A074707
exp(Sum_{n>0} x^n/n^2) = Sum_{n>=0} a(n)*x^n/n!^2.
Original entry on oeis.org
1, 1, 3, 19, 214, 3846, 102206, 3795758, 188648664, 12125977944, 980768153832, 97599402813672, 11728820086206576, 1675575398007779568, 280777781273956353648, 54556438705747006028016, 12169284368165386948212096, 3088938134820388829982311808
Offset: 0
- R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, second edition, Addison Wesley, pp. 464-466.
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a(n)=if(n<2,1,(n-1)!^2+n!^2/n*sum(k=1,n-1,a(k)/k!^2/(n-k)))
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{a(n)=n!^2*polcoeff(exp(sum(m=1, n, x^m/m^2)+x*O(x^n)), n)}
A336260
a(0) = 1; a(n) = (n!)^4 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^4.
Original entry on oeis.org
1, 1, 17, 1474, 404768, 271581776, 377987513392, 974814164752800, 4289222350867156992, 30232332223815625555968, 324796212685273837095714816, 5108947647642107040382284423168, 113818571142935411070742114448769024, 3492592855002964381945529723625305210880
Offset: 0
-
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-i)/i^4, i=1..n))
end:
a:= n-> n!^4*b(n):
seq(a(n), n=0..14); # Alois P. Heinz, Jan 04 2024
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a[0] = 1; a[n_] := a[n] = (n!)^4 Sum[a[k]/(k! (n - k))^4, {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
nmax = 13; CoefficientList[Series[1/(1 - PolyLog[4, x]), {x, 0, nmax}], x] Range[0, nmax]!^4
A193436
exp( Sum_{n>=1} x^n/n^3 ) = Sum_{n>=0} a(n)*x^n/n!^3.
Original entry on oeis.org
1, 1, 5, 71, 2276, 144724, 16688884, 3249507820, 1005334796864, 468967172341824, 315409074574480704, 294510517409159769024, 369877735410388416241920, 608401340784471133062837504, 1281569707473914769353921666304, 3391681347749396029674738480747264
Offset: 0
A(x) = 1 + x + 5*x^2/2!^3 + 71*x^3/3!^3 + 2276*x^4/4!^3 +...
where
log(A(x)) = x + x^2/8 + x^3/27 + x^4/64 + x^5/125 + x^6/216 +...
A336441
a(n) = (n!)^n * [x^n] exp(Sum_{k>=1} x^k / k^n).
Original entry on oeis.org
1, 1, 3, 71, 30232, 435772624, 357189846148256, 25740403176657987904960, 234446578865185870182814945640448, 363178754511398964104990417951192651478859776, 122088173887703514886799765831338556792096849201928981184512
Offset: 0
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Table[(n!)^n SeriesCoefficient[Exp[Sum[x^k/k^n, {k, 1, n}]], {x, 0, n}], {n, 0, 10}]
b[n_, k_] := If[n == 0, 1, (1/n) Sum[(Binomial[n, j] (n - j - 1)!)^k (n - j) b[j, k], {j, 0, n - 1}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 10}]
Showing 1-4 of 4 results.
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