A300617 O.g.f. A(x) satisfies [x^n] exp( n * A(x) ) = n^2 * [x^(n-1)] exp( n * A(x) ) for n >= 1.

1, 3, 30, 550, 15375, 601398, 31299268, 2093655600, 175312873125, 17987972309725, 2221603804365924, 325310016974127276, 55749742122979646105, 11056914755618659399500, 2513208049272148754203200, 649086459674801585681092992, 189044817293654530855544266209, 61671809408989968268084102641075, 22399957973327602630210233608217250, 9009223131975798265447660437783058050

Offset

1,2

Comments

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

It is conjectured that this sequence consists entirely of integers.

a(n) is divisible by n*(n+1)/2 (conjecture); A300589(n) = a(n) / (n*(n+1)/2).

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

O.g.f. equals the logarithm of the e.g.f. of A300616.

O.g.f. A(x) satisfies: [x^n] exp(n*A(x)) * (1 - n^2*x) = 0, for n > 0. - Paul D. Hanna, Oct 15 2018 [Corrected by Seiichi Manyama, Apr 12 2026]

a(n) ~ c * (n!)^2, where c = 1.685041722777551007711429045295022018562828... - Vaclav Kotesovec, Mar 10 2018

From Seiichi Manyama, Apr 12 2026: (Start)

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

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

and e_n(0) = 1, e_n(k) = (n/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 + 3*x^2 + 30*x^3 + 550*x^4 + 15375*x^5 + 601398*x^6 + 31299268*x^7 + 2093655600*x^8 + 175312873125*x^9 + 17987972309725*x^10 + ...
where
exp(A(x)) = 1 + x + 7*x^2/2! + 199*x^3/3! + 14065*x^4/4! + 1924201*x^5/5! + 445859911*x^6/6! + 161145717727*x^7/7! + 85790577700129*x^8/8! + ... + A300616(n)*x^n/n! + ...
such that: [x^n] exp( n * A(x) ) = n^2 * [x^(n-1)] exp( n * A(x) ).
RELATED SEQUENCES.
The sequence A300589(n) = a(n) / (n*(n+1)/2) begins:
[1, 1, 5, 55, 1025, 28638, 1117831, 58157100, 3895841625, 327054041995, ...].
The table of coefficients in x^k/k! in exp(n*A(x)) * (1 - n^2*x) begins:
  n=1: [1,   0,    5,    178,   13269,   1853876,   434314705, 158024698350, ...];
  n=2: [1,  -2,    0,    248,   22976,   3416592,   822150016, 303575549440, ...];
  n=3: [1,  -6,  -27,      0,   21861,   4129758,  1079984097, 415322613324, ...];
  n=4: [1, -12,  -88,   -848,       0,   3286304,  1109402752, 469332346368, ...];
  n=5: [1, -20, -195,  -2650,  -55675,         0,   794678425, 438768342850, ...];
  n=6: [1, -30, -360,  -5832, -161856,  -6828624,           0, 293555007360, ...];
  n=7: [1, -42, -595, -10892, -339339, -18549958, -1433676839,            0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(#A-1)); A[#A] = ((#A-1)^2*V[#A-1] - V[#A])/(#A-1) ); polcoeff( log(Ser(A)), n)}
for(n=1, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = -Vec( exp(m^1*x*Ser(A))*(1-m^2*x +x^2*O(x^m))^(1))[m+1]/m ); A[n]}
for(n=1, 20, print1(a(n), ", "))
(Ruby)
def A300617(n)
  a = [0]
  (1..n).each{|i|
    c = [0] + (1..i - 1).map{|k| a[k] - i ** (2 * k - 1) / k.to_r}
    e = [1]
    (1..i - 1).each{|k| e << i / k.to_r * (1..k).inject(0){|s, j| s + j * c[j] * e[k - j]}}
    a << i ** (2 * i - 2) - (1..i - 1).inject(0){|s, k| s + k * c[k] * e[i - k]}.to_i / i
  }
  a[1..-1]
end
p A300617(20) # Seiichi Manyama, Apr 12 2026

Crossrefs

Cf. A300616, A300589, A296171, A300591, A300593, A300595, A300597.

Sequence in context: A375871 A080527 A370057 * A379685 A365653 A301663

Adjacent sequences: A300614 A300615 A300616 * A300618 A300619 A300620

Keyword

nonn

Author

Paul D. Hanna, Mar 10 2018

Status

approved