A395022 - OEIS

%I #33 Jun 18 2026 06:35:55

%S 1,1,2,2,5,7,21,37,126,264,1001,2433,10129,27913,126006,386906,

%T 1877593,6346119,32888637,121159373,666757418,2655174768,15442626557,

%U 66028903633,404271491249,1845579100993,11855277004330,57506847262162,386411610169757,1983312152411351,13904155500245141

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

%C 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).

%C 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).

%C Also generated by iterated binomial transforms in the following way:

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

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

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

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

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

%H Andrii Husiev, <a href="https://arxiv.org/abs/2605.06689">Extended Central Factorial Numbers and the Flickering Operator</a>, arXiv:2605.06689 [math.GM], 2026. See pp. 10-11.

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

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

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

%F 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).

%F 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)) + ...

%F 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 ).

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

%o (Python)

%o from functools import cache

%o @cache

%o def T(n, k):

%o     if k == 1 or k == n: return 1

%o     if k < 1 or k > n: return 0

%o     if k % 2 == 0:

%o         if n % 2 != 0:

%o             return 0

%o         else:

%o             return (2 * T(n, k-1)) // k

%o     else:

%o         if n % 2 == 0:

%o             return T(n-1, k) * (k + 1) // 2

%o         else:

%o             term1 = T(n-1, k) * (k + 1) // 2

%o             term2 = T(n-1, k-2) * 2 // (k - 1)

%o             return term1 + term2

%o def a(n):

%o     if n == 0: return 1

%o     return sum(T(n, k) for k in range(1, n + 1))

%o print([a(n) for n in range(0, 31)])

%Y Cf. A394813, A394582, A008957.

%Y Row sums of A395021.

%K nonn

%O 0,3

%A _Husiev Andrii Alekseevich_, Apr 10 2026