cp's OEIS Frontend

Inspired by a formula in the reference, the study of the singular points of planar differential systems leads to 3 two-dimensional polynomial families, one ordinary (degenerate case, considered in one dimension, see A129326) and two odd (the second, considered in one dimension, see A129587).

The first is in one dimension P(2n-1,x)=(n+1+x^n)*sum_{i=0..n-1} x^i/(i+1), n>=1.

The table of coefficients of P() with 2n coefficients per row starts:

2, 1;

3, 3/2, 1, 1/2;

4, 2, 4/3, 1, 1/2, 1/3;.. .

Rows multiplied by n!, the table becomes Q():

2, 1;

6, 3, 2, 1;

24, 12, 8, 6, 3, 2;

120, 60, 40, 30, 24, 12, 8, 6;

720, 360, 240, 180, 144,...

The sequence gives the alternating row sums of this table Q, positive sign for coefficients in front of even and negative sign for coefficients in front of odd powers of x.

The row sums of Q are (n+2)*A000254(n)= 3, 12, 55, 300...

Adding the alternating and ordinary row sums yields the sequence 4, 16, 70, 384....

The sequence of sums of antidiagonals in the Q table starts 2, 6+1=7, 24+3=27, 120+12+1=134.

p^2 divides a(2p-2) for prime p>3. a(2p-2)/p^2 = A061002(n) = A001008(p-1)/p^2 for prime p>2. - Alexander Adamchuk, Jul 07 2006

Generalized harmonic numbers are defined as H(m,k) = Sum_{i=1..m} 1/i^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{i=1..m} (-1)^(i+1)*1/i^k.

a(18)..a(24) = {77,104,77,136,104,370,136}. a(26)..a(27) = {77,104}.

a(n) is currently unknown for n = {16,17,25,...}. See more details in Comments at A128672 and A125581.

For the denominators see A262022.

The reordered alternating harmonic series considered here is 1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + 1/9 + 1/11 - 1/6 + ... + ... - ...

The limit n -> infinity of the partial sums s(n) = a(n)/A262031(n) is 3*log(2)/2, approximately 1.03972077083991... For the decimal expansion see A262023.

Combining three consecutive terms of this series leads to the series b(0) + b(1) + ..., with b(k) = (1/2)*(8*k+5)/((4*k+1)*(4*k+3)*(k+1)). This produces partial sums 5/6, 13/140, 7/198, 29/1560, 37/3230, ..., which are given by s(3*n+2), n = 0, 1, .... Therefore, the limit is the same as the one given above, and it is obtained from Sum_{k=0..n} b(k) = (1/4)*Psi(n+5/4) + (1/4)*Psi(n+7/4) - (1/2)*Psi(n+2) + (3/2)*log(2), with the digamma function Psi(x).

This reordered alternating harmonic series appears as an example in the famous Dirichlet article, p. 319 (Werke I). Martin Ohm showed that for the reordering with alternating m consecutive positive terms followed by n negative terms (here n = 2 and m = 1) the sum becomes log(2) + (1/2)*log(m/n). See the reference, paragraph 8. p. 12-14. See also the Pringsheim reference.

a(n)/n! is the invert transform of [1, 1 - 1/2, 1 - 1/2 + 1/3, 1 - 1/2 + 1/3 - 1/4, 1 - 1/2 + 1/3 - 1/4 + 1/5, ...].

The length of row n is ceiling(n/2) = A008619(n-1).

The numbers for these cycles of permutations of [n], appear in the solution of the Locker Problem. See the link, p. 25.

For the probability of failures with the strategy used in the locker problem with n lockers and opening of up to floor(n/2) lockers see A058313(n)/A058312(n), for n > = 1. For n = 1 the one team member is not allowed to open the one locker (with the member's wallet) because (n/2) = 0; so certainly a failure.

For the probability of success in this locker problem for n lockers see A119248(n)/A058312(n), for n >= 1.

Also the ratio of the numerators of n*H'(n) = A119787(n) and H'(n) = A058313(n) when they are different. (H'(n) is the alternating harmonic number H'(n) = Sum_{k=1..n} (-1)^(k+1)*1/k.)

The ratio of numerators A119787(n)/A058313(n) for n = 1..400 is given in A119788(n).

It appears that most a(n) are prime divisors of the corresponding indices A121594(n).

The first and only composite a(n) up to A119788(6000) is a(31) = 49 corresponding to A119788(1470).

It appears that all a(n) belong to A092579(n), which is a sieve using the Fibonacci sequence over the integers >= 2. [Edited by Petros Hadjicostas, May 11 2020]

Note that (-1)^n/n+(-1)^(n+1)/(n+1) = (-1)^n/(n*(n+1)) for any n > 0.

Conjecture: (i) Suppose that Sum_{i=j..k} (-1)^i/i and Sum_{r=s..t} (-1)^r/r with 0 < min{2,k} <= j <= k, 0 < min{2,t} <= s <= t and j <= s have the same fractional part, but the ordered pairs (j,k) and (s,t) are different. Then Sum_{i=j..k} (-1)^i/i = Sum_{r=s..t} (-1)^r/r. Moreover, if j is odd, then j > 1, k = j*(j+1) and (s,t) = (j+2,j*(j+1)-1); if j is even, then either (k = j+1 and s = t = j*(j+1)), or (k = j*(j+1)-1 and (s,t) = (j+2,j*(j+1))).

(ii) Let a > b >= 0 and m > 0 be integers with gcd(a,b) = 1 < max{a,m}. For each r = 0,1, the numbers Sum_{i=j..k} (-1)^(i-r*j)/(a*i-b)^m with 1 <= j <= k and (j > 1 if k > a-b = 1) have pairwise distinct fractional parts.

This is an analog of the conjecture in A261878. Part (i) of the conjecture implies that a(n) = n*(n-1)/2 + 2 - floor((sqrt(4n+1)-1)/2) - floor((sqrt(4n+1)-1)/4) for all n > 1.