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A279840
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E.g.f.: exp( Integral exp(x^2) dx ).
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2
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1, 1, 1, 3, 9, 33, 153, 723, 4209, 25377, 172689, 1269699, 9918009, 84824577, 755458281, 7273792467, 73106578017, 778521070017, 8706817538721, 101639490754563, 1247219636693481, 15865740131343201, 211222989431067321, 2910911923076727123, 41712768080815125969, 618850476497056820193, 9493258647299740012593, 150683229897137204994243, 2464182867193114878735129, 41617827328955209795843137, 722857076727380399275752969
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OFFSET
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0,4
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COMMENTS
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Conjectures:
(1) a(n) is divisible by 3^floor((n+6)/9), for n>=0.
(2) a(9*n+2) is the final term with exactly n factors of 3, for n>=0.
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LINKS
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Paul D. Hanna, Table of n, a(n) for n = 0..1000
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FORMULA
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E.g.f.: exp( -sqrt(Pi)/2 * i * erf(i*x) ) (corrected by Vaclav Kotesovec, Sep 03 2017).
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EXAMPLE
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E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 33*x^5/5! + 153*x^6/6! + 723*x^7/7! + 4209*x^8/8! + 25377*x^9/9! + 172689*x^10/10! + 1269699*x^11/11! + 9918009*x^12/12! +...
where A(x) = exp( Integral exp(x^2) dx ).
Related series.
log(A(x)) = x + x^3/(3*1!) + x^5/(5*2!) + x^7/(7*3!) + x^9/(9*4!) + x^11/(11*5!) + x^13/(13*6!) + x^15/(15*7!) + x^17/(17*8!) +...
which equals Integral exp(x^2) dx.
Cosh( Integral exp(x^2) dx ) = 1 + x^2/2! + 9*x^4/4! + 153*x^6/6! + 4209*x^8/8! + 172689*x^10/10! +...
Cosh( Integral exp(x^2) dx )^2 = 1 + 2*x^2/2! + 24*x^4/4! + 576*x^6/6! + 22656*x^8/8! + 1302528*x^10/10! + 101763072*x^12/12! + 10295230464*x^14/14! + 1303603347456*x^16/16! +...+ A280794(n)*x^(2*n)/(2*n)! +...
Initial series log(A(x))^n/n! begin:
log(A(x)) = x + 2*x^3/3! + 12*x^5/5! + 120*x^7/7! + 1680*x^9/9! + 30240*x^11/11! + 665280*x^13/13! +...+ A001813(n-1)*x^(2*n-1)/(2*n-1)! +...
log(A(x))^2/2! = x^2/2! + 8*x^4/4! + 112*x^6/6! + 2304*x^8/8! + 63744*x^10/10! + 2242560*x^12/12! +...
log(A(x))^3/3! = x^3/3! + 20*x^5/5! + 532*x^7/7! + 18656*x^9/9! + 830544*x^11/11! +...
log(A(x))^4/4! = x^4/4! + 40*x^6/6! + 1792*x^8/8! + 97280*x^10/10! + 6375424*x^12/12! +...
log(A(x))^5/5! = x^5/5! + 70*x^7/7! + 4872*x^9/9! + 384560*x^11/11! +...
log(A(x))^6/6! = x^6/6! + 112*x^8/8! + 11424*x^10/10! + 1253120*x^12/12! +...
...
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MATHEMATICA
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CoefficientList[Series[E^(-Sqrt[Pi]/2*I*Erf[I*x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 03 2017 *)
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PROG
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(PARI) {a(n) = n!*polcoeff( exp( intformal( exp(x^2 +x*O(x^n) ) ) ), n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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Cf. A280794, A001813.
Sequence in context: A012584 A101899 A180632 * A009220 A294035 A007489
Adjacent sequences: A279837 A279838 A279839 * A279841 A279842 A279843
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Jan 15 2017
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STATUS
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approved
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