A256126
Coefficients in asymptotic expansion of sequence A179327.
Original entry on oeis.org
1, 1, 3, 11, 50, 278, 1860, 14793, 138166, 1494034, 18422609, 255359957, 3929301362, 66412322717, 1222216175058, 24314268876147, 519701698551031, 11874016816562299, 288722141589331161, 7442890569982739838, 202733505298293899570, 5817564888930184685708
Offset: 0
A179327(n) / (n-1)! ~ 1 + 1/n + 3/n^2 + 11/n^3 + 50/n^4 + 278/n^5 + 1860/n^6 + ...
A062363
a(n) = Sum_{d|n} d!.
Original entry on oeis.org
0, 1, 3, 7, 27, 121, 729, 5041, 40347, 362887, 3628923, 39916801, 479002353, 6227020801, 87178296243, 1307674368127, 20922789928347, 355687428096001, 6402373706091609, 121645100408832001, 2432902008180268947, 51090942171709445047, 1124000727777647596803
Offset: 0
The divisors of 3 are 1 and 3 so 1! + 3! = 7. The divisors of 4 are 1, 2 and 4 so 1! + 2! + 4! = 27.
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nmax=20; CoefficientList[Series[Sum[m!*x^m/(1-x^m),{m,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 14 2015 *)
Join[{0},Table[Total[Divisors[n]!],{n,20}]] (* Harvey P. Dale, Aug 15 2024 *)
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a(n)=if(n<1, 0, sumdiv(n, d, d!));
A107895
Euler transform of n!.
Original entry on oeis.org
1, 1, 3, 9, 36, 168, 961, 6403, 49302, 430190, 4199279, 45326013, 535867338, 6884000262, 95453970483, 1420538043009, 22579098396600, 381704267100888, 6837775526561031, 129377310771795789, 2578101967764973314, 53965231260126083854, 1183813954026245944519
Offset: 0
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EulerTrans := proc(p) local b; b := proc(n) option remember; local d, j;
`if`(n=0,1, add(add(d*p(d),d=numtheory[divisors](j)) *b(n-j),j=1..n)/n) end end:
A107895 := EulerTrans(n->n!): seq(A107895(n),n=0..20);
# After Alois P. Heinz, A000335. [Peter Luschny, Jul 07 2011]
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EulerTrans[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]]; b]; A107895 = EulerTrans[Factorial]; Table[A107895[n], {n, 0, 22}] (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)
A261047
Euler transform of (n+1)!.
Original entry on oeis.org
1, 2, 9, 40, 212, 1248, 8400, 63576, 540858, 5132564, 53952742, 623324184, 7855144818, 107224120980, 1575511525794, 24784246515256, 415435624535225, 7389692971336602, 138992875726543381, 2755750468146310688, 57433108983590606292
Offset: 0
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a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
(d+1)!, d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jul 28 2017
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nmax=20; CoefficientList[Series[Product[1/(1-x^k)^((k+1)!), {k, 1, nmax}], {x, 0, nmax}], x]
A305867
Expansion of Product_{k>=1} 1/(1 - x^k)^(2*k-1)!!.
Original entry on oeis.org
1, 1, 4, 19, 130, 1120, 11960, 151595, 2230550, 37361755, 701873371, 14610774346, 333746628499, 8298025724194, 223049950124065, 6444634486214748, 199165237980655863, 6555102341516877027, 228905611339161301812, 8452656930719845696590, 329075775511339959533232, 13471099892869946627980017
Offset: 0
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N:= 25:
S:=series(mul((1-x^k)^(-doublefactorial(2*k-1)),k=1..N),x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Jun 12 2018
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nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(2 k - 1)!!, {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (2 d - 1)!!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]
A321522
Expansion of Product_{k>=1} (1 + x^k)^((k-1)!).
Original entry on oeis.org
1, 1, 1, 3, 8, 32, 153, 883, 5980, 46660, 411861, 4057263, 44104688, 524243696, 6762188285, 94055795999, 1403061499362, 22342571084082, 378257158227079, 6783952072695685, 128481050502464062, 2562250926987454694, 53668572808754641369, 1177957644341460946099
Offset: 0
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)*binomial((i-1)!, j), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..23); # Alois P. Heinz, Aug 10 2021
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nmax = 23; CoefficientList[Series[Product[(1 + x^k)^((k - 1)!), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 23}]
Showing 1-6 of 6 results.
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