A193374
E.g.f.: A(x) = exp( Sum_{n>=1} x^(n*(n+1)/2) / (n*(n+1)/2) ).
Original entry on oeis.org
1, 1, 1, 3, 9, 21, 201, 1191, 4593, 36009, 620721, 5297931, 40360761, 474989373, 4345942329, 122776895151, 2118941145441, 21344580276561, 303071564084193, 4476037678611219, 59935820004483561, 3838519441659950181, 78361805638079449641, 949279542954821272503
Offset: 0
E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 21*x^5/5! + 201*x^6/6! +...
where
log(A(x)) = x + x^3/3 + x^6/6 + x^10/10 + x^15/15 + x^21/21 +...
-
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(issqr(8*j+1),
a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n))
end:
seq(a(n), n=0..25); # Alois P. Heinz, May 12 2016
-
a[n_] := a[n] = If[n == 0, 1, Sum[If[IntegerQ @ Sqrt[8*j + 1], a[n - j]*(j - 1)!*Binomial[n - 1, j - 1], 0], {j, 1, n}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)
-
{a(n)=n!*polcoeff(exp(sum(m=1,sqrtint(2*n+1),x^(m*(m+1)/2)/(m*(m+1)/2)+x*O(x^n))),n)}
A273994
Number of endofunctions on [n] whose cycle lengths are Fibonacci numbers.
Original entry on oeis.org
1, 1, 4, 27, 250, 2975, 43296, 744913, 14797036, 333393345, 8403026320, 234300271811, 7161316358616, 238108166195263, 8556626831402560, 330494399041444425, 13654219915946513296, 600870384794864432897, 28060233470995898505024, 1386000542545570348128235
Offset: 0
-
b:= proc(n) option remember; local r, f, g;
if n=0 then 1 else r, f, g:= $0..2;
while f<=n do r:= r+(f-1)!*b(n-f)*
binomial(n-1, f-1); f, g:= g, f+g
od; r fi
end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);
-
b[n_] := b[n] = Module[{r, f, g}, If[n == 0, 1, {r, f, g} = {0, 1, 2}; While[f <= n, r = r + (f - 1)!*b[n - f]*Binomial[n - 1, f - 1]; {f, g} = {g, f + g}]; r]];
a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 06 2018, from Maple *)
A273996
Number of endofunctions on [n] whose cycle lengths are factorials.
Original entry on oeis.org
1, 1, 4, 25, 218, 2451, 33952, 560407, 10750140, 235118665, 5775676496, 157448312649, 4716609543736, 154007821275595, 5443783515005760, 207093963680817511, 8436365861409555728, 366403740283162634193, 16900793597898691865920, 825115046704241167668025
Offset: 0
-
b:= proc(n) option remember; local r, f, g;
if n=0 then 1 else r, f, g:= $0..2;
while f<=n do r:= r+(f-1)!*b(n-f)*
binomial(n-1, f-1); f, g:= f*g, g+1
od; r fi
end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);
-
b[n_] := b[n] = Module[{r, f, g}, If[n == 0, 1, {r, f, g} = {0, 1, 2}; While[f <= n, r = r + (f - 1)!*b[n - f]*Binomial[n - 1, f - 1]; {f, g} = {f*g, g + 1}]; r]];
a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 06 2018, from Maple *)
A273998
Number of endofunctions on [n] whose cycle lengths are primes.
Original entry on oeis.org
1, 0, 1, 8, 75, 904, 13255, 229536, 4587961, 103971680, 2634212961, 73787255200, 2264440519891, 75563445303072, 2724356214102055, 105546202276277504, 4373078169296869425, 192970687573630633216, 9035613818754820178689, 447469496697658409400960
Offset: 0
-
b:= proc(n) option remember; local r, p;
if n=0 then 1 else r, p:=0, 2;
while p<=n do r:= r+(p-1)!*b(n-p)*
binomial(n-1, p-1); p:= nextprime(p)
od; r fi
end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);
-
b[n_] := b[n] = Module[{r, p}, If[n == 0, 1, {r, p} = {0, 2}; While[p <= n, r = r + (p - 1)!*b[n - p]*Binomial[n-1, p-1]; p = NextPrime[p]]; r]];
a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 06 2018, from Maple *)
A205799
E.g.f.: exp( Sum_{n>=1} x^(n*(n+1)/2) / (n*(n+1)/2)! ).
Original entry on oeis.org
1, 1, 1, 2, 5, 11, 32, 113, 365, 1373, 6072, 25279, 115633, 606321, 3051413, 16344785, 98402881, 576283953, 3523586227, 23840955908, 158428389359, 1085566420290, 8128568533790, 60203101002122, 455911264482697, 3734114950288571, 30413492882578846
Offset: 0
E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 11*x^5/5! + 32*x^6/6! +...
where
log(A(x)) = x + x^3/3! + x^6/6! + x^10/10! + x^15/15! + x^21/21! +...
-
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
issqr(8*j+1), a(n-j)*binomial(n-1, j-1), 0), j=1..n))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jun 10 2018
-
m = 30;
CoefficientList[Exp[Sum[x^(n(n+1)/2)/(n(n+1)/2)!, {n, 1, m}]] + O[x]^m, x]* Range[0, m-1]! (* Jean-François Alcover, Mar 05 2021 *)
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{a(n)=n!*polcoeff(exp(sum(m=1, sqrtint(2*n+1), x^(m*(m+1)/2)/(m*(m+1)/2)!+x*O(x^n))), n)}
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