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A304322 O.g.f. A(x) satisfies: [x^n] exp( n^2 * x ) / A(x) = 0 for n>0. 11
1, 1, 5, 54, 935, 22417, 685592, 25431764, 1106630687, 55174867339, 3097872254493, 193283918695494, 13260815963831108, 991928912663646012, 80325879518096889760, 7000127337189146831092, 653156403671376068448047, 64963788042207845593775999, 6861040250464949653809027311, 766815367797924824316405828466, 90417908118862070187113849296815 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

It is conjectured that the coefficients of o.g.f. A(x) consist entirely of integers.

Equals row 2 of table A304320.

O.g.f. A(x) = 1/(1 - x*B(x)), where B(x) is the o.g.f. of A107668.

Logarithmic derivative of o.g.f. A(x), A'(x)/A(x), equals o.g.f. of A304312.

Conjecture: given o.g.f. A(x), the coefficient of x^n in A'(x)/A(x) is the number of connected n-state finite automata with 2 inputs (A006691).

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..400

FORMULA

a(n) ~ sqrt(1-c) * 2^(2*n - 1/2) * n^(n - 1/2) / (sqrt(Pi) * c^n * (2-c)^n * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020

EXAMPLE

O.g.f.: A(x) = 1 + x + 5*x^2 + 54*x^3 + 935*x^4 + 22417*x^5 + 685592*x^6 + 25431764*x^7 + 1106630687*x^8 + 55174867339*x^9 + 3097872254493*x^10 + ...

ILLUSTRATION OF DEFINITION.

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

n=0: [1, -1, -8, -270, -19584, -2427000, -455544000, -120136161600, ...];

n=1: [1, 0, -9, -296, -20715, -2527704, -470405285, -123376631664, ...];

n=2: [1, 3, 0, -350, -24672, -2867256, -518870528, -133753337280, ...];

n=3: [1, 8, 55, 0, -29547, -3559056, -614943333, -153534305160, ...];

n=4: [1, 15, 216, 2674, 0, -4291704, -783235520, -187656684864, ...];

n=5: [1, 24, 567, 12880, 251541, 0, -948897125, -243358236600, ...];

n=6: [1, 35, 1216, 41634, 1372320, 38884296, 0, -295870371264, ...];

n=7: [1, 48, 2295, 109000, 5106453, 230531544, 8944955227, 0, ...];

n=8: [1, 63, 3960, 248050, 15443328, 949131144, 56257429312, 2865412167360, 0, ...]; ...

in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^2*x ) / A(x) = 0 for n>=0.

LOGARITHMIC DERIVATIVE.

The logarithmic derivative of A(x) yields the o.g.f. of A304312:

A'(x)/A(x) = 1 + 9*x + 148*x^2 + 3493*x^3 + 106431*x^4 + 3950832*x^5 + 172325014*x^6 + 8617033285*x^7 + 485267003023*x^8 + 30363691715629*x^9 + ... + A304312(n)*x^n +...

INVERT TRANSFORM.

1/A(x) = 1 - x*B(x), where B(x) is the o.g.f. of A107668:

B(x) = 1 + 4*x + 45*x^2 + 816*x^3 + 20225*x^4 + 632700*x^5 + 23836540*x^6 + 1048592640*x^7 + 52696514169*x^8 + ... + A107668(n)*x^n + ...

PROG

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

for(n=0, 25, print1( a(n), ", "))

CROSSREFS

Cf. A304320, A304312, A304321, A304323, A304324, A304325.

Cf. A006691, A107668.

Sequence in context: A110430 A110432 A215542 * A132619 A251583 A223896

Adjacent sequences:  A304319 A304320 A304321 * A304323 A304324 A304325

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 11 2018

STATUS

approved

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Last modified March 5 11:57 EST 2021. Contains 341823 sequences. (Running on oeis4.)