A166343 -id:A166343 - cp's OEIS frontend

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) = xD( 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.

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

Roger L. Bagula, Oct 12 2009

			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;

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

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

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

Cf. A004466, A123125.

(* 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 *)

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

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)

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) = xD( 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

Roger L. Bagula, Oct 12 2009

			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;

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

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

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

Cf. A007588, A123125.

(* 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 *)

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

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)

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) = xD( 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

Roger L. Bagula, Oct 12 2009

			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;

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

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

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

Cf. A000447, A123125.

(* 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 *)

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

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)

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) = xD( 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

Roger L. Bagula, Oct 12 2009

			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;

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

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

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

Cf. A005900, A123125.

(* 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 *)

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

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)

Edited by G. C. Greubel, Mar 11 2022

A168518 Expansion of g.f. (1/2)( a(1+x)^n + b(1-x)^(n+2)LerchPhi(x, -n-1, 1) + c2^(n+1)(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -4, b = 2, and c = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 51, 51, 1, 1, 170, 514, 170, 1, 1, 521, 3646, 3646, 521, 1, 1, 1552, 22247, 49472, 22247, 1552, 1, 1, 4591, 125565, 534995, 534995, 125565, 4591, 1, 1, 13590, 677776, 5058698, 9506078, 5058698, 677776, 13590, 1, 1, 40341, 3560448, 43870968, 140136690, 140136690, 43870968, 3560448, 40341, 1

Offset: 0

Roger L. Bagula, Nov 28 2009

			Triangle begins as:
  1;
  1,     1;
  1,    12,       1;
  1,    51,      51,        1;
  1,   170,     514,      170,         1;
  1,   521,    3646,     3646,       521,         1;
  1,  1552,   22247,    49472,     22247,      1552,        1;
  1,  4591,  125565,   534995,    534995,    125565,     4591,       1;
  1, 13590,  677776,  5058698,   9506078,   5058698,   677776,   13590,     1;
  1, 40341, 3560448, 43870968, 140136690, 140136690, 43870968, 3560448, 40341, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A142460, A155491, A155495, A157273, A166343.

Cf. A168517, A168549, A168551, A168552.

p[x_, n_, a_, b_, c_]= (1/2)*(a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi[x,-n-1,1] + c*2^(n+1)*(1-x)^(n+1)*LerchPhi[x,-n,1/2]);
Table[CoefficientList[p[x,n,-4,2,2], x], {n,0,10}]//Flatten (* modified by G. C. Greubel, Mar 31 2022 *)

def A168518(n,k,a,b,c): return (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) )
flatten([[A168518(n,k,-4,2,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 31 2022

G.f.: (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1 - x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -4, b = 2, and c = 2.
From G. C. Greubel, Mar 31 2022: (Start)
T(n, k) = (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) ), with a = -4, b = 2, and c = 2.
T(n, n-k) = T(n, k). (End)

Edited by G. C. Greubel, Mar 31 2022

cp's OEIS Frontend

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.

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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.

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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.

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Extensions

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.

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A168518 Expansion of g.f. (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -4, b = 2, and c = 2, read by rows.

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Examples

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Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

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) = xD( 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.

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) = xD( 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.

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) = xD( 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.

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) = xD( 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.

A168518 Expansion of g.f. (1/2)( a(1+x)^n + b(1-x)^(n+2)LerchPhi(x, -n-1, 1) + c2^(n+1)(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -4, b = 2, and c = 2, read by rows.