A300591 O.g.f. A(x) satisfies [x^n] exp( n^2 * A(x) ) = n^2 * [x^(n-1)] exp( n^2 * A(x) ) for n >= 1.

1, 2, 27, 736, 30525, 1715454, 123198985, 10931897664, 1172808994833, 149774206572050, 22487782439633786, 3927856758905547936, 790620718368726490063, 181836026214536919343314, 47416473117145116482171400, 13920906749656695367066255360, 4572270908185359745686931830057, 1670388578072378805032472463218378, 675225859431899136993903503004997481, 300576566118865697499246162737030656800

Offset

1,2

Comments

Compare to: [x^n] exp( n^2 * x ) = n * [x^(n-1)] exp( n^2 * x ) for n>=1.

It is conjectured that this sequence consists entirely of integers.

a(n) is divisible by n (conjecture): A300598(n) = a(n)/n for n>=1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Formula

O.g.f. equals the logarithm of the e.g.f. of A300590.

a(n) ~ c * n!^2 * n^2, where c = 0.1354708370957778563796... - Vaclav Kotesovec, Oct 13 2020

From Seiichi Manyama, Apr 13 2026: (Start)

G.f. A(x) satisfies [x^n] exp(n^2*A(x)) * (1 - n^2*x) = 0, for n >= 1.

a(n) = n^(2*n-3) - (1/n) * Sum_{k=1..n-1} k * c_n(k) * e_n(n-k),

where c_n(k) = a(k) - n^(2*k-2)/k for 1 <= k <= n-1,

and e_n(0) = 1, e_n(k) = (n^2/k) * Sum_{j=1..k} j * c_n(j) * e_n(k-j) for 1 <= k <= n-1. (End)

Example

O.g.f.: A(x) = x + 2*x^2 + 27*x^3 + 736*x^4 + 30525*x^5 + 1715454*x^6 + 123198985*x^7 + 10931897664*x^8 + 1172808994833*x^9 + 149774206572050*x^10 + ...
where
exp(A(x)) = 1 + x + 5*x^2/2! + 175*x^3/3! + 18385*x^4/4! + 3759701*x^5/5! + 1258735981*x^6/6! + 630063839035*x^7/7! + 445962163492385*x^8/8! + ... + A300590(n)*x^n/n! + ...
such that: [x^n] exp( n^2 * A(x) ) = n^2 * [x^(n-1)] exp( n^2 * A(x) ).

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^2)); A[#A] = ((#A-1)^2*V[#A-1] - V[#A])/(#A-1)^2 ); polcoeff( log(Ser(A)), n)}
for(n=1, 30, print1(a(n), ", "))
(Ruby)
def A300591(n)
  a = [0]
  (1..n).each{|i|
    c = [0] + (1..i - 1).map{|k| a[k] - i ** (2 * k - 2) / k.to_r}
    e = [1]
    (1..i - 1).each{|k| e << i ** 2 / k.to_r * (1..k).inject(0){|s, j| s + j * c[j] * e[k - j]}}
    a << i ** (2 * i - 3) - (1..i - 1).inject(0){|s, k| s + k * c[k] * e[i - k]}.to_i / i
  }
  a[1..-1]
end
p A300591(20) # Seiichi Manyama, Apr 13 2026

Crossrefs

Cf. A300590, A300598, A300871, A296171, A300593.

Cf. A088716, A300595, A300597.

Sequence in context: A153850 A138458 A090248 * A377893 A251693 A182934

Adjacent sequences: A300588 A300589 A300590 * A300592 A300593 A300594

Keyword

nonn

Author

Paul D. Hanna, Mar 09 2018

Status

approved