A118930
E.g.f.: A(x) = exp( Sum_{n>=0} x^(2^n)/2^(2^n-1) ).
Original entry on oeis.org
1, 1, 2, 4, 13, 41, 166, 652, 3494, 18118, 114076, 681176, 5016892, 35377564, 288204008, 2232198256, 21124254181, 191779964597, 2011347229114, 19840403629108, 231266808172181, 2553719667653281, 31743603728993542
Offset: 0
E.g.f. A(x) = exp( x + x^2/2 + x^4/2^3 + x^8/2^7 + x^16/2^15 +...)
= 1 + 1*x + 2*x^2/2! + 4*x^3/3! + 13*x^4/4! + 41*x^5/5!+ 166*x^6/6!+...
Using coefficients A100861(n,k) = n!/[k!(n-2k)!*2^k]:
a(5) = 1*a(0) +10*a(1) +15*a(2) = 1*1 +10*1 +15*2 = 41.
a(6) = 1*a(0) +15*a(1) +45*a(2) +15*a(3) = 1*1 +15*1 +45*2 +15*4 = 166.
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A118930 := proc(n)
option remember;
if n<= 1 then
1 ;
else
n!*add(procname(k)/k!/(n-2*k)!/2^k,k=0..n/2) ;
end if;
end proc;
seq(A118930(n),n=0..10) ; # R. J. Mathar, Aug 19 2014
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a[n_] := a[n] = If[n==0, 1, Sum[Binomial[n, 2k] (2k-1)!! a[k], {k, 0, n/2}]];
a /@ Range[0, 22] (* Jean-François Alcover, Mar 26 2020 *)
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{a(n)=if(n==0,1,sum(k=0,n\2,n!/(k!*(n-2*k)!*2^k)*a(k)))}
for(n=0,30,print1(a(n),", "))
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/* Defined by E.G.F.: */
{a(n)=n!*polcoeff( exp(sum(k=0,#binary(n),x^(2^k)/2^(2^k-1))+x*O(x^n)),n,x)}
for(n=0,30,print1(a(n),", "))
A118932
E.g.f.: A(x) = exp( Sum_{n>=0} x^(3^n)/3^((3^n -1)/2) ).
Original entry on oeis.org
1, 1, 1, 3, 9, 21, 81, 351, 1233, 10249, 75841, 388411, 3733401, 33702813, 215375889, 1984583511, 19181083041, 141963117201, 1797976123393, 22534941675379, 202605151063081, 2992764505338021, 43182110678814801, 445326641624332623
Offset: 0
E.g.f. A(x) = exp( x + x^3/3 + x^9/3^4 + x^27/3^13 + x^81/3^40 + ...)
= 1 + 1*x + 1*x^2/2! + 3*x^3/3! + 9*x^4/4! + 21*x^5/5! + 81*x^6/6! + ...
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function a(n)
F:=Factorial;
if n eq 0 then return 1;
else return (&+[F(n)*a(j)/(3^j*F(j)*F(n-3*j)): j in [0..Floor(n/3)]]);
end if; return a; end function;
[a(n): n in [0..25]]; // G. C. Greubel, Mar 07 2021
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a[n_]:= a[n]= If[n==0, 1, Sum[n!*a[k]/(3^k*k!*(n-3*k)!), {k, 0, Floor[n/3]}] ];
Table[a[n], {n, 0, 25}] (* G. C. Greubel, Mar 07 2021 *)
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{a(n) = if(n==0,1,sum(k=0,n\3,n!/(k!*(n-3*k)!*3^k)*a(k)))}
for(n=0,30,print1(a(n),", "))
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/* Defined by E.G.F.: */
{a(n) = n!*polcoeff( exp(sum(k=0,ceil(log(n+1)/log(3)),x^(3^k)/3^((3^k-1)/2))+x*O(x^n)),n,x)}
for(n=0,30,print1(a(n),", "))
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@CachedFunction
def a(n):
f=factorial;
if n==0: return 1
else: return sum( f(n)*a(k)/(3^k*f(k)*f(n-3*k)) for k in (0..n/3))
[a(n) for n in (0..25)] # G. C. Greubel, Mar 07 2021
A118933
Triangle, read by rows, where T(n,k) = n!/(k!*(n-4*k)!*4^k) for n>=4*k>=0.
Original entry on oeis.org
1, 1, 1, 1, 1, 6, 1, 30, 1, 90, 1, 210, 1, 420, 1260, 1, 756, 11340, 1, 1260, 56700, 1, 1980, 207900, 1, 2970, 623700, 1247400, 1, 4290, 1621620, 16216200, 1, 6006, 3783780, 113513400, 1, 8190, 8108100, 567567000, 1, 10920, 16216200, 2270268000, 3405402000
Offset: 0
Triangle begins:
1;
1;
1;
1;
1, 6;
1, 30;
1, 90;
1, 210;
1, 420, 1260;
1, 756, 11340;
1, 1260, 56700;
1, 1980, 207900;
1, 2970, 623700, 1247400; ...
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F:= Factorial;
[n lt 4*k select 0 else F(n)/(4^k*F(k)*F(n-4*k)): k in [0..Floor(n/4)], n in [0..20]]; // G. C. Greubel, Mar 07 2021
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T[n_, k_]:= If[n<4*k, 0, n!/(4^k*k!*(n-4*k)!)];
Table[T[n, k], {n,0,20}, {k,0,n/4}]//Flatten (* G. C. Greubel, Mar 07 2021 *)
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T(n,k)=if(n<4*k,0,n!/(k!*(n-4*k)!*4^k))
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f=factorial;
flatten([[0 if n<4*k else f(n)/(4^k*f(k)*f(n-4*k)) for k in [0..n/4]] for n in [0..20]]) # G. C. Greubel, Mar 07 2021
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