^(n-2)*A063521(k), b(n, 0) = 1, and b(1, k) = 1.
T(n, n-k) = T(n, k). (End)

Extensions

Edited by G. C. Greubel, Mar 11 2022

A166340 Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+8*x+x^2)/(1-x)^4, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 19, 19, 1, 1, 42, 114, 42, 1, 1, 89, 510, 510, 89, 1, 1, 184, 1975, 4080, 1975, 184, 1, 1, 375, 7029, 26195, 26195, 7029, 375, 1, 1, 758, 23712, 146954, 261950, 146954, 23712, 758, 1, 1, 1525, 77200, 753800, 2191474, 2191474, 753800, 77200, 1525, 1
Offset: 1

Views

Author

Roger L. Bagula, Oct 12 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    8,     1;
  1,   19,    19,      1;
  1,   42,   114,     42,       1;
  1,   89,   510,    510,      89,       1;
  1,  184,  1975,   4080,    1975,     184,      1;
  1,  375,  7029,  26195,   26195,    7029,    375,     1;
  1,  758, 23712, 146954,  261950,  146954,  23712,   758,    1;
  1, 1525, 77200, 753800, 2191474, 2191474, 753800, 77200, 1525, 1;

References

  • Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91

Links

Crossrefs

Cf. A166341, A166343, A166344, A166345, A166346, A166349.
Cf. A004466, A123125.

Programs

  • Mathematica
    (* First program *)
p[x_, 1]:= x/(1-x)^2;
p[x_, 2]:= x*(1+x)/(1-x)^3;
p[x_, 3]:= x*(1+8*x+x^2)/(1-x)^4;
p[x_, n_]:= p[x, n]= x*D[p[x, n-1], x]
Table[CoefficientList[(1-x)^(n+1)*p[x, n]/x, x], {n,12}]//Flatten
(* Second program *)
b[n_, k_, m_]:= If[n<2, 1, If[k==0, 0, k^(n-1)*((m+3)*k^2 - m)/3]];
t[n_, k_, m_]:= t[n, k]= Sum[(-1)^(k-j)*Binomial[n+1, k-j]*b[n,j,m], {j,0,k}];
T[n_, k_, m_]:= T[n, k, m]= If[k==1, 1, t[n-1, k, m] - t[n-1, k-1, m]];
Table[T[n, k, 2], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 11 2022 *)
  • Sage
    def b(n,k,m):
    if (n<2): return 1
    elif (k==0): return 0
    else: return k^(n-1)*((m+3)*k^2 - m)/3
@CachedFunction
def t(n,k,m): return sum( (-1)^(k-j)*binomial(n+1, k-j)*b(n,j,m) for j in (0..k) )
def A166340(n,k): return 1 if (k==1) else t(n-1,k,2) - t(n-1,k-1,2)
flatten([[A166340(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 11 2022

Formula

T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+8*x+x^2)/(1-x)^4.
From G. C. Greubel, Mar 11 2022: (Start)
T(n, k) = t(n-1, k) - t(n-1, k-1), T(n,1) = 1, where t(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n+1, k-j)*b(n, j), b(n, k) = k^(n-2)*A004466(k), b(n, 0) = 1, and b(1, k) = 1.
T(n, n-k) = T(n, k). (End)

Extensions

Edited by G. C. Greubel, Mar 11 2022

A166341 Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+10*x+x^2)/(1-x)^4, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 23, 23, 1, 1, 50, 138, 50, 1, 1, 105, 614, 614, 105, 1, 1, 216, 2367, 4912, 2367, 216, 1, 1, 439, 8397, 31483, 31483, 8397, 439, 1, 1, 886, 28264, 176314, 314830, 176314, 28264, 886, 1, 1, 1781, 91880, 903104, 2632034, 2632034, 903104, 91880, 1781, 1
Offset: 1

Views

Author

Roger L. Bagula, Oct 12 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,    1;
  1,   10,     1;
  1,   23,    23,      1;
  1,   50,   138,     50,       1;
  1,  105,   614,    614,     105,       1;
  1,  216,  2367,   4912,    2367,     216,      1;
  1,  439,  8397,  31483,   31483,    8397,    439,     1;
  1,  886, 28264, 176314,  314830,  176314,  28264,   886,    1;
  1, 1781, 91880, 903104, 2632034, 2632034, 903104, 91880, 1781, 1;

References

  • Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91

Links

Crossrefs

Cf. A166340, A166343, A166344, A166345, A166346, A166349.
Cf. A007588, A123125.

Programs

  • Mathematica
    (* First program *)
p[x_, 1]:= x/(1-x)^2;
p[x_, 2]:= x*(1+x)/(1-x)^3;
p[x_, 3]:= x*(1+10*x+x^2)/(1-x)^4;
p[x_, n_]:= p[x, n]= x*D[p[x, n-1], x]
Table[CoefficientList[(1-x)^(n+1)*p[x, n]/x, x], {n,12}]//Flatten
(* Second program *)
b[n_, k_, m_]:= If[n<2, 1, If[k==0, 0, k^(n-1)*((m+3)*k^2 - m)/3]];
t[n_, k_, m_]:= t[n,k,m]= Sum[(-1)^(k-j)*Binomial[n+1, k-j]*b[n,j,m], {j,0,k}];
T[n_, k_, m_]:= T[n,k,m]= If[k==1, 1, t[n-1,k,m] - t[n-1,k-1,m]];
Table[T[n,k,3], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 11 2022 *)
  • Sage
    def b(n,k,m):
    if (n<2): return 1
    elif (k==0): return 0
    else: return k^(n-1)*((m+3)*k^2 - m)/3
@CachedFunction
def t(n,k,m): return sum( (-1)^(k-j)*binomial(n+1, k-j)*b(n,j,m) for j in (0..k) )
def A166341(n,k): return 1 if (k==1) else t(n-1,k,3) - t(n-1,k-1,3)
flatten([[A166341(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 11 2022

Formula

T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+10*x+x^2)/(1-x)^4.
From G. C. Greubel, Mar 11 2022: (Start)
T(n, k) = t(n-1, k) - t(n-1, k-1), T(n,1) = 1, where t(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n+1, k-j)*b(n, j), b(n, k) = k^(n-2)*A007588(k), b(n, 0) = 1, and b(1, k) = 1.
T(n, n-k) = T(n, k). (End)

Extensions

Edited by G. C. Greubel, Mar 11 2022

A166344 Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x)^4, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 15, 15, 1, 1, 34, 90, 34, 1, 1, 73, 406, 406, 73, 1, 1, 152, 1583, 3248, 1583, 152, 1, 1, 311, 5661, 20907, 20907, 5661, 311, 1, 1, 630, 19160, 117594, 209070, 117594, 19160, 630, 1, 1, 1269, 62520, 604496, 1750914, 1750914, 604496, 62520, 1269, 1
Offset: 1

Views

Author

Roger L. Bagula, Oct 12 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    6,     1;
  1,   15,    15,      1;
  1,   34,    90,     34,       1;
  1,   73,   406,    406,      73,       1;
  1,  152,  1583,   3248,    1583,     152,      1;
  1,  311,  5661,  20907,   20907,    5661,    311,     1;
  1,  630, 19160, 117594,  209070,  117594,  19160,   630,    1;
  1, 1269, 62520, 604496, 1750914, 1750914, 604496, 62520, 1269, 1;

References

  • Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91

Links

Crossrefs

Cf. A166340, A166341, A166343, A166345, A166346, A166349.
Cf. A000447, A123125.

Programs

  • Mathematica
    (* First program *)
p[x_, 1]:= x/(1-x)^2;
p[x_, 2]:= x*(1+x)/(1-x)^3;
p[x_, 3]:= x*(1+6*x+x^2)/(1-x)^4;
p[x_, n_]:= p[x, n]= x*D[p[x, n-1], x]
Table[CoefficientList[(1-x)^(n+1)*p[x, n]/x, x], {n,12}]//Flatten
(* Second program *)
b[n_, k_, m_]:= If[n<2, 1, If[k==0, 0, k^(n-1)*((m+3)*k^2 - m)/3]];
t[n_, k_, m_]:= t[n,k,m]= Sum[(-1)^(k-j)*Binomial[n+1, k-j]*b[n,j,m], {j,0,k}];
T[n_, k_, m_]:= T[n,k,m]= If[k==1, 1, t[n-1,k,m] - t[n-1,k-1,m]];
Table[T[n,k,1], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 11 2022 *)
  • Sage
    def b(n,k,m):
    if (n<2): return 1
    elif (k==0): return 0
    else: return k^(n-1)*((m+3)*k^2 - m)/3
@CachedFunction
def t(n,k,m): return sum( (-1)^(k-j)*binomial(n+1, k-j)*b(n,j,m) for j in (0..k) )
def A166344(n,k): return 1 if (k==1) else t(n-1,k,1) - t(n-1,k-1,1)
flatten([[A166344(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 11 2022

Formula

T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x)^4.
From G. C. Greubel, Mar 11 2022: (Start)
T(n, k) = t(n-1, k) - t(n-1, k-1), T(n,1) = 1, where t(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n+1, k-j)*b(n, j), b(n, k) = k^(n-2)*A000447(k), b(n, 0) = 1, and b(1, k) = 1.
T(n, n-k) = T(n, k). (End)

Extensions

Edited by G. C. Greubel, Mar 11 2022

A166345 Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+2*x+x^2)/(1-x)^4, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 7, 7, 1, 1, 18, 42, 18, 1, 1, 41, 198, 198, 41, 1, 1, 88, 799, 1584, 799, 88, 1, 1, 183, 2925, 10331, 10331, 2925, 183, 1, 1, 374, 10056, 58874, 103310, 58874, 10056, 374, 1, 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1
Offset: 1

Views

Author

Roger L. Bagula, Oct 12 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   2,     1;
  1,   7,     7,      1;
  1,  18,    42,     18,      1;
  1,  41,   198,    198,     41,      1;
  1,  88,   799,   1584,    799,     88,      1;
  1, 183,  2925,  10331,  10331,   2925,    183,     1;
  1, 374, 10056,  58874, 103310,  58874,  10056,   374,   1;
  1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1;

References

  • Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91

Links

Crossrefs

Cf. A166340, A166341, A166343, A166344, A166346, A166349.
Cf. A005900, A123125.

Programs

  • Mathematica
    (* First program *)
p[x_, 1]:= x/(1-x)^2;
p[x_, 2]:= x*(1+x)/(1-x)^3;
p[x_, 3]:= x*(1+10*x+x^2)/(1-x)^4;
p[x_, n_]:= p[x, n]= x*D[p[x, n-1], x]
Table[CoefficientList[(1-x)^(n+1)*p[x, n]/x, x], {n,12}]//Flatten
(* Second program *)
b[n_, k_, m_]:= If[n<2, 1, If[k==0, 0, k^(n-1)*((m+3)*k^2 - m)/3]];
t[n_, k_, m_]:= t[n,k,m]= Sum[(-1)^(k-j)*Binomial[n+1, k-j]*b[n,j,m], {j,0,k}];
T[n_, k_, m_]:= T[n,k,m]= If[k==1, 1, t[n-1,k,m] - t[n-1,k-1,m]];
Table[T[n,k,-1], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 11 2022 *)
  • Sage
    def b(n,k,m):
    if (n<2): return 1
    elif (k==0): return 0
    else: return k^(n-1)*((m+3)*k^2 - m)/3
@CachedFunction
def t(n,k,m): return sum( (-1)^(k-j)*binomial(n+1, k-j)*b(n,j,m) for j in (0..k) )
def A166345(n,k): return 1 if (k==1) else t(n-1,k,-1) - t(n-1,k-1,-1)
flatten([[A166345(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 11 2022

Formula

T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+2*x+x^2)/(1-x)^4.
From G. C. Greubel, Mar 11 2022: (Start)
T(n, k) = t(n-1, k) - t(n-1, k-1), T(n,1) = 1, where t(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n+1, k-j)*b(n, j), b(n, k) = k^(n-2)*A005900(k), b(n, 0) = 1, and b(1, k) = 1.
T(n, n-k) = T(n, k). (End)

Extensions

Edited by G. C. Greubel, Mar 11 2022
Showing 1-5 of 5 results.

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