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)).

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

0,5

Comments

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.

Links

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

Wikipedia, Lerch zeta function

Wikipedia, Polylogarithm

Formula

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

Example

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

Mathematica

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

Prog

(Maxima)
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 */
(Magma)
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
(SageMath)
# 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

Crossrefs

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

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

Sequence in context: A220595 A154335 A142467 * A142597 A156137 A152972

Adjacent sequences: A142172 A142173 A142174 * A142176 A142177 A142178

Keyword

nonn,easy,tabl

Author

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

Extensions

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

Status

approved