A187742 -id:A187742 - cp's OEIS frontend

A204064 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + nx) / (1 + kx + n*x^2).

1, 1, 2, 5, 14, 44, 152, 572, 2324, 10124, 47012, 231572, 1204964, 6599444, 37924292, 228033332, 1431128804, 9354072404, 63548071172, 447923400692, 3270361265444, 24696229801364, 192625876675652, 1549890430643252, 12849460733123684, 109647468132256724

Offset: 0

Paul D. Hanna, Jan 09 2013

nonn

Compare to the identity:
Sum_{n>=0} x^n * Product_{k=1..n} (k + x) / (1 + k*x + x^2) = (1+x^2)/(1-x).
Compare to the g.f. of A187741:
Sum_{n>=0} x^n * (1 + n*x)^n / (1 + x + n*x^2)^n  =  1/2 + (1+2*x)*Sum_{n>=0} (n+1)!*x^(2*n)/2.

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 44*x^5 + 152*x^6 + 572*x^7 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1+2*x)*(2+2*x)/((1+x+2*x^2)*(1+2*x+2*x^2)) + x^3*(1+3*x)*(2+3*x)*(3+3*x)/((1+x+3*x^2)*(1+2*x+3*x^2)*(1+3*x+3*x^2)) + x^4*(1+4*x)*(2+4*x)*(3+4*x)*(4+4*x)/((1+x+4*x^2)*(1+2*x+4*x^2)*(1+3*x+4*x^2)*(1+4*x+4*x^2)) +...
Also, we have the identity (cf. A229046):
A(x) = 1/2 + (1/2)*(1+x)/(1+x) + (2!/2)*x*(1+x)^2/((1+x)*(1+2*x)) + (3!/2)*x^2*(1+x)^3/((1+x)*(1+2*x)*(1+3*x)) + (4!/2)*x^3*(1+x)^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + (5!/2)*x^4*(1+x)^5/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)*(1+5*x)) +...

Alois P. Heinz, Table of n, a(n) for n = 0..597

Cf. A229046, A204066, A187741, A187742, A220181.

b:= proc(n, k) option remember; `if`(n<1, 1, `if`(k>
      ceil(n/2), 0, add((k-j)*b(n-1-j, k-j), j=0..1)))
    end:
a:= n-> ceil(add(b(n+2, k), k=1..1+ceil(n/2))/2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 26 2018

b[n_, k_] := b[n, k] = If[n < 1, 1, If[k > Ceiling[n/2], 0, Sum[(k - j) b[n - 1 - j, k - j], {j, 0, 1}]]];
a[n_] := Ceiling[Sum[b[n + 2, k], {k, 1, 1 + Ceiling[n/2]}]/2];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)

{a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1,m,(k+m*x)/(1+k*x+m*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=polcoeff( 1/2 + sum(m=1, n+1, m!/2*x^(m-1)*(1+x)^m/prod(k=1, m, 1+k*x +x*O(x^n))), n)}
for(n=0,30,print1(a(n),", "))

{a(n)=if(n<0,0,if(n<1,1,(1/2)*sum(k=0, floor((n+1)/2), sum(i=0, k, (-1)^i*binomial(k, i)*(k-i+1)^(n-k+1)))))} \\ Paul D. Hanna, Jul 13 2014
for(n=0, 30, print1(a(n), ", "))

G.f.: 1/2 + Sum_{n>=1} n!/2 * x^(n-1) * (1+x)^n / Product_{k=1..n} (1 + k*x). - Paul D. Hanna, Oct 27 2013
a(n) = A229046(n+1)/2 for n>0.
a(n) = (1/2)*Sum_{k=0..floor((n+1)/2)} Sum_{i=0..k} (-1)^i * C(k,i) * (k-i+1)^(n-k+1) for n>1. (See Paul Barry's formula in A105795). - Paul D. Hanna, Jul 13 2014

A187741 G.f.: Sum_{n>=0} (1 + nx)^n x^n / (1 + x + n*x^2)^n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 24, 60, 120, 360, 720, 2520, 5040, 20160, 40320, 181440, 362880, 1814400, 3628800, 19958400, 39916800, 239500800, 479001600, 3113510400, 6227020800, 43589145600, 87178291200, 653837184000, 1307674368000, 10461394944000, 20922789888000

Offset: 0

Paul D. Hanna, Jan 03 2013

nonn

This is an enumeration of the disjoint union (with repetition) of A001710(n), for n > 0, and A000142(n), for n > 0. The first lists the orders of the alternating groups; the second lists the orders of the symmetric groups. - Hal M. Switkay, Mar 13 2023

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 24*x^7 + 60*x^8 +...
where
A(x) = 1 + (1+x)*x/(1+x+x^2) + (1+2*x)^2*x^2/(1+x+2*x^2)^2 + (1+3*x)^3*x^3/(1+x+3*x^2)^3 + (1+4*x)^4*x^4/(1+x+4*x^2)^4 + (1+5*x)^5*x^5/(1+x+5*x^2)^5 +...

Cf. A000142, A001710, A187742, A187735, A208236, A204064, A361382.

a[n_] := If[OddQ[n], ((n + 1)/2)!, (n/2 + 1)!/2]; a[0] = 1; Array[a, 32, 0] (* Amiram Eldar, Dec 11 2022 *)

{a(n)=polcoeff( sum(m=0, n, (x+m*x^2)^m / (1 + x+m*x^2 +x*O(x^n))^m), n)}
for(n=0, 40, print1(a(n), ", "))

{a(n)=if(n==0,1, if(n%2==0, ((n+2)/2)!/2, ((n+1)/2)! ))}
for(n=0, 30, print1(a(n), ", "))

G.f.: 1/2 + (1+2*x) * Sum_{n>=0} (n+1)!*x^(2*n)/2.
a(2*n) = (n+1)!/2,  a(2*n-1) = n!,  for n>0 with a(0)=1.
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 3*e - 4.
Sum_{n>=0} (-1)^n/a(n) = e - 2. (End)

A204066 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (n + kx) / (1 + nx + k*x^2).

Original entry on oeis.org

1, 1, 4, 16, 82, 502, 3574, 29002, 264166, 2668666, 29612014, 358025986, 4684916902, 65966957722, 994546450174, 15984888286642, 272845934899606, 4929166716321706, 93963635086523374, 1884915966747571906, 39691711412770983622, 875410001054417122042, 20180907494704416823774

Offset: 0

Paul D. Hanna, Jan 09 2013

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 82*x^4 + 502*x^5 + 3574*x^6 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(2+x)*(2+2*x)/((1+2*x+x^2)*(1+2*x+2*x^2)) + x^3*(3+x)*(3+2*x)*(3+3*x)/((1+3*x+x^2)*(1+3*x+2*x^2)*(1+3*x+3*x^2)) + x^4*(4+x)*(4+2*x)*(4+3*x)*(4+4*x)/((1+4*x+x^2)*(1+4*x+2*x^2)*(1+4*x+3*x^2)*(1+4*x+4*x^2)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A204064, A187741, A187742, A220181.

{a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (m+k*x)/(1+m*x+k*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

a(n) ~ exp(1/2) * n! * n/2. - Vaclav Kotesovec, Nov 02 2014

A187746 G.f.: Sum_{n>=0} (2n+x)^n x^n / (1 + 2nx + x^2)^n.

Original entry on oeis.org

1, 2, 13, 100, 984, 11712, 163200, 2603520, 46771200, 934133760, 20530298880, 492355584000, 12793813401600, 358063276032000, 10737974299852800, 343513154086502400, 11676590580695040000, 420271561157640192000, 15967576932074127360000

Offset: 0

Paul D. Hanna, Jan 03 2013

nonn

			G.f.: A(x) = 1 + 2*x + 13*x^2 + 100*x^3 + 984*x^4 + 11712*x^5 +...
where
A(x) = 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 +...

Cf. A187742, A187735.

{a(n)=polcoeff( sum(m=0, n, (2*m+x)^m*x^m/(1+2*m*x+x^2 +x*O(x^n))^m), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=if(n==0,1,if(n==1,2,(2*n^2+2*n+1)*2^(n-2)*(n-1)!))}
for(n=0, 30, print1(a(n), ", "))

{a(n)=n!*polcoeff(1/2 + 1/(2*(1-2*x)^2) - x/2 - log(1-2*x +x*O(x^n))/4, n)}
for(n=0, 30, print1(a(n), ", "))

a(n) = (2*n^2+2*n+1) * 2^(n-2) * (n-1)! for n>1 with a(0)=1, a(1)=2.
E.g.f.: 1/2 + 1/(2*(1-2*x)^2) - x/2 - log(1-2*x)/4.
E.g.f.: Sum_{n>=0} a(n+1)*x^n/n! = 2/(1-2*x)^3 + x/(1-2*x).

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

Paul D. Hanna, Jan 09 2013

nonn

			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 +...

Cf. A187742, A187741, A187735, A187746.

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 *)

{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), ", "))

{a(n)=if(n==0||n==1, 1, (n-1)*(n+2)/2 * (n-1)!)}
for(n=0, 30, print1(a(n), ", "))

{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), ", "))

a(n) = (n-1)*(n+2)/2 * (n-1)!, for n>1 with a(0)=a(1)=1.
E.g.f.: 1/2 + 1/(2*(1-x)^2) + x + log(1-x).
E.g.f.: Sum_{n>=0} a(n+1)*x^n/n! = 1/(1-x)^3 - x/(1-x).
From Amiram Eldar, Dec 23 2022: (Start)
Sum_{n>=0} 1/a(n) = Pi^2/9 + 43/27.
Sum_{n>=0} (-1)^n/a(n) = Pi^2/18 - 4*log(2)/9 + 5/27. (End)

A243227 G.f.: Sum_{n>=0} n^(2n) x^n / (1 + n^2*x)^n.

Original entry on oeis.org

1, 1, 15, 602, 46620, 5921520, 1118557440, 294293759760, 102896614941120, 46150861752777600, 25832386565857872000, 17651395149921751680000, 14460364581345685626624000, 13990151265412450143375360000, 15782226575197809064309171200000, 20533602558350213132577801792768000

Offset: 0

Paul D. Hanna, Aug 21 2014

nonn

Compare to: Sum_{n>=0} n^n * x^n / (1 + n*x)^n = 1 + Sum_{n>=1} (n+1)!/2 * x^n.

			G.f.: A(x) = 1 + x + 15*x^2 + 602*x^3 + 46620*x^4 + 5921520*x^5 +...
where
A(x) = 1 + x/(1+x) + 4^2*x^2/(1+4*x)^2 + 9^3*x^3/(1+9*x)^3 + 16^4*x^4/(1+16*x)^4 + 25^5*x^5/(1+25*x)^5 + 36^6*x^6/(1+36*x)^6 + 49^7*x^7/(1+49*x)^7 +...

Cf. A187742, A247238.

Flatten[{1, Table[(n-1)! * StirlingS2[2*n+1, n],{n,1,20}]}] (* Vaclav Kotesovec, Nov 05 2014 *)

{a(n)=polcoeff( sum(m=0, n, m^(2*m)*x^m/(1+m^2*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=if(n==0,1,sum(k=0, n-1, (-1)^(n-k-1) * binomial(n-1,k) * (k+1)^(2*n)))}
for(n=0,20,print1(a(n),", "))

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0,1, (n-1)! * Stirling2(2*n+1, n) )}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0,1,(2*n+1)!/n * polcoeff(((exp(x + x*O(x^(2*n+1))) - 1)^n), 2*n+1))}
for(n=0, 20, print1(a(n), ", "))

a(n) = Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n-1,k) * (k+1)^(2*n) for n>0 with a(0)=1.
a(n) = (n-1)! * Stirling2(2*n+1, n) for n>0 with a(0)=1.
a(n) = (2*n+1)!/n * [x^(2*n+1)] (exp(x) - 1)^n for n>0 with a(0)=1.
a(n) ~ 2^(2*n+1) * n^(2*n) / (sqrt(1-c) * exp(2*n) * c^n * (2-c)^(n+1)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... (see A226775). - Vaclav Kotesovec, Nov 05 2014

A203799 G.f.: Sum_{n>=0} (n-2x)^n x^n / (1 + nx - 2x^2)^n.

Original entry on oeis.org

1, 1, 1, 8, 48, 312, 2280, 18720, 171360, 1733760, 19232640, 232243200, 3033676800, 42631142400, 641383142400, 10287038361600, 175228365312000, 3159341273088000, 60111175348224000, 1203646256676864000, 25302180885037056000, 557134559872450560000, 12823826485099069440000

Offset: 0

Paul D. Hanna, Jan 09 2013

nonn

			G.f.: A(x) = 1 + x + x^2 + 8*x^3 + 48*x^4 + 312*x^5 + 2280*x^6 +...
where
A(x) = 1 + (1-2*x)*x/(1+x-2*x^2) + (2-2*x)^2*x^2/(1+2*x-2*x^2)^2 + (3-2*x)^3*x^3/(1+3*x-2*x^2)^3 + (4-2*x)^4*x^4/(1+4*x-2*x^2)^4 + (5-2*x)^5*x^5/(1+5*x-2*x^2)^5 +...

Cf. A202365, A187742.

{a(n)=polcoeff( sum(m=0, n, (m-2*x)^m*x^m/(1+m*x-2*x^2 +x*O(x^n))^m), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=if(n==0||n==1, 1, (n^2 + n - 4)/2 * (n-1)!)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=n!*polcoeff(1/2 + 1/(2*(1-x)^2) + 2*x + 2*log(1-x +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

a(n) = (n^2 + n - 4)/2 * (n-1)!, for n>1 with a(0)=a(1)=1.
E.g.f.: 1/2 + 1/(2*(1-x)^2) + 2*x + 2*log(1-x).
E.g.f.: Sum_{n>=0} a(n+1)*x^n/n! = 1/(1-x)^3 - 2*x/(1-x).

cp's OEIS Frontend

A204064 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + n*x) / (1 + k*x + n*x^2).

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A187741 G.f.: Sum_{n>=0} (1 + n*x)^n * x^n / (1 + x + n*x^2)^n.

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A204066 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (n + k*x) / (1 + n*x + k*x^2).

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A187746 G.f.: Sum_{n>=0} (2*n+x)^n * x^n / (1 + 2*n*x + x^2)^n.

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A202365 G.f.: Sum_{n>=0} (n-x)^n * x^n / (1 + n*x - x^2)^n.

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A243227 G.f.: Sum_{n>=0} n^(2*n) * x^n / (1 + n^2*x)^n.

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A203799 G.f.: Sum_{n>=0} (n-2*x)^n * x^n / (1 + n*x - 2*x^2)^n.

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A204064 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + nx) / (1 + kx + n*x^2).

A187741 G.f.: Sum_{n>=0} (1 + nx)^n x^n / (1 + x + n*x^2)^n.

A204066 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (n + kx) / (1 + nx + k*x^2).

A187746 G.f.: Sum_{n>=0} (2n+x)^n x^n / (1 + 2nx + x^2)^n.

A243227 G.f.: Sum_{n>=0} n^(2n) x^n / (1 + n^2*x)^n.

A203799 G.f.: Sum_{n>=0} (n-2x)^n x^n / (1 + nx - 2x^2)^n.