A347210 Expansion of the e.g.f. (1 - 2*x - 2*log(1 - x) - exp(2*x)*(1 - x)^2) / 4 - 1.

-1, 0, 1, 2, 3, 4, 20, 216, 2072, 18880, 177984, 1805440, 19935872, 239445504, 3113377280, 43588830208, 653836446720, 10461393240064, 177843710148608, 3201186844016640, 60822550184493056, 1216451004043755520, 25545471085755629568, 562000363888584687616

Offset

0,4

Comments

For all p prime, a(p) == -1 (mod p).

For n > 1, a(n) == 0 (mod (n-1)).

Links

Table of n, a(n) for n=0..23.

Formula

a(n) = Sum_{k=0..floor(n/2)} (-1)^(k-1)*ceiling(2^(k-2))*A106828(n, k).

a(n) ~ (n-1)!/2. - Vaclav Kotesovec, Dec 09 2021

Example

E.g.f.: -1 + x^2/2! + 2*x^3/3! + 3*x^4/4! + 4*x^5/5! + 20*x^6/6! + 216*x^7/7! + 2072*x^8/8! + 18880*x^9/9! + ...
a(19) = Sum_{k=1..9} (-1)^(k-1)*ceiling(2^(k-2))*A106828(19, k) = 3201186844016640.
For k = 1, (-1)^(1-1)*ceiling(2^(1-2))*A106828(19, 1) == -1 (mod 19), because (-1)^(1-1)*ceiling(2^(1-2)) = 1 and A106828(19, 1) = (19-1)!
For k >= 2, (-1)^(k-1)*ceiling(2^(k-2))*A106828(19, k) == 0 (mod 19), because A106828(19, k) == 0 (mod 19), result a(19) == -1 (mod 19).
a(10) = Sum_{k=1..5}  (-1)^(k-1)*ceiling(2^(k-2))*A106828(10, k) = 177984.
a(10) == 0 (mod (10-1)), because for k >= 1, A106828(10, k) == 0 (mod 9).

Maple

a := series((1-2*x-2*log(1-x)-exp(2*x)*(1-x)^2)/4-1, x=0, 25):
seq(n!*coeff(a, x, n), n=0..23);
# second program:
a := n -> add((-1)^(k-1)*ceil(2^(k-2))*A106828(n, k), k=0..iquo(n, 2)):
seq(a(n), n=0..23);

Mathematica

CoefficientList[Series[(1 - 2*x - 2*Log[1 - x] - E^(2*x)*(1 - x)^2)/4 - 1, {x, 0, 23}], x]*Range[0, 23]!

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace((1-2*x-2*log(1-x)-exp(2*x)*(1-x)^2)/4 - 1)) \\ Michel Marcus, Aug 23 2021

Crossrefs

Cf. A106828, A343482, A345697, A345969, A346119.

Sequence in context: A012286 A302610 A331558 * A012284 A244673 A012574

Adjacent sequences: A347207 A347208 A347209 * A347211 A347212 A347213

Keyword

sign

Author

Mélika Tebni, Aug 23 2021

Status

approved