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A326603
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E.g.f.: Sum_{n>=0} (x^(2*n-1) + 1)^n * x^n * exp(-x^(2*n+1)) / n!, an even function.
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2
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1, 4, -22, 2, 60482, -4838398, 119750402, 7264857602, -348713164798, 4268249137152002, -2020660279013375998, 327387513566969856002, -17234677825923317759998, 20004536762232422400002, 12702000644400049997414400002, -4420880996869850977271807999998, 55184528880536814259678111334400002, -98410947805609541237532965260492799998, 61998887798316869577999908025139200000002
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!, where
(1) A(x) = Sum_{n>=0} (x^(2*n-1) + 1)^n * x^n * exp(-x^(2*n+1)) / n!,
(2) A(x) = Sum_{n>=0} (x^(2*n-1) - 1)^n * x^n * exp(+x^(2*n+1)) / n!.
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EXAMPLE
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E.g.f.: A(x) = 1 + 4*x^2/2! - 22*x^4/4! + 2*x^6/6! + 60482*x^8/8! - 4838398*x^10/10! + 119750402*x^12/12! + 7264857602*x^14/14! - 348713164798*x^16/16! + 4268249137152002*x^18/18! + ...
such that
A(x) = exp(-x) + (1 + x)*x*exp(-x^3) + (1 + x^3)^2*x^2*exp(-x^5)/2! + (1 + x^5)^3*x^3*exp(-x^7)/3! + (1 + x^7)^4*x^4*exp(-x^9)/4! + (1 + x^9)^5*x^5*exp(-x^11)/5! + (1 + x^11)^6*x^6*exp(-x^13)/6! + ...
also
A(x) = exp(+x) - (1 - x)*x*exp(+x^3) + (1 - x^3)^2*x^2*exp(+x^5)/2! - (1 - x^5)^3*x^3*exp(+x^7)/3! + (1 - x^7)^4*x^4*exp(+x^9)/4! - (1 - x^9)^5*x^5*exp(+x^11)/5! + (1 - x^11)^6*x^6*exp(+x^13)/6! + ...
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PROG
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(PARI) {a(n) = my(A = sum(m=0, 2*n+1, (x^(2*m-1) + 1 +O(x^(2*n+2)) )^m * x^m * exp(-x^(2*m+1) +O(x^(2*n+2))) / m! ));
(2*n)!*polcoeff(A, 2*n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A = sum(m=0, 2*n+1, (x^(2*m-1) - 1 +O(x^(2*n+2)) )^m * x^m * exp(+x^(2*m+1) +O(x^(2*n+2))) / m! ));
(2*n)!*polcoeff(A, 2*n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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