A349959 a(n) = Sum_{k=0..floor(n/2)} (k-1)^2*A106828(n, k).

1, 0, 0, 0, 3, 20, 190, 1764, 17773, 192632, 2250036, 28254600, 380304639, 5468906508, 83750505826, 1361579283596, 23431400945145, 425669127018416, 8142731710207432, 163636478165355408, 3447201944202849819, 75973975479088955460, 1748531872985454054246, 41951755708613404583732

Offset

0,5

Comments

For all p prime, a(p) == 0 (mod p*(p-1)).

Links

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

Formula

E.g.f.: (-2 - x + (3 + log((1 - x)^(1 + 2*x)) + (log(1 - x))^2) / (1 - x)) / exp(x).

a(n) ~ n! * exp(-1) * log(n)^2 * (1 + (2*gamma - 3)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 09 2021

Example

E.g.f.: 1 + 3*x^4/4! + 20*x^5/5! + 190*x^6/6! + 1764*x^7/7! + 17773*x^8/8! + 192632*x^9/9! + ...
a(13) = Sum_{k=0..6} (k-1)^2*A106828(13, k).
a(13) =  1*0 + 0*479001600 + 1*967524480 + 4*647536032 + 9*177331440 + 16*18858840 + 25*540540 = 5468906508.
For k = 0, A106828(13, 0) = 0.
For k = 1, (1-1)^2 = 0.
For 2 <= k <= 6, A106828(13, k) == 0 (mod 13*12).
Result a(13) == 0 (mod 13*12).

Maple

a := n -> add((k-1)^2*A106828(n, k), k=0..iquo(n, 2)):
seq(a(n), n=0..23);
# second program:
a := series((-2-x+(3+log((1-x)^(1+2*x))+(log(1-x))^2)/(1-x))/exp(x), x=0, 24):
seq(n!*coeff(a, x, n), n=0..23);

Mathematica

CoefficientList[Series[(-2-x+(3+Log[(1-x)^(1+2*x)]+(Log[1-x])^2)/(1-x))/Exp[x], {x, 0, 23}], x]*Range[0, 23]!

Prog

(PARI) E2(n, m) = sum(k=0, n-m, (-1)^(n+k)*binomial(2*n+1, k)*stirling(2*n-m-k+1, n-m-k+1, 1)); \\ A008517
ast1(n, k) = if ((n==0) && (k==0), 1, sum(j=0, n-k, binomial(j, n-2*k)*E2(n-k, j+1))); \\ A106828
a(n) = sum(k=0, n\2, (k-1)^2*ast1(n, k)); \\ Michel Marcus, Dec 07 2021

Crossrefs

Cf. A106828, A347210, A347571, A348208.

Sequence in context: A286794 A176043 A108206 * A120485 A087152 A158833

Adjacent sequences: A349956 A349957 A349958 * A349960 A349961 A349962

Keyword

nonn

Author

Mélika Tebni, Dec 07 2021

Status

approved