A128231
Expansion of exp(x)/(1 - x^3/3!), where a(n) = 1 + binomial(n,3)*a(n-3).
Original entry on oeis.org
1, 1, 1, 2, 5, 11, 41, 176, 617, 3445, 21121, 101806, 757901, 6040607, 37057385, 344844956, 3382739921, 25199021801, 281393484097, 3277874983450, 28726884853141, 374253333849011, 5047927474513001, 50875313074912712
Offset: 0
E.g.f.: exp(x)/(1 - x^3/3!) = 1 + x + 1*x^2/2! + 2*x^3/3! + 5*x^4/4! + 11*x^5/5! + 41*x^6/6! + ... + a(n)*x^n/n! + ...
where a(n) = 1 + n*(n-1)*(n-2)*a(n-3)/3!.
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restart: G(x):=2*exp(-x)/(x^3/3!+1): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(-f[n-1],x) od: x:=0: seq(f[n]/2,n=0..23); # Zerinvary Lajos, Apr 03 2009
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a(n)=n!*polcoeff(exp(x+x*O(x^n))/(1-x^3/3! +x*O(x^n)),n)
A128232
Expansion of exp(x)/(1 - x^4/4!), where a(n) = 1 + C(n,4)*a(n-4).
Original entry on oeis.org
1, 1, 1, 1, 2, 6, 16, 36, 141, 757, 3361, 11881, 69796, 541256, 3364362, 16217566, 127028721, 1288189281, 10294947721, 62859285817, 615454153246, 7709812846786, 75307542579116, 556618975909536, 6539815832391997
Offset: 0
E.g.f.: exp(x)/(1 - x^4/4!) = 1 + x + 1*x^2/2! + 1*x^3/3! + 2*x^4/4! + 6*x^5/5! + 16*x^6/6! +... + a(n)*x^n/n! +...
where a(n) = 1 + n*(n-1)*(n-2)*(n-3)*a(n-4)/4!.
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G(x):=exp(x)/(1-x^4/4!): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..24); # Zerinvary Lajos, Apr 03 2009
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a(n)=n!*polcoeff(exp(x+x*O(x^n))/(1-x^4/4! +x*O(x^n)),n)
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/* Recurrence: */ a(n)=if(n<0,0,if(n<4,1,1 + n*(n-1)*(n-2)*(n-3)*a(n-4)/4!))
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