|
|
A142175
|
|
T(n,k) = (1/4)*A007318(n,k) - (3/2)*A008292(n+1,k+1) + (9/4)*A060187(n+1,k+1), triangle read by rows (0 <= k <= n).
|
|
9
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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;
|
|
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);
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|