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
Examples
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
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
- Wikipedia, Lerch zeta function
- Wikipedia, Polylogarithm
Crossrefs
Programs
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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
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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 -
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 */
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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
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
Extensions
Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 19 2018
Comments