A395023 Expansion of the o.g.f. Sum_{k >= 0} ((k + 1) * x^k) / Product_{j=1..k} (1 - j^2 * x).

1, 2, 5, 21, 126, 1001, 10129, 126006, 1877593, 32888637, 666757418, 15442626557, 404271491249, 11855277004330, 386411610169757, 13904155500245141, 549062380938559062, 23670411643929747681, 1108835228310948875297, 56206315123997801696158, 3071280303942573508668753, 180294553591510412499169117

Offset

0,2

Comments

a(n) is equal to the sum of the number of partitions of the set {1,1',2,2',...,n,n'} 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) and the total number of blocks (nonempty subsets V_j) across all such partitions (This matches A394822).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..335

Formula

a(n) are the row sums of A395021 for even-indexed rows.

a(n) = A135920(n) + A394822(n) for n>=1.

a(n) = A395022(2*n).

a(n) = (2*n)! * [x^(2*n)] (cosh(2*sinh(x/2)) + sinh(x/2) * sinh(2*sinh(x/2))).

O.g.f.: 1 + 2*x/(1-x) + 3*x^2/((1-x)*(1-4*x)) + 4*x^3/((1-x)*(1-4*x)*(1-9*x)) + ...

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:
        return 0 if n % 2 != 0 else (2 * T(n, k-1)) // k
    else:
        term1 = T(n-1, k) * (k + 1) // 2
        if n % 2 == 0:
            return term1
        else:
            term2 = T(n-1, k-2) * 2 // (k - 1)
            return term1 + term2
def a(n):
    if n == 0: return 1
    return sum(T(2 * n, k) for k in range(1, 2 * n + 1))
print([a(n) for n in range(0, 22)])

Crossrefs

Bisection of A395022.

Cf. A394813, A394582, A008957, A395021, A135920, A394822.

Sequence in context: A243272 A139153 A180998 * A383938 A144271 A193189

Adjacent sequences: A395020 A395021 A395022 * A395024 A395025 A395027

Keyword

nonn

Author

Husiev Andrii Alekseevich, Apr 10 2026

Status

approved