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)}
A336259
a(0) = 1; a(n) = (n!)^3 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^3.
Original entry on oeis.org
1, 1, 9, 278, 20464, 2948824, 735078968, 291153023664, 172201253334528, 145044581320046592, 167609226267379703808, 257816558769660828601344, 514890814087717253133447168, 1307445058678686737908660752384, 4146656933568759002389401276616704
Offset: 0
-
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-i)/i^3, i=1..n))
end:
a:= n-> n!^3*b(n):
seq(a(n), n=0..14); # Alois P. Heinz, Jan 04 2024
-
a[0] = 1; a[n_] := a[n] = (n!)^3 Sum[a[k]/(k! (n - k))^3, {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]
nmax = 14; CoefficientList[Series[1/(1 - PolyLog[3, x]), {x, 0, nmax}], x] Range[0, nmax]!^3
A217145
exp( Sum_{n>=1} x^n/n^4 ) = Sum_{n>=0} a(n)*x^n/n!^4.
Original entry on oeis.org
1, 1, 9, 313, 30232, 6874776, 3355094696, 3302015131304, 6189229701416448, 20757720442141804032, 116803259505967824465408, 1039413737809909553149398528, 13914325979093456341597993070592, 268988472559744572003351007811825664
Offset: 0
A(x) = 1 + x + 9*x^2/2!^4 + 313*x^3/3!^4 + 30232*x^4/4!^4 + 6874776*x^5/5!^4 +...
where
log(A(x)) = x + x^2/2^4 + x^3/3^4 + x^4/4^4 + x^5/5^3 + x^6/6^4 +...
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{a(n)=n!^4*polcoeff(exp(sum(m=1, n, x^m/m^4)+x*O(x^n)), n)}
for(n=0,20,print1(a(n),", "))
A193435
exp( Sum_{n>=1} x^n / (n*(n+1)/2) ) = Sum_{n>=0} a(n)*x^n / (n!*(n+1)!).
Original entry on oeis.org
1, 2, 10, 96, 1528, 36720, 1248560, 57272320, 3417283968, 257711328000, 23999961081600, 2707648512307200, 364172045286804480, 57600985355595601920, 10589369533424230348800, 2239779182794304126976000, 540207965396186411279155200
Offset: 0
A(x) = 1 + 2*x/(1!*2!) + 10*x^2/(2!*3!) + 96*x^3/(3!*4!) + 1528*x^4/(4!*5!) +...
where
log(A(x)) = x + x^2/3 + x^3/6 + x^4/10 + x^5/15 + x^6/21 + x^7/28 +...
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 1-5 of 5 results.
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