A306065 E.g.f. A(x) satisfies: Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n+1-k) - (k+1)*A(x) = 0.

1, 1, 2, 9, 60, 545, 6240, 86499, 1407840, 26328105, 556338240, 13110436845, 340916083200, 9696978168657, 299505048041472, 9982704111951375, 357144207270359040, 13651153329833408145, 555203925284795043840, 23940076477993593415857, 1090918710974007336960000, 52384418915257291697680545

Offset

0,3

Comments

a(n) = (-1)^(n+1) * A306066(n+1)/(n+1) for n >= 0.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..133 (terms 0..100 from Paul D. Hanna)

Formula

E.g.f. A(x) satisfies:

(1) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n+1-k) - (k+1)*A(x) = 0.

(2) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n+1-k - m) - (k+1)*A(x) = -m * A(x)^(2*m-2) * x^(m-2) / (A(x) - 1)^(m-2).

(3) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n+1-k) - (k+1 - p)*A(x) = p/A(x) * x^(p-2) / (A(x) - 1)^(p-2).

(4) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n+1-k - m) - (k+1 - p)*A(x) = -(m -p*A(x)) * A(x)^(2*m-2) * x^(m+p-2) / (A(x) - 1)^(m+p-2).

(5) A(x) = 1 / A(-x*A(x)).

(6) Sum_{n>=0} (-x)^n/n! * Product_{k=1..n} (n-k) - k*A(x) = A(x).

(7) Sum_{n>=0} (-x)^n/n! * Product_{k=1..n} (n-k+1) - (k-1)*A(x) = 1/A(x).

(8) Sum_{n>=0} (-x/2)^n/n! * Product_{k=1..n} (2*(n-k)+1) - (2*k-1)*A(x) = 1.

(9) Sum_{n>=0} (-x/2)^n/n! * Product_{k=1..n} (2*(n-k)+1 - m) - (2*k-1)*A(x) = A(x)^( m/(1 + A(x)) ).

(10) Sum_{n>=0} (-x/2)^n/n! * Product_{k=1..n} (2*(n-k)+1) - (2*k-1 - p)*A(x) = A(x)^( -p*A(x)/(1 + A(x)) ).

(11) Sum_{n>=0} (-x/2)^n/n! * Product_{k=1..n} (2*(n-k)+1 - m) - (2*k-1 - p)*A(x) = A(x)^( (m - p*A(x))/(1 + A(x)) ).

a(n)/n! ~ c * d^n / n^(3/2), where d = 2.4559812888215554548... and c = 0.65281176845553367... - Vaclav Kotesovec, Jul 12 2018

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 60*x^4/4! + 545*x^5/5! + 6240*x^6/6! + 86499*x^7/7! + 1407840*x^8/8! + 26328105*x^9/9! + 556338240*x^10/10! + ...
such that
0  =  (1 - A(x)) - (2 - A(x))*(1 - 2*A(x))*x/1! + (3 - A(x))*(2 - 2*A(x))*(1 - 3*A(x))*x^2/2! - (4 - A(x))*(3 - 2*A(x))*(2 - 3*A(x))*(1 - 4*A(x))*x^3/3! + (5 - A(x))*(4 - 2*A(x))*(3 - 3*A(x))*(2 - 4*A(x))*(1 - 5*A(x))*x^4/4! + ...
Also,
1  =  1 - (1 - A(x))*x/(1!*2) + (3 - A(x))*(1 - 3*A(x))*x^2/(2!*2^2) - (5 - A(x))*(3 - 3*A(x))*(1 - 5*A(x))*x^3/(3!*2^3) + (7 - A(x))*(5 - 3*A(x))*(3 - 5*A(x))*(1 - 7*A(x))*x^4/(4!*2^4) - (9 - A(x))*(7 - 3*A(x))*(5 - 5*A(x))*(3 - 7*A(x))*(1 - 9*A(x))*x^5/(5!*2^5) + ...
More generally, the e.g.f. A(x) satisfies the following sums.
Define
S1(m,p) = Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n+1-k - m) - (k+1 - p)*A(x),
then
S1(m,p) = -(m - p*A(x)) * A(x)^(2*m-2) * x^(m+p-2) / (A(x) - 1)^(m+p-2).
Define
S2(m,p) = Sum_{n>=0} (-x/2)^n/n! * Product_{k=1..n} (2*(n-k)+1 - m) - (2*k-1 - p)*A(x),
then
S2(m,p) = A(x)^( (m - p*A(x))/(1 + A(x)) ).
RELATED SERIES.
The e.g.f. also satisfies A(x) = 1/A(-x*A(x)), where:
A(-x*A(x)) = 1/A(x) = 1 - x - 3*x^3/3! - 12*x^4/4! - 125*x^5/5! - 1320*x^6/6! - 18249*x^7/7! - 290976*x^8/8! - 5385393*x^9/9! - 112642560*x^10/10! + ...
Also,
(A(x) - 1)/x  =  1 + x + 3*x^2/2! + 15*x^3/3! + 109*x^4/4! + 1040*x^5/5! + 12357*x^6/6! + 175980*x^7/7! + 2925345*x^8/8! + 55633824*x^9/9! + 1191857895*x^10/10! + 28409673600*x^11/11! + 745921397589*x^12/12! + ...
appears commonly in formulas for e.g.f. A(x).

Prog

(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, (-x)^m/m!*prod(k=0, m, (m+1-k) - (k+1)*Ser(A) ) ), #A-1)); n!*A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A306066, A306090, A306067.

Sequence in context: A161391 A120014 A036774 * A166882 A268937 A269634

Adjacent sequences: A306062 A306063 A306064 * A306066 A306067 A306068

Keyword

nonn

Author

Paul D. Hanna, Jun 30 2018

Status

approved