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A260878 Number of set partitions of {1, 2, ..., 2*n} with sizes in {[n, n], [2n]}.

%I A260878 #62 May 02 2024 11:49:07
%S A260878 2,2,4,11,36,127,463,1717,6436,24311,92379,352717,1352079,5200301,
%T A260878 20058301,77558761,300540196,1166803111,4537567651,17672631901,
%U A260878 68923264411,269128937221,1052049481861,4116715363801,16123801841551,63205303218877,247959266474053
%N A260878 Number of set partitions of {1, 2, ..., 2*n} with sizes in {[n, n], [2n]}.
%C A260878 Third column in A260876.
%H A260878 Alois P. Heinz, <a href="/A260878/b260878.txt">Table of n, a(n) for n = 0..1000</a>
%H A260878 Hans Salié, <a href="https://doi.org/10.1002/mana.19550140405">Über die Koeffizienten der Blasiusschen Reihe</a>, Math. Nachr. 1955, (14, 4-6), 241--248.
%F A260878 G.f.: (4*x^2 - 13*x + 3 + sqrt((1 - 4*x)*(x - 1)^2))/(2*(4*x - 1)*(x - 1)). - _Alois P. Heinz_, Aug 06 2015
%F A260878 a(n) = Binomial(2*n-1, n) + 1. - _Vladimir Kruchinin_, Feb 26 2017
%F A260878 The generating function G(x) satisfies the differential equation x^3 + 2*x = (4*x^4 - 9*x^3 + 6*x^2 - x)*diff(G(x), x) + (2*x^3 - 4*x^2 + 2*x)*G(x). - _Peter Luschny_, Feb 12 2019
%F A260878 From _Peter Luschny_, Aug 02 2019: (Start)
%F A260878 a(n) = ((4*n - 2)*a(n-1) - 3*n + 2)/n for n >= 2.
%F A260878 a(n) = (2*n)! * [x^(2*n)] exp(exp(x)*(1 - (Gamma(n,x)/Gamma(n)))) for n >= 2.
%F A260878 a(n) ~ 4^n/sqrt(4*Pi*n). More precise asymptotic estimates are:
%F A260878 1 + (4^n/sqrt(n*Pi)) * (1/2 - 1/(16*n) * (1 - 1/(16*n))), and
%F A260878 1 + 4^n*(2 - 2/N^2 + 21/N^4 - 671/N^6) / sqrt(2*N*Pi) with N = 8*n + 2.
%F A260878 Let b(n) = binomial(2*(n-1), n-1) + 1 = A323230(n) for n >= 0. Then by Salié:
%F A260878 p divides a(p+k) - b(k+1) if p is a prime > k and 0 <= k <= 4.
%F A260878 Conjecture: p divides a(p+5) - b(6) if p is a prime > b(6).
%F A260878 If p is a prime divisor of n then a(n) == a(n/p) (mod p) (by Salié, theorem 2).
%F A260878 (End)
%F A260878 From _Peter Bala_, Apr 20 2024: (Start)
%F A260878 a(n) = Sum_{k = 0..n} (-1)^k * 3*n/(2*n + k) * binomial(2*n+k, n-k) for n >= 1.
%F A260878 a(n) = Sum_{k = 0..n} (-1)^k * 3*n/(n + 2*k) * binomial(2*n+k-1, n-k) for n >= 1.
%F A260878 (-1)^n * a(n) equals the n-th order Taylor polynomial (centered at 0) of 1/c(x)^(3*n) evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. (End)
%e A260878 The set partitions counted by a(3) = 11 are: {{1, 2, 3, 4, 5, 6}},
%e A260878 {{1, 2, 4}, {3, 5, 6}}, {{1, 2, 3}, {4, 5, 6}}, {{1, 3, 4}, {2, 5, 6}},
%e A260878 {{1, 3, 5}, {2, 4, 6}}, {{1, 4, 5}, {2, 3, 6}}, {{1, 5, 6}, {2, 3, 4}},
%e A260878 {{1, 4, 6}, {2, 3, 5}}, {{1, 3, 6}, {2, 4, 5}}, {{1, 2, 6}, {3, 4, 5}},
%e A260878 {{1, 2, 5}, {3, 4, 6}}.
%p A260878 a := proc(n) option remember;
%p A260878 if n < 2 then [2, 2][n+1] else ((4*n - 2)*a(n-1) - 3*n + 2)/n fi end:
%p A260878 seq(a(n), n=0..26); # Or:
%p A260878 egf := n -> exp(exp(x)*(1 - (GAMMA(n,x)/GAMMA(n)))):
%p A260878 a := n -> `if`(n<2, 2, (2*n)!*coeff(series(egf(n), x, 2*n+1), x, 2*n)):
%p A260878 seq(a(n), n=0..26); # _Peter Luschny_, Aug 02 2019
%t A260878 Table[Binomial[2 n - 1, n] + 1, {n, 0, 26}] (* or *)
%t A260878 CoefficientList[Series[(4 x^2 - 13 x + 3 + Sqrt[(1 - 4 x) (x - 1)^2])/(2 (4 x - 1) (x - 1)), {x, 0, 26}], x] (* _Michael De Vlieger_, Feb 26 2017 *)
%o A260878 (Sage)
%o A260878 print([A260876(n,2) for n in (0..30)])
%o A260878 (Sage) # Alternative:
%o A260878 def A260878():
%o A260878     a, f, s, n = 2, 2, 1, 1
%o A260878     yield a
%o A260878     while True:
%o A260878         yield a
%o A260878         f += 4; s += 3; n += 1
%o A260878         a = (f*a - s)/n
%o A260878 a = A260878()
%o A260878 print([next(a) for n in range(27)]) # _Peter Luschny_, Aug 02 2019
%Y A260878 a(n) = A112849(n) for n >= 2. - _Alois P. Heinz_, Aug 06 2015
%Y A260878 a(n) = A052473(n+2) - 1.
%Y A260878 a(n) = A088218(n) + 1.
%Y A260878 a(n) = (-1)^n*A110556(n) + 1.
%Y A260878 a(n+1) - a(n) = A097613(n+1) for n > 0.
%Y A260878 Cf. A260876, A001700.
%Y A260878 Cf. A323230 (d=0), this sequence (d=1), A323229 (d=2).
%K A260878 nonn,easy
%O A260878 0,1
%A A260878 _Peter Luschny_, Aug 02 2015