A187742
G.f.: Sum_{n>=0} (n+x)^n * x^n / (1 + n*x + x^2)^n.
Original entry on oeis.org
1, 1, 4, 14, 66, 384, 2640, 20880, 186480, 1854720, 20321280, 243129600, 3153427200, 44068147200, 660064204800, 10548573235200, 179151388416000, 3222109642752000, 61178237632512000, 1222853377794048000, 25667116186263552000, 564433265896980480000, 12977099311614197760000
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 14*x^3 + 66*x^4 + 384*x^5 + 2640*x^6 +...
where
A(x) = 1 + (1+x)*x/(1+x+x^2) + (2+x)^2*x^2/(1+2*x+x^2)^2 + (3+x)^3*x^3/(1+3*x+x^2)^3 + (4+x)^4*x^4/(1+4*x+x^2)^4 + (5+x)^5*x^5/(1+5*x+x^2)^5 +...
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List([3..11], n->Size(OrbitsDomain(Group((1,2)),SymmetricGroup(IsPermGroup, n), \^))); # Attila Egri-Nagy, Aug 15 2014
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a[0] = 1; a[1] = 1; a[n_] := (n^2 + n + 2)*(n - 1)!/2; Table[a[n], {n, 0, 20}] (* Wesley Ivan Hurt, Aug 15 2014 *)
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{a(n)=polcoeff( sum(m=0, n, (m+x)^m*x^m/(1+m*x+x^2 +x*O(x^n))^m), n)}
for(n=0, 30, print1(a(n), ", "))
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{a(n)=if(n>=0&n<=1,1,(n^2+n+2)*(n-1)!/2)}
for(n=0, 30, print1(a(n), ", "))
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{a(n)=n!*polcoeff(1/2 + 1/(2*(1-x)^2) - x - log(1-x +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
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x='x+O('x^66); concat([1], Vec(serlaplace(1/(1-x)^3 + x/(1-x)))) \\ Joerg Arndt, Aug 15 2014
A202365
G.f.: Sum_{n>=0} (n-x)^n * x^n / (1 + n*x - x^2)^n.
Original entry on oeis.org
1, 1, 2, 10, 54, 336, 2400, 19440, 176400, 1774080, 19595520, 235872000, 3073593600, 43110144000, 647610163200, 10374216652800, 176536039680000, 3180264062976000, 60466862776320000, 1210048630382592000, 25423825985445888000, 559567461880627200000, 12874917427270778880000
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 54*x^4 + 336*x^5 + 2400*x^6 +...
where
A(x) = 1 + (1-x)*x/(1+x-x^2) + (2-x)^2*x^2/(1+2*x-x^2)^2 + (3-x)^3*x^3/(1+3*x-x^2)^3 + (4-x)^4*x^4/(1+4*x-x^2)^4 + (5-x)^5*x^5/(1+5*x-x^2)^5 +...
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a[n_] := Switch[n, 0|1, 1, _, (n-1)*(n+2)/2*(n-1)!];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 24 2022 *)
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{a(n)=polcoeff( sum(m=0, n, (m-x)^m*x^m/(1+m*x-x^2 +x*O(x^n))^m), n)}
for(n=0, 30, print1(a(n), ", "))
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{a(n)=if(n==0||n==1, 1, (n-1)*(n+2)/2 * (n-1)!)}
for(n=0, 30, print1(a(n), ", "))
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{a(n)=n!*polcoeff(1/2 + 1/(2*(1-x)^2) + x + log(1-x +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
A226045
G.f.: ( Sum_{n>=0} (2*n+x)^n * x^n / (1 + 2*n*x + x^2)^n )^(1/2).
Original entry on oeis.org
1, 1, 6, 44, 430, 5162, 72890, 1178978, 21449704, 433116488, 9606793874, 232145293502, 6070097785376, 170763070370848, 5142963967765530, 165115679014587758, 5629558857460143814, 203146937778126705662, 7735490130309647256862
Offset: 0
G.f.: A(x) = 1 + x + 6*x^2 + 44*x^3 + 430*x^4 + 5162*x^5 +...
where the square of the g.f. begins
A(x)^2 = 1 + 2*x + 13*x^2 + 100*x^3 + 984*x^4 + 11712*x^5 + 163200*x^6 +...+ A187746(n)*x^n +...
and equals the series
A(x)^2 = 1 + (2+x)*x/(1+2*x+x^2) + (4+x)^2*x^2/(1+4*x+x^2)^2 + (6+x)^3*x^3/(1+6*x+x^2)^3 + (8+x)^4*x^4/(1+8*x+x^2)^4 + (10+x)^5*x^5/(1+10*x+x^2)^5 +...
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{a(n)=polcoeff( sum(m=0, n, (2*m+x)^m*x^m/(1+2*m*x+x^2 +x*O(x^n))^m)^(1/2), n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.
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