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a(n+1) = Sum_{k=0..n} (n!/k!)*binomial(n,k)*a(k).
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5
1, 1, 2, 8, 50, 442, 5212, 78664, 1472756, 33378740, 898227944, 28253387104, 1025373023848, 42467845178632, 1988513519453360, 104413376937507488, 6104596110052561808, 394921638012548722576, 28112685278602155590944, 2191142414957886078590080
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add(
a(n-i)*binomial(n-1, i-1)^2*(i-1)!, i=1..n))
end:
MATHEMATICA
nmax=20; b = ConstantArray[0, nmax+2]; b[[1]]=1; Do[b[[n+2]] = Sum[n!/k!*Binomial[n, k]*b[[k+1]], {k, 0, n}], {n, 0, nmax}]; b (* Vaclav Kotesovec, Mar 02 2014 *)
E.g.f. A(x) satisfies: A(x/(1-x)) = x*A'(x).
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1
1, 2, 9, 68, 760, 11664, 233828, 5905696, 182846592, 6792372480, 297550188672, 15153482847744, 886517886778368, 58975120009537536, 4422337095720648960, 370957479138591903744, 34576037926690499493888, 3559813114275891217760256
FORMULA
a(n) = n*(n-2)!* Sum_{k=1..n-1} C(n-1,k-1)* a(k)/k! for n>1 with a(1)=1.
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 68*x^4/4! + 760*x^5/5! +...
Related expansions:
A(x/(1-x)) = x + 4*x^2/2! + 27*x^3/3! + 272*x^4/4! + 3800*x^5/5! +...
x*A'(x) = x + 4*x^2/2! + 27*x^3/3! + 272*x^4/4! + 3800*x^5/5! +...
PROG
(PARI) {a(n)=local(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A]=Vec(subst(F, x, x/(1-x)))[#A]/(#A-1)); if(n<1, 0, n!*A[n])}
(PARI) {a(n)=if(n<1, 0, if(n==1, 1, n!/(n-1)*sum(k=1, n-1, binomial(n-1, k-1)*a(k)/k!)))}
E.g.f. A(x) satisfies: d/dx A(x) = 1 + A(x/(1 + x)).
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1
1, 1, -1, -1, 23, -197, 1093, 5377, -374863, 9934889, -195976201, 2134159519, 67270069831, -6730482201869, 365726633654957, -15494292929032063, 458469986808144737, 2828723973314497873, -2067360599320208561297, 238687490584532161293631, -20425982116564721266720009
FORMULA
Recurrence: a(n+1) = Sum_{k=1..n} (-1)^(n-k)*binomial(n-1,k-1)*a(k)*n!/k!.
MATHEMATICA
terms = 21; A[_] = 0; Do[A[x_] = Normal[Integrate[1 + A[x/(1 + x) + O[x]^(terms + 1)], x] + O[x]^(terms + 1)], terms]; Rest[CoefficientList[A[x], x] Range[0, terms]!]
a[n_] := a[n] = Sum[(-1)^(n - k - 1) Binomial[n - 2, k - 1] a[k] (n - 1)!/k!, {k, 1, n - 1}]; a[1] = 1; Table[a[n], {n, 1, 21}]
E.g.f. A(x) satisfies: d/dx A(x) = 1 + (1/(1 - x)) * A(x/(1 - x)).
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1
1, 1, 5, 32, 280, 3280, 49480, 927560, 21037640, 566134160, 17803754560, 646052181520, 26757321804880, 1252934215973600, 65791336312915520, 3846554938702140320, 248841434876849499040, 17713758333248102781760, 1380631354206969100115200
FORMULA
a(1) = 1; a(n+1) = Sum_{k=0..n-1} binomial(n,k)^2 * k! * a(n-k).
MATHEMATICA
terms = 20; A[_] = 0; Do[A[x_] = Normal[Integrate[1 + 1/(1 - x) A[x/(1 - x) + O[x]^(terms + 1)], x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]! // Rest
a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k]^2 k! a[n - k - 1], {k, 0, n - 2}]; Table[a[n], {n, 1, 20}]
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