A168288 -id:A168288 - cp's OEIS frontend

A142175 Triangle read by rows: T(n,k) = (1/4)(A007318(n,k) - 6A008292(n+1,k+1) + 9*A060187(n+1,k+1)).

1, 1, 1, 1, 8, 1, 1, 36, 36, 1, 1, 133, 420, 133, 1, 1, 449, 3334, 3334, 449, 1, 1, 1446, 21939, 49364, 21939, 1446, 1, 1, 4534, 130044, 560957, 560957, 130044, 4534, 1, 1, 13991, 724222, 5459561, 10284514, 5459561, 724222, 13991, 1, 1, 42747, 3880014, 48160170, 154214412, 154214412, 48160170, 3880014, 42747, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 16 2008

Row n gives the coefficients in the expansion of (1/4)*(1 + x)^n + (9/4)*2^n*(1 - x)^(1 + n)*Phi(x, -n, 1/2) - (3/2)*(1 - x)^(n + 2)*Phi(x, -1 - n, 1), where Phi is the Lerch transcendant.

			Triangle begins:
     1;
     1,    1;
     1,    8,      1;
     1,   36,     36,      1;
     1,  133,    420,    133,      1;
     1,  449,   3334,   3334,    449,      1;
     1, 1446,  21939,  49364,  21939,   1446,    1;
     1, 4534, 130044, 560957, 560957, 130044, 4534, 1;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Wikipedia, Lerch zeta function
Wikipedia, Polylogarithm

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A168287, A168288, A168289, A168290, A168291, A168292, A168293.

A060187:= func< n,k | (&+[(-1)^(k-j)*Binomial(n, k-j)*(2*j-1)^(n-1): j in [1..k]]) >;
A142175:= func< n,k | (Binomial(n,k) - 6*EulerianNumber(n+1,k) + 9*A060187(n+1,k+1))/4 >;
[A142175(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 30 2024

p[x_, n_] = 1/4*(1 + x)^n + 9/4*2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2] - 3/2*(1 - x)^(2 + n)*PolyLog[-1 - n, x]/x;
Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 0, 10}]// Flatten

A008292(n, k) := sum((-1)^j*(k - j)^n*binomial(n + 1, j), j, 0, k)$
A060187(n, k) := sum((-1)^(k - j)*binomial(n, k - j)*(2*j - 1)^(n - 1), j, 1, k)$
T(n, k) := (binomial(n, k) - 6*A008292(n + 1, k + 1) + 9*A060187(n + 1, k + 1))/4$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 20 2018 */

# from sage.all import * # (use for Python)
from sage.combinat.combinat import eulerian_number
def A060187(n,k): return sum(pow(-1, k-j)*binomial(n, k-j)*pow(2*j-1, n-1) for j in range(1,k+1))
def A142175(n,k): return (binomial(n,k) - 6*eulerian_number(n+1,k) +9*A060187(n+1,k+1))//4
print(flatten([[A142175(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Dec 30 2024

E.g.f.: (exp((1 + x)*y) - 6*(1 - x)^2*exp(y*(1 - x))/(1 - x*exp(y*(1 - x)))^2 + 9*(1 - x)*exp((1 - x)*y)/(1 - x*exp(2*(1 - x)*y)))/4. - Franck Maminirina Ramaharo, Oct 20 2018

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 19 2018

A142147 Irregular triangle read by rows: first row is 1, and the n-th row gives the coefficients in the expansion of (1/2x)(1 - 2x(1 - x))^(n + 1)Li(-n, 2x*(1 - x)), where Li(n, z) is the polylogarithm.

Original entry on oeis.org

1, 1, -1, 1, 1, -4, 2, 1, 7, -12, -4, 12, -4, 1, 21, 0, -102, 100, 4, -32, 8, 1, 51, 160, -532, -24, 904, -672, 48, 80, -16, 1, 113, 980, -1094, -5128, 8760, -736, -6224, 3920, -432, -192, 32, 1, 239, 4284, 5276, -43964, 19764, 90272, -114080, 19824, 36304

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 15 2008

			Triangle begins:
     1;
     1, -1;
     1,  1,  -4,    2;
     1,  7, -12,   -4,  12,  -4;
     1, 21,   0, -102, 100,   4,  -32,  8;
     1, 51, 160, -532, -24, 904, -672, 48, 80, -16;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

Eric Weisstein's World of Mathematics, Polylogarithm

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142175, A168287, A168288, A168289, A168290, A168291, A168292, A168293.

p[x_, n_] = If[n == 0, 1, (1 + 2*(-1 + x)*x)^(n + 1)*PolyLog[-n, 2*x*(1 - x)]/(2*x)];
Table[CoefficientList[FullSimplify[Expand[p[x, n]]], x], {n, 0, 10}]//Flatten

E.g.f.: ((1 - x)*(1 - 2*x)*exp(t*(1 + 2*x^2)) + x*exp(2*t*x))/(exp(2*t*x) - 2*x*(1 - x)*exp(t*(1 + 2*x^2))). - Franck Maminirina Ramaharo, Oct 22 2018

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 21 2018

A168287 T(n,k) = 2*A046802(n+1,k+1) - A007318(n,k), triangle read by rows (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 26, 60, 26, 1, 1, 57, 252, 252, 57, 1, 1, 120, 931, 1746, 931, 120, 1, 1, 247, 3201, 10187, 10187, 3201, 247, 1, 1, 502, 10534, 53542, 89788, 53542, 10534, 502, 1, 1, 1013, 33698, 262466, 688976, 688976, 262466, 33698, 1013

Offset: 0

Roger L. Bagula, Nov 22 2009

			Triangle begins:
     1;
     1,    1;
     1,    4,     1;
     1,   11,    11,     1;
     1,   26,    60,    26,     1;
     1,   57,   252,   252,    57,     1;
     1,  120,   931,  1746,   931,   120,     1;
     1,  247,  3201, 10187, 10187,  3201,   247,   1;
     1,  502, 10534, 53542, 89788, 53542, 10534, 502, 1;
     ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A142175, A168288, A168289, A168290, A168291, A168292, A168293.

p[t_] = 2*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - Exp[t*(1 + x)];
Table[CoefficientList[FullSimplify[n!*SeriesCoefficient[Series[p[t], {t, 0, n}], n]], x], {n, 0, 10}]//Flatten

A046802(n, k) := sum(binomial(n - 1, r)*sum(j!*(-1)^(k - j - 1)*stirling2(r, j)*binomial(r - j, k - j - 1), j, 0, k - 1), r, k - 1, n - 1)$
T(n, k) := 2*A046802(n + 1, k + 1) - binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

E.g.f.: 2*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - exp(t*(1 + x)).

Edited, and new name by Franck Maminirina Ramaharo, Oct 21 2018

A168289 T(n,k) = 4A046802(n+1,k+1) - 3A007318(n,k), triangle read by rows (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 19, 19, 1, 1, 48, 114, 48, 1, 1, 109, 494, 494, 109, 1, 1, 234, 1847, 3472, 1847, 234, 1, 1, 487, 6381, 20339, 20339, 6381, 487, 1, 1, 996, 21040, 107028, 179506, 107028, 21040, 996, 1, 1, 2017, 67360, 524848, 1377826, 1377826, 524848

Offset: 0

Roger L. Bagula, Nov 22 2009

			Triangle begins:
     1;
     1,   1;
     1,   6,     1;
     1,  19,    19,      1;
     1,  48,   114,     48,      1;
     1, 109,   494,    494,    109,      1;
     1, 234,  1847,   3472,   1847,    234,     1;
     1, 487,  6381,  20339,  20339,   6381,   487,   1;
     1, 996, 21040, 107028, 179506, 107028, 21040, 996, 1;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A142175, A168287, A168288, A168290, A168291, A168292, A168293.

p[t_] = 4*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 3*Exp[t*(1 + x)];
Table[CoefficientList[FullSimplify[n!*SeriesCoefficient[Series[p[t], {t, 0, n}], n]], x], {n, 0, 10}]//Flatten

A046802(n, k) := sum(binomial(n - 1, r)*sum(j!*(-1)^(k - j - 1)*stirling2(r, j)*binomial(r - j, k - j - 1), j, 0, k - 1), r, k - 1, n - 1)$
T(n, k) := 4*A046802(n + 1, k + 1) - 3*binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

E.g.f: 4*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 3*exp(t*(1 + x)).

Edited, and new name by Franck Maminirina Ramaharo, Oct 21 2018

A168290 T(n,k) = 5A046802(n+1,k+1) - 4A007318(n,k), triangle read by rows (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 23, 23, 1, 1, 59, 141, 59, 1, 1, 135, 615, 615, 135, 1, 1, 291, 2305, 4335, 2305, 291, 1, 1, 607, 7971, 25415, 25415, 7971, 607, 1, 1, 1243, 26293, 133771, 224365, 133771, 26293, 1243, 1, 1, 2519, 84191, 656039, 1722251, 1722251, 656039

Offset: 0

Roger L. Bagula, Nov 22 2009

			Triangle begins:
    1;
    1,    1;
    1,    7,     1;
    1,   23,    23,      1;
    1,   59,   141,     59,      1;
    1,  135,   615,    615,    135,      1;
    1,  291,  2305,   4335,   2305,    291,     1;
    1,  607,  7971,  25415,  25415,   7971,   607,    1;
    1, 1243, 26293, 133771, 224365, 133771, 26293, 1243, 1;
     ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A142175, A168287, A168288, A168289, A168291, A168292, A168293.

p[t_] = 5*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 4*Exp[t*(1 + x)];
Table[CoefficientList[FullSimplify[n!*SeriesCoefficient[Series[p[ t], {t, 0, n}], n]], x], {n, 0, 10}]//Flatten

A046802(n, k) := sum(binomial(n - 1, r)*sum(j!*(-1)^(k - j - 1)*stirling2(r, j)*binomial(r - j, k - j - 1), j, 0, k - 1), r, k - 1, n - 1)$
T(n, k) := 5*A046802(n + 1, k + 1) - 4*binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

E.g.f.: 5*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 4*exp(t*(1 + x)).

Edited, new name by Franck Maminirina Ramaharo, Oct 21 2018

A168291 T(n,k) = 4A046802(n+1,k+1) - 2A008518(n,k) - A007318(n,k), triangle read by rows (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 15, 15, 1, 1, 32, 82, 32, 1, 1, 65, 330, 330, 65, 1, 1, 130, 1159, 2304, 1159, 130, 1, 1, 259, 3801, 13195, 13195, 3801, 259, 1, 1, 516, 12016, 67316, 117170, 67316, 12016, 516, 1, 1, 1029, 37212, 319332, 889230, 889230, 319332, 37212

Offset: 0

Roger L. Bagula, Nov 22 2009

			Triangle begins:
     1;
     1,   1;
     1,   6,     1;
     1,  15,    15,     1;
     1,  32,    82,    32,      1;
     1,  65,   330,   330,     65,     1;
     1, 130,  1159,  2304,   1159,   130,     1;
     1, 259,  3801, 13195,  13195,  3801,   259,   1;
     1, 516, 12016, 67316, 117170, 67316, 12016, 516, 1;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A142175, A168287, A168288, A168289, A168290, A168292, A168293.

A123125(n, k) := sum((-1)^(k - j)*(binomial(n - j, k - j))*stirling2(n, j)*j!, j, 0, k)$
A046802(n, k) := sum(binomial(n - 1, r)*A123125(r, k - 1), r, k - 1, n - 1)$
A008518(n, k) := A123125(n, k) + A123125(n, k + 1)$
T(n, k) := 4*A046802(n + 1, k + 1) - 2*A008518(n, k) - binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

E.g.f.: 4*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 2*(exp(t) - x*exp(t*x))/(exp(t*x) - x*exp(t)) - exp(t*(1 + x)).

Edited, new name by Franck Maminirina Ramaharo, Oct 21 2018

A168292 T(n,k) = 24A046802(n+1,k+1) - 9A008518(n,k) - 8*A007318(n,k), triangle read by rows (0 <= k <= n).

Original entry on oeis.org

7, 7, 7, 7, 38, 7, 7, 99, 99, 7, 7, 220, 546, 220, 7, 7, 461, 2236, 2236, 461, 7, 7, 942, 8001, 15596, 8001, 942, 7, 7, 1903, 26697, 89921, 89921, 26697, 1903, 7, 7, 3824, 85660, 463520, 796594, 463520, 85660, 3824, 7, 7, 7665, 268530, 2224350, 6068400

Offset: 0

Roger L. Bagula, Nov 22 2009

			Triangle begins:
     7;
     7,    7;
     7,   38,     7;
     7,   99,    99,      7;
     7,  220,   546,    220,      7;
     7,  461,  2236,   2236,    461,      7;
     7,  942,  8001,  15596,   8001,    942,     7;
     7, 1903, 26697,  89921,  89921,  26697,  1903,    7;
     7, 3824, 85660, 463520, 796594, 463520, 85660, 3824, 7;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A142175, A168287, A168288, A168289, A168290, A168291, A168293.

A123125(n, k) := sum((-1)^(k - j)*(binomial(n - j, k - j))*stirling2(n, j)*j!, j, 0, k)$
A046802(n, k) := sum(binomial(n - 1, r)*A123125(r, k - 1), r, k - 1, n - 1)$
A008518(n, k) := A123125(n, k) + A123125(n, k + 1)$
T(n, k) := 24*A046802(n + 1, k + 1) - 9*A008518(n, k) - 8*binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

E.g.f.: 24*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 9*(exp(t) - x*exp(t*x))/(exp(t*x) - x*exp(t)) - 8*exp(t*(1 + x)).

Edited, new name from Franck Maminirina Ramaharo, Oct 21 2018

A168293 T(n,k) = 12A046802(n+1,k+1) - 9A008518(n,k) - 2*A007318(n,k), triangle read by rows (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 14, 1, 1, 33, 33, 1, 1, 64, 186, 64, 1, 1, 119, 724, 724, 119, 1, 1, 222, 2415, 5120, 2415, 222, 1, 1, 421, 7491, 28799, 28799, 7491, 421, 1, 1, 812, 22456, 142268, 257866, 142268, 22456, 812, 1, 1, 1587, 66342, 649554, 1934544, 1934544, 649554

Offset: 0

Roger L. Bagula, Nov 22 2009

			Triangle begins:
    1;
    1,   1;
    1,  14,     1;
    1,  33,    33,      1;
    1,  64,   186,     64,      1;
    1, 119,   724,    724,    119,      1;
    1, 222,  2415,   5120,   2415,    222,     1;
    1, 421,  7491,  28799,  28799,   7491,   421,   1;
    1, 812, 22456, 142268, 257866, 142268, 22456, 812, 1:
     ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A142175, A168287, A168288, A168289, A168290, A168291, A168292.

A123125(n, k) := sum((-1)^(k - j)*(binomial(n - j, k - j))*stirling2(n, j)*j!, j, 0, k)$
A046802(n, k) := sum(binomial(n - 1, r)*A123125(r, k - 1), r, k - 1, n - 1)$
A008518(n, k) := A123125(n, k) + A123125(n, k + 1)$
T(n, k) := 12*A046802(n + 1, k + 1) - 9*A008518(n, k) - 2*binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

E.g.f.: 12*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 9*(exp(t) - x*exp(t*x))/(exp(t*x) - x*exp(t)) - 2*exp(t*(1 + x)).

Edited, and new name by Franck Maminirina Ramaharo, Oct 21 2018

cp's OEIS Frontend

A142175 Triangle read by rows: T(n,k) = (1/4)*(A007318(n,k) - 6*A008292(n+1,k+1) + 9*A060187(n+1,k+1)).

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A142147 Irregular triangle read by rows: first row is 1, and the n-th row gives the coefficients in the expansion of (1/2*x)*(1 - 2*x*(1 - x))^(n + 1)*Li(-n, 2*x*(1 - x)), where Li(n, z) is the polylogarithm.

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A168287 T(n,k) = 2*A046802(n+1,k+1) - A007318(n,k), triangle read by rows (0 <= k <= n).

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A168289 T(n,k) = 4*A046802(n+1,k+1) - 3*A007318(n,k), triangle read by rows (0 <= k <= n).

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A168290 T(n,k) = 5*A046802(n+1,k+1) - 4*A007318(n,k), triangle read by rows (0 <= k <= n).

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Maxima

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A168291 T(n,k) = 4*A046802(n+1,k+1) - 2*A008518(n,k) - A007318(n,k), triangle read by rows (0 <= k <= n).

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A168292 T(n,k) = 24*A046802(n+1,k+1) - 9*A008518(n,k) - 8*A007318(n,k), triangle read by rows (0 <= k <= n).

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A142175 Triangle read by rows: T(n,k) = (1/4)(A007318(n,k) - 6A008292(n+1,k+1) + 9*A060187(n+1,k+1)).

A142147 Irregular triangle read by rows: first row is 1, and the n-th row gives the coefficients in the expansion of (1/2x)(1 - 2x(1 - x))^(n + 1)Li(-n, 2x*(1 - x)), where Li(n, z) is the polylogarithm.

A168289 T(n,k) = 4A046802(n+1,k+1) - 3A007318(n,k), triangle read by rows (0 <= k <= n).

A168290 T(n,k) = 5A046802(n+1,k+1) - 4A007318(n,k), triangle read by rows (0 <= k <= n).

A168291 T(n,k) = 4A046802(n+1,k+1) - 2A008518(n,k) - A007318(n,k), triangle read by rows (0 <= k <= n).

A168292 T(n,k) = 24A046802(n+1,k+1) - 9A008518(n,k) - 8*A007318(n,k), triangle read by rows (0 <= k <= n).

A168293 T(n,k) = 12A046802(n+1,k+1) - 9A008518(n,k) - 2*A007318(n,k), triangle read by rows (0 <= k <= n).