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A304858 O.g.f. A(x) satisfies: [x^n] exp( x*A(x) ) * ((n+1)^2 - A(x)) = 0 for n >= 0. 1
1, 3, 25, 387, 9474, 335586, 16227069, 1027133019, 82431484482, 8178962850738, 983083429763946, 140755187306751102, 23669632316780047476, 4619101994508500503524, 1035316517544442028328093, 264139038734628383862962067, 76103931635294576516782940034, 24590219611525203919532065976898, 8855119878, 497758892543591063493054, 3534094294690990688065264913160020874, 1555343504708375398483400548093409427804 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:

(1) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0,

(2) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0.

It is remarkable that this sequence should consist entirely of integers.

LINKS

Table of n, a(n) for n=0..21.

FORMULA

E(n) = (1/(n*(n+2))) * Sum_{k=1..n} n!/(n-k)! * a(k) * E(n-k) for n > 0 with E(0) = 1, where exp(x*A(x)) = Sum_{n>=0} E(n) * x^n/n!.

EXAMPLE

O.g.f.: A(x) = 1 + 3*x + 25*x^2 + 387*x^3 + 9474*x^4 + 335586*x^5 + 16227069*x^6 + 1027133019*x^7 + 82431484482*x^8 + 8178962850738*x^9 + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k/k! in exp( x*A(x) ) * ((n+1)^2 - A(x)) begins:

n=0: [0, -3, -56, -2535, -240792, -41804735, -11985858768,  ...];

n=1: [3, 0, -35, -2028, -210693, -38221472, -11236742595, ...];

n=2: [8, 5, 0, -1183, -160528, -32249367, -9988215640, ...];

n=3: [15, 12, 49, 0, -90297, -23888420, -8240277903, ...];

n=4: [24, 21, 112, 1521, 0, -13138631, -5992929384, ...];

n=5: [35, 32, 189, 3380, 110363, 0, -3246170083, ...];

n=6: [48, 45, 280, 5577, 240792, 15527473, 0, ...];

n=7: [63, 60, 385, 8112, 391287, 33443788, 3745580865, 0, ...]; ...

in which the main diagonal is all zeros, illustrating the exponential property that [x^n] exp( x*A(x) ) * ((n+1)^2 - A(x)) = 0 for n >= 0.

RELATED SERIES.

exp(x*A(x)) = 1 + x + 7*x^2/2! + 169*x^3/3! + 10033*x^4/4! + 1194421*x^5/5! + 249705391*x^6/6! + 83700203917*x^7/7! + 42129949171969*x^8/8! + ... + E(n)*x^n/n! + ... where E(n) = (1/(n*(n+2))) * Sum_{k=1..n} n!/(n-k)! * a(k) * E(n-k) for n > 0.

PROG

(PARI) {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( x*(Ser(A)) ) * (m^2 - Ser(A)) )[m] ); A[n+1]}

for(n=0, 20, print1(a(n), ", "))

/* Routine to generate E(n) where exp(x*A(x)) = Sum_{n>=0} E(n) * x^n/n! */

{E(n) = if(n==0, 1, (1/(n*(n+2))) * sum(k=1, n, n!/(n-k)! * a(k) * E(n-k) ))}

for(n=0, 20, print1(E(n), ", "))

CROSSREFS

Cf. A304859, A305110, A305112, A305113, A305114, A305115, A305116.

Sequence in context: A241703 A251569 A319122 * A192552 A143925 A245309

Adjacent sequences:  A304855 A304856 A304857 * A304859 A304860 A304861

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 26 2018

STATUS

approved

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Last modified January 24 18:53 EST 2022. Contains 350565 sequences. (Running on oeis4.)