A323693 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = (n+1)^2 * [x^(n-1)] A(x)^(n+1) for n >= 1 with A'(0) = 1.

1, 2, 14, 228, 6332, 255800, 13862744, 962576816, 83146713104, 8746885895136, 1102050352603232, 163997224386523712, 28480503345597714112, 5711832009893579651456, 1310680283957123653000064, 341305200596595166803458816, 100122955976950431349888239872, 32871729257928892872345863470592, 12007438407819424861612909690881536, 4854069613493626427129286480218215424

Offset

0,2

Comments

a(n) / 2^floor((n+1)/2) is odd for n >= 0 (conjecture).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

a(n) ~ c * n^2 * n!^2, where c = 1.00095034093694419071357524982261... - Vaclav Kotesovec, Feb 11 2026, corrected Mar 15 2026

From Paul D. Hanna, Mar 08 2026: (Start)

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) [x^n] A(x)^(n+1) * (1 - (n+1)^2*x) = 0 for n >= 1 (by definition).

(2) A(x) = 1 + x*A(x)^2 * (2*A(x)^2 - x*A(x)*A'(x) - x^2*A'(x)^2 + x^2*A(x)*A''(x)) / (A(x) - x*A'(x))^3.

(3) A(x) = x/Series_Reversion(G(x)) where G(x) = x + x*d/dx ( x*d/dx G(x)^2/2 ) is the g.f. of A393629, so that G(x) = x*A(G(x)).

(End)

Example

G.f.: A(x) = 1 + 2*x + 14*x^2 + 228*x^3 + 6332*x^4 + 255800*x^5 + 13862744*x^6 + 962576816*x^7 + 83146713104*x^8 + 8746885895136*x^9 + ...
The table of coefficients of x^k in A(x)^n starts as
  n=1: [1,  2,  14,  228,  6332,  255800,  13862744, ...];
  n=2: [1,  4,  32,  512, 13772,  543312,  28977968, ...];
  n=3: [1,  6,  54,  860, 22488,  866448,  45462704, ...];
  n=4: [1,  8,  80, 1280, 32664, 1229568,  63445984, ...];
  n=5: [1, 10, 110, 1780, 44500, 1637512,  83069960, ...];
  n=6: [1, 12, 144, 2368, 58212, 2095632, 104491088, ...];
  n=7: [1, 14, 182, 3052, 74032, 2609824, 127881376, ...]; ...
RELATED SEQUENCES.
In the above table, the main diagonal (A216733) begins
[1, 4, 54, 1280, 44500, 2095632, 127881376, 9819500544, ...]
which, when divided by (n+1)^2, yields the secondary diagonal (A323694):
[1, 1, 6, 80, 1780, 58212, 2609824, 153429696, 11457990000, ...].
The sequence a(n) / 2^floor((n+1)/2) appears to consist only of odd numbers:
[1, 1, 7, 57, 1583, 31975, 1732843, 60161051, 5196669569, 273340184223, 34439073518851, 2562456631039433, 445007864774964283, ...].
RELATED SERIES.
Series G(x) = x + x*d/dx (x*d/dx G(x)^2/2) is the g.f. of A393629, which begins
G(x) = x + 2*x^2 + 18*x^3 + 320*x^4 + 8900*x^5 + 349272*x^6 + 18268768*x^7 + 1227437568*x^8 + ... + A393629(n)*x^n + ...
where G(x) = x*A(G(x)).
Other properties of G(x) include
(i) G(x/A(x)) = x,
(ii) G'(x/A(x)) = A(x)^2/(A(x) - x*A'(x)),
(iii) G''(x/A(x)) = A(x)^3 * (2*A(x)*A'(x) - 2*x*A'(x)^2 + x*A(x)*A''(x)) / (A(x) - x*A'(x))^3,
(iv) G(x) = x + x*G(x)*G'(x) + x^2*G'(x)^2 + x^2*G(x)*G''(x).
If we substitute x/A(x) for x in (iv) then simplify by using (i)-(iii), we obtain the formula A(x) = 1 + x*A(x)^2 * (2*A(x)^2 - x*A(x)*A'(x) - x^2*A'(x)^2 + x^2*A(x)*A''(x)) / (A(x) - x*A'(x))^3.

Prog

(PARI) {a(n) = my(A=[1], V); for(m=2, n+1, A=concat(A, 0); V=Vec(Ser(A)^m); A[#A] = V[#A-1]*m - V[#A]/m ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Informal method of obtaining N terms: */
N=30; A=[1]; for(n=2, N, A=concat(A, 0); V=Vec(Ser(A)^n); A[#A] = V[#A-1]*n - V[#A]/n ); A
(PARI) \\ A(x) = x/Series_Reversion(G(x))
{a(n) = my(G=x+x^2, A=x); for(i=0, n, G = x + x*deriv(x*deriv(G^2/2 +x^2*O(x^n)))); A = x/serreverse(G); polcoef(GF=A, n)}
{upto(n) = a(n); Vec(GF)}
upto(25) \\ Paul D. Hanna, Mar 08 2026
(PARI) {upto(N) = my(G=x+x^2, A=x); for(i=0, N, G = x + x*deriv(x*deriv(G^2/2 +x^2*O(x^N)))); A = x/serreverse(G); Vec(A) }
upto(25) \\ Joerg Arndt, Mar 09 2026

Crossrefs

Cf. A323694, A295766, A393629, A216733.

Sequence in context: A105749 A251692 A338187 * A118086 A048163 A093548

Adjacent sequences: A323690 A323691 A323692 * A323694 A323695 A323696

Keyword

nonn

Author

Paul D. Hanna, Feb 20 2019

Status

approved