A121860
a(n) = Sum_{d|n} n!/(d!*(n/d)!).
Original entry on oeis.org
1, 2, 2, 8, 2, 122, 2, 1682, 10082, 30242, 2, 7318082, 2, 17297282, 3632428802, 36843206402, 2, 2981705126402, 2, 1690185726028802, 3379030566912002, 28158588057602, 2, 76941821303636889602, 1077167364120207360002
Offset: 1
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with(numtheory): seq(n!*add(1/(d!*(n/d)!), d in divisors(n)), n = 1..25); # Peter Bala, Aug 04 2025
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f[n_] := Block[{d = Divisors@n}, Plus @@ (n!/(d! (n/d)!))]; Array[f, 25] (* Robert G. Wilson v, Sep 11 2006 *)
Table[DivisorSum[n, n!/(#!*(n/#)!) &], {n, 25}] (* Michael De Vlieger, Sep 12 2018 *)
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a(n) = sumdiv(n, d, n!/(d!*(n/d)!)); \\ Michel Marcus, Sep 13 2018
A258900
E.g.f.: S(x) = Series_Reversion( Integral 1/(1-x^4)^(1/4) dx ), where the constant of integration is zero.
Original entry on oeis.org
1, -6, -1764, -7700616, -147910405104, -8310698364852576, -1085420895640591777344, -284168646775526186095019136, -134459287943928269154814258953984, -106506405136317713669903020280294647296
Offset: 0
E.g.f. with offset 0 is C(x) and e.g.f. with offset 1 is S(x) where:
C(x) = 1 - 6*x^4/4! - 1764*x^8/8! - 7700616*x^12/12! - 147910405104*x^16/16! -...
S(x) = x - 6*x^5/5! - 1764*x^9/9! - 7700616*x^13/13! - 147910405104*x^17/17! -...
such that C(x)^4 + S(x)^4 = 1:
C(x)^4 = 1 - 24*x^4/4! + 8064*x^8/8! + 2128896*x^12/12! + 52932870144*x^16/16! +...
S(x)^4 = 24*x^4/4! - 8064*x^8/8! - 2128896*x^12/12! - 52932870144*x^16/16! -...
Related Expansions.
(1) The series reversion of S(x) is Integral 1/(1-x^4)^(1/4) dx:
Series_Reversion(S(x)) = x + 6*x^5/5! + 6300*x^9/9! + 56133000*x^13/13! +...
1/(1-x^4)^(1/4) = 1 + 6*x^4/4! + 6300*x^8/8! + 56133000*x^12/12! + 1992160170000*x^16/16! +...+ A258899(n)*x^(4*n)/(4*n)! +...
(2) d/dx S(x)/C(x) = 1/C(x)^4:
1/C(x)^4 = 1 + 24*x^4/4! + 32256*x^8/8! + 285272064*x^12/12! +...
S(x)/C(x) = x + 24*x^5/5! + 32256*x^9/9! + 285272064*x^13/13! + 8967114326016*x^17/17! +...+ A258901(n)*x^(4*n+1)/(4*n+1)! +...
where
Series_Reversion(S(x)/C(x)) = x - x^5/5 + x^9/9 - x^13/13 + x^17/17 - x^21/21 +...
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/* E.g.f. Series_Reversion(Integral 1/(1-x^4)^(1/4) dx): */
{a(n)=local(S=x); S = serreverse( intformal( 1/(1-x^4 +x*O(x^(4*n)))^(1/4) )); (4*n+1)!*polcoeff(S,4*n+1)}
for(n=0,15,print1(a(n),", "))
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/* E.g.f. C(x) with offset 0: */
{a(n)=local(S=x, C=1+x); for(i=1, n, S=intformal(C +x*O(x^(4*n))); C=1-intformal(S^3/C^2 +x*O(x^(4*n))); ); (4*n)!*polcoeff(C, 4*n)}
for(n=0, 15, print1(a(n), ", "))
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/* E.g.f. S(x) with offset 1: */
{a(n)=local(S=x, C=1+x); for(i=1, n+1, S=intformal(C +x*O(x^(4*n))); C=1-intformal(S^3/C^2 +x*O(x^(4*n+1))); ); (4*n+1)!*polcoeff(S, 4*n+1)}
for(n=0, 15, print1(a(n), ", "))
A258903
E.g.f.: 2 - exp(2) + Sum_{n>=1} 2^n * exp(3*x^n) / n!.
Original entry on oeis.org
1, 6, 30, 78, 426, 582, 12450, 4758, 407010, 2218182, 19172370, 360438, 4755166050, 3213222, 85631151090, 5099958831318, 54483404779650, 258673542, 11939347971403410, 2326095798, 5556296851712151330, 35398724239897109862, 10235928407592878130, 188311523478, 758680053859872239555010
Offset: 0
E.g.f.: A(x) = 1 + 6*x + 30*x^2/2! + 78*x^3/3! + 426*x^4/4! + 582*x^5/5! +...
where
A(x) = 2 - exp(2) + 2*exp(3*x) + 2^2*exp(3*x^2)/2! + 2^3*exp(3*x^3)/3! + 2^4*exp(3*x^4)/4! + 2^5*exp(3*x^5)/5! +...
A(x) = 2 - exp(3) + 3*exp(2*x) + 3^2*exp(2*x^2)/2! + 3^3*exp(2*x^3)/3! + 3^4*exp(2*x^4)/4! + 3^5*exp(2*x^5)/5! +...
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with(numtheory): seq(n!*add(2^d*3^(n/d)/(d!*(n/d)!), d in divisors(n)), n = 1..25); # Peter Bala, Aug 08 2025
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{a(n) = local(A=1); A = 2-exp(2) + sum(m=1,n,2^m/m!*exp(3*x^m +x*O(x^n))); if(n==0,1, n!*polcoeff(A,n))}
for(n=0,30, print1(a(n),", "))
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{a(n) = local(A=1); A = 2-exp(3) + sum(m=1,n,3^m/m!*exp(2*x^m +x*O(x^n))); if(n==0,1, n!*polcoeff(A,n))}
for(n=0,30, print1(a(n),", "))
Showing 1-3 of 3 results.
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