A209884
E.g.f. A(x) satisfies: A(x/(1-x))/(1-x) = (1/x) * d/dx x^2*A(x)/2.
Original entry on oeis.org
1, 2, 10, 88, 1172, 21688, 527576, 16224640, 612742784, 27786137856, 1486036616448, 92365550416896, 6591134271264000, 534423153792751104, 48801114693060804096, 4980221379342136676352, 564172247490745614434304, 70522894891787889987747840
Offset: 0
E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 88*x^3/3! + 1172*x^4/4! + 21688*x^5/5! +...
Related expansions:
A(x/(1-x))/(1-x) = 1 + 3*x + 20*x^2/2! + 220*x^3/3! + 3516*x^4/4! +...
A(x) + x*A'(x)/2 = 1 + 3*x + 20*x^2/2! + 220*x^3/3! + 3516*x^4/4! +...
Also, a(n) appears in the expansion:
B(x) = 1 + 2*x + 10*x^2/2!^2 + 88*x^3/3!^2 + 1172*x^4/4!^2 + 21688*x^5/5!^2 +...
such that
log(B(x)) = 2*x + 2*x^2/(2*2!) + 2*x^3/(3*3!) + 2*x^4/(4*4!) + 2*x^5/(5*5!) +...
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Table[Sum[BellY[n, k, 2/Range[n]], {k, 0, n}] n!, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
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{a(n)=local(A=1+x, B); for(i=1, n, B=subst(A, x, x/(1-x+x*O(x^n)))/(1-x); A=1+2*intformal((B-A)/x)); n!*polcoeff(A, n)}
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{a(n)=if(n<0, 0, if(n==0, 1, (n-1)!*sum(k=0, n-1, 2*binomial(n, k)*a(k)/k!)))}
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{a(n)=n!^2*polcoeff(exp(sum(m=1, n, 2*x^m/(m*m!))+x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))
A209917
E.g.f. A(x) satisfies: A(x/(1-x))/(1-x) = (1/x^2) * d/dx x^3*A(x)/3.
Original entry on oeis.org
1, 3, 21, 249, 4356, 103932, 3213216, 124146432, 5834291328, 326570493312, 21408981213888, 1621281984076224, 140205279698051328, 13711076231477352192, 1503606581609959001088, 183562416179374733411328, 24787906630769478567297024
Offset: 0
E.g.f.: A(x) = 1 + 3*x + 21*x^2/2! + 249*x^3/3! + 4356*x^4/4! + 103932*x^5/5! + ...
Related expansions:
A(x/(1-x))/(1-x) = 1 + 4*x + 35*x^2/2! + 498*x^3/3! + 10164*x^4/4! + ...
A(x) + x*A'(x)/3 = 1 + 4*x + 35*x^2/2! + 498*x^3/3! + 10164*x^4/4! + ...
Also, a(n) appears in the expansion:
B(x) = 1 + 3*x + 21*x^2/2!^2 + 249*x^3/3!^2 + 4356*x^4/4!^2 + 103932*x^5/5!^2 + ...
such that
log(B(x)) = 3*x + 3*x^2/(2*2!) + 3*x^3/(3*3!) + 3*x^4/(4*4!) + 3*x^5/(5*5!) + ...
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a = ConstantArray[0,21]; a[[1]]=1; a[[2]]=3; Do[a[[n+1]] = (n-1)!*Sum[3*Binomial[n, k]*a[[k+1]]/k!,{k,0,n-1}],{n,2,20}]; a (* Vaclav Kotesovec, Feb 23 2014 *)
Table[Sum[BellY[n, k, 3/Range[n]], {k, 0, n}] n!, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
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{a(n)=local(A=1+x, B); for(i=1, n, B=subst(A, x, x/(1-x+x*O(x^n)))/(1-x); A=1+3*intformal((B-A)/x)); n!*polcoeff(A, n)}
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{a(n)=if(n<0, 0, if(n==0, 1, (n-1)!*sum(k=0, n-1, 3*binomial(n, k)*a(k)/k!)))}
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{a(n)=n!^2*polcoeff(exp(sum(m=1, n, 3*x^m/(m*m!))+x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))
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@CachedFunction
def a(n): return 1 if (n==0) else factorial(n-1)*sum( 3*binomial(n, j)*a(j)/factorial(j) for j in (0..n-1) )
[a(n) for n in (0..20)] # G. C. Greubel, Jun 23 2021
A193160
E.g.f. A(x) satisfies: A(x/(1-x)) = x*A'(x).
Original entry on oeis.org
1, 2, 9, 68, 760, 11664, 233828, 5905696, 182846592, 6792372480, 297550188672, 15153482847744, 886517886778368, 58975120009537536, 4422337095720648960, 370957479138591903744, 34576037926690499493888, 3559813114275891217760256
Offset: 1
E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 68*x^4/4! + 760*x^5/5! +...
Related expansions:
A(x/(1-x)) = x + 4*x^2/2! + 27*x^3/3! + 272*x^4/4! + 3800*x^5/5! +...
x*A'(x) = x + 4*x^2/2! + 27*x^3/3! + 272*x^4/4! + 3800*x^5/5! +...
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{a(n)=local(A=[1],F=x);for(i=1,n,A=concat(A,0);F=x*Ser(A);A[#A]=Vec(subst(F,x,x/(1-x)))[#A]/(#A-1));if(n<1,0,n!*A[n])}
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{a(n)=if(n<1,0,if(n==1,1,n!/(n-1)*sum(k=1,n-1,binomial(n-1,k-1)*a(k)/k!)))}
A336243
a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * (n-k-1)! * a(k).
Original entry on oeis.org
1, 1, 5, 56, 1114, 34624, 1549648, 94402356, 7511324448, 756406501200, 94039208461584, 14146468841290752, 2532586289913605088, 532113978869395649856, 129662518122880634567232, 36270261084908437106586624, 11543682123659880166705099776
Offset: 0
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a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[1/(1 - Sum[x^k/(k k!), {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2
nmax = 16; Assuming[x > 0, CoefficientList[Series[1/(1 + EulerGamma - ExpIntegralEi[x] + Log[x]), {x, 0, nmax}], x]] Range[0, nmax]!^2
A336290
a(0) = 1; a(n) = n! * Sum_{k=1..n} binomial(n-1,k-1) * H(k) * a(n-k) / (n-k)!, where H(k) is the k-th harmonic number.
Original entry on oeis.org
1, 1, 5, 44, 628, 12994, 363548, 13141974, 593579712, 32644440048, 2141946861312, 164937634714896, 14703536203936512, 1500149281670010048, 173464224256287048576, 22541427301008492798144, 3267767649638304967827456, 525055667919614566758512640
Offset: 0
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a[0] = 1; a[n_] := a[n] = n! Sum[Binomial[n - 1, k - 1] HarmonicNumber[k] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[Sum[HarmonicNumber[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2
nmax = 17; Assuming[x > 0, CoefficientList[Series[Exp[Exp[x] (EulerGamma - ExpIntegralEi[-x] + Log[x])], {x, 0, nmax}], x]] Range[0, nmax]!^2
Showing 1-5 of 5 results.