A395022 Expansion of 1 + Sum_{k >= 1} (x^(2k-1) + (k + 1) * x^(2k)) / Product_{j=1..k} (1 - j^2 * x^2).

1, 1, 2, 2, 5, 7, 21, 37, 126, 264, 1001, 2433, 10129, 27913, 126006, 386906, 1877593, 6346119, 32888637, 121159373, 666757418, 2655174768, 15442626557, 66028903633, 404271491249, 1845579100993, 11855277004330, 57506847262162, 386411610169757, 1983312152411351, 13904155500245141

Offset

0,3

Comments

For odd n = 2*k-1: a(n) is equal to the number of partitions of the set {1,1',2,2',...,k,k'} into disjoint nonempty subsets V1,...,Vk (1 <= k <= n) such that, for each 1 <= j <= k, if i is the least integer such that either i or i' belongs to Vj then {i,i'} is a subset of Vj (this matches A135920).

For even n = 2*k: a(n) is equal to the sum of the number of such partitions (A135920), and the total number of blocks (nonempty subsets V_j) across all such partitions (A394822).

Also generated by iterated binomial transforms in the following way:

a = BINOMIAL([1, 1, 2, 2, 5, 7, 21, 37, 126, ...]);

a = BINOMIAL^3([1, -1, 2, -6, 21, -75, 269, ...]);

a = BINOMIAL^5([1, -3, 10, -38, 165, -797, 4125, ...]);

a = BINOMIAL^7([1, -5, 26, -142, 821, -5039, 32709, ...]);

a = BINOMIAL^9([1, -7, 50, -366, 2757, -21441, 172421, ...]).

Links

Table of n, a(n) for n=0..30.

Andrii Husiev, Extended Central Factorial Numbers and the Flickering Operator, arXiv:2605.06689 [math.GM], 2026. See pp. 10-11.

Formula

If n is odd (n = 2*k-1): a(n) = A135920(k).

If n is even (n = 2*k): a(n) = A135920(k) + A394822(k).

a(n) = A135920((n+1)/2) if n is odd; a(n) = A135920(n/2) + A394822(n/2) if n is even.

a(n) = (2*k)! * [x^(2*k)] (cosh(2*s) + (1 - (n mod 2)) * s * sinh(2*s)) where k = floor((n+1)/2) and s = sinh(x/2).

O.g.f.: A(x) = (x+2*x^2)/(1-x^2) + (x^3+3*x^4)/((1-x^2)*(1-4*x^2)) + (x^5+4*x^6)/((1-x^2)*(1-4*x^2)*(1-9*x^2)) + (x^7+5*x^8)/((1-x^2)*(1-4*x^2)*(1-9*x^2)*(1-16*x^2)) + ...

a(n) = (1/2) * Sum_{p>=0} Sum_{q>=0} (1/(p! * q!)) * ( ((-1)^q - (-1)^p) * ((p - q + 1)/2)^n + ((-1)^q + (-1)^p) * ((p - q)/2)^n ).

E.g.f.: cosh(2*sinh(x/2)) + exp(x/2)*sinh(2*sinh(x/2)).

Prog

(Python)
from functools import cache
@cache
def T(n, k):
    if k == 1 or k == n: return 1
    if k < 1 or k > n: return 0
    if k % 2 == 0:
        if n % 2 != 0:
            return 0
        else:
            return (2 * T(n, k-1)) // k
    else:
        if n % 2 == 0:
            return T(n-1, k) * (k + 1) // 2
        else:
            term1 = T(n-1, k) * (k + 1) // 2
            term2 = T(n-1, k-2) * 2 // (k - 1)
            return term1 + term2
def a(n):
    if n == 0: return 1
    return sum(T(n, k) for k in range(1, n + 1))
print([a(n) for n in range(0, 31)])

Crossrefs

Cf. A394813, A394582, A008957.

Row sums of A395021.

Sequence in context: A356434 A300439 A208818 * A221835 A074476 A224610

Adjacent sequences: A395019 A395020 A395021 * A395023 A395024 A395025

Keyword

nonn

Author

Husiev Andrii Alekseevich, Apr 10 2026

Status

approved