Search: a281964 -id:a281964
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A282132
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Imaginary part of n!*Sum_{k=1..n} i^(k-1)/k, where i is sqrt(-1).
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+10
3
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0, 1, 3, 6, 30, 300, 2100, 11760, 105840, 1421280, 15634080, 147692160, 1919998080, 33106993920, 496604908800, 6638004172800, 112846070937600, 2386916704972800, 45351417394483200, 785383247480832000, 16493048197097472000, 413938002507853824000
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) ~ log(sqrt(2)) * n!.
a(1) = 0, a(n+1) = a(n)*(n+1) + n!*sin(Pi*n/2).
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EXAMPLE
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For n=5, a(5) = 30, which is the imaginary part of 5!*(1/1 + i/2 - 1/3 - i/4 + 1/5) = 104+30*i.
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PROG
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(PARI) a(n) = imag(n!*sum(k=1, n, I^(k-1)/k));
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CROSSREFS
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The corresponding real part is A281964.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A282564
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Real part of A000178(n) * Sum_{k=0..n} i^k/k!, where i = sqrt(-1).
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+10
2
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1, 1, 1, 6, 156, 18720, 13443840, 67756953600, 2732085780480000, 991419288020582400000, 3597660477435617162035200000, 143607093745702043133526671360000000, 68788027941331539080620236035063808000000000, 428344480781652673551035086691251861743206400000000000
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OFFSET
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0,4
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LINKS
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FORMULA
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a(0) = 1, a(n) = n!*a(n-1) + A000178(n-1)*cos(Pi/2*n).
Lim_{n->infinity} a(n)/G(n+2) = cos(1), where G(z) is the Barnes G-function.
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EXAMPLE
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For n = 4, a(4) = 156, which is the real part of A000178(4)*(1/0! + i/1! - 1/2! - i/3! + 1/4!) = 156+240*i.
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PROG
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(PARI) a(n) = real(prod(k=0, n, k!) * sum(k=0, n, I^k/k!));
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CROSSREFS
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The corresponding imaginary part is A282567.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A296726
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Expansion of e.g.f. arcsin(x)/(1 - x).
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+10
2
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0, 1, 2, 7, 28, 149, 894, 6483, 51864, 477801, 4778010, 53451135, 641413620, 8446433085, 118250063190, 1792012416075, 28672198657200, 491536207523025, 8847651735414450, 169292834944205175, 3385856698884103500, 71531660838216529125, 1573696538440763640750
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OFFSET
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0,3
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LINKS
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FORMULA
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E.g.f.: -i*log(i*x + sqrt(1 - x^2))/(1 - x), where i is the imaginary unit.
a(2*n+1) = (2*n+1)*(2*n)*a(2*n-1) + ((2*n-1)!!)^2, using the double factorial notation from A001147. - Greg Dresden, Apr 11 2024
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EXAMPLE
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arcsin(x)/(1 - x) = x/1! + 2*x^2/2! + 7*x^3/3! + 28*x^4/4! + 149*x^5/5! + ...
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MAPLE
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a:=series(arcsin(x)/(1 - x), x=0, 23): seq(n!*coeff(a, x, n), n=0..22); # Paolo P. Lava, Mar 27 2019
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MATHEMATICA
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nmax = 22; CoefficientList[Series[ArcSin[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[-I Log[I x + Sqrt[1 - x^2]]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
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PROG
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(PARI) first(n) = x='x+O('x^n); Vec(serlaplace(asin(x)/(1 - x)), -n) \\ Iain Fox, Dec 19 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A296727
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Expansion of e.g.f. arcsinh(x)/(1 - x).
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+10
2
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0, 1, 2, 5, 20, 109, 654, 4353, 34824, 324441, 3244410, 34795485, 417545820, 5536151685, 77506123590, 1144330385625, 18309286170000, 315366695240625, 5676600514331250, 106667957800963125, 2133359156019262500, 45229212438054868125, 995042673637207098750, 22696937952367956440625
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graph;
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history;
text;
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OFFSET
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0,3
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LINKS
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FORMULA
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E.g.f.: log(x + sqrt(1 + x^2))/(1 - x).
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EXAMPLE
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arcsinh(x)/(1 - x) = x/1! + 2*x^2/2! + 5*x^3/3! + 20*x^4/4! + 109*x^5/5! + ...
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MAPLE
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a:=series(arcsinh(x)/(1 - x), x=0, 24): seq(n!*coeff(a, x, n), n=0..23); # Paolo P. Lava, Mar 27 2019
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MATHEMATICA
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nmax = 23; CoefficientList[Series[ArcSinh[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[x + Sqrt[1 + x^2]]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
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PROG
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(PARI) first(n) = x='x+O('x^n); Vec(serlaplace(asinh(x)/(1 - x)), -n) \\ Iain Fox, Dec 19 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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