A187741 -id:A187741 - 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

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

Paul D. Hanna, Jan 03 2013

nonn

For values of n between 3 and 11 (possibly continuing) the number of conjugacy classes of the symmetric group S_n when conjugating by a single transposition. - Attila Egri-Nagy, Aug 15 2014

			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 +...

Cf. A202365, A187741, A187735, A187746.

List([3..11], n->Size(OrbitsDomain(Group((1,2)),SymmetricGroup(IsPermGroup, n), \^))); # Attila Egri-Nagy, Aug 15 2014

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 *)

{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^2+n+2)*(n-1)!/2)}
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), ", "))

x='x+O('x^66); concat([1], Vec(serlaplace(1/(1-x)^3 + x/(1-x)))) \\ Joerg Arndt, Aug 15 2014

a(n) = (n^2+n+2) * (n-1)!/2, 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).

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

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)

A208236 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + nkx) / (1 + x + nkx^2).

Original entry on oeis.org

1, 1, 1, 4, 10, 50, 208, 1290, 7456, 55982, 411796, 3650514, 32484460, 332970374, 3468625588, 40420787250, 481757564956, 6295577910182, 84407459209876, 1223095585594674, 18208380720893980, 289843786627539014, 4741844351895315028, 82269590167564595250

Offset: 0

Paul D. Hanna, Jan 10 2013

nonn

Compare to the identity:
Sum_{n>=0} x^n * Product_{k=1..n} (1 + t*k*x) / (1 + x + t*k*x^2) = (1+x)/(1-t*x^2).
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.
Limit n->infinity (a(n)/n!)^(1/n) = 1/(2*log(2)). - Vaclav Kotesovec, Nov 03 2014

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 10*x^4 + 50*x^5 + 208*x^6 + 1290*x^7 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1+2*1*x)*(1+2*2*x)/((1+x+2*1*x^2)*(1+x+2*2*x^2)) + x^3*(1+3*1*x)*(1+3*2*x)*(1+3*3*x)/((1+x+3*1*x^2)*(1+x+3*2*x^2)*(1+x+3*3*x^2)) + x^4*(1+4*1*x)*(1+4*2*x)*(1+4*3*x)*(1+4*4*x)/((1+x+4*1*x^2)*(1+x+4*2*x^2)*(1+x+4*3*x^2)*(1+x+4*4*x^2)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A204064, A187741, A204066.

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

A361382 The orders, with repetition, of subset-transitive permutation groups.

Original entry on oeis.org

1, 2, 3, 6, 12, 20, 24, 60, 120, 120, 360, 720, 2520, 5040, 20160, 40320, 181440, 362880, 1814400, 3628800, 19958400, 39916800, 239500800, 479001600, 3113510400, 6227020800, 43589145600, 87178291200, 653837184000, 1307674368000, 10461394944000, 20922789888000

Offset: 1

Hal M. Switkay, Mar 09 2023

nonn

If G is a permutation group on k letters, k > 0, then G induces a permutation of the subsets of size j for 0 <= j <= k. We call G subset-transitive if it has only one orbit of subsets for each j. G is subset-transitive if and only if it is (at least) floor(k/2)-transitive.
This restrictive condition admits only 1) symmetric groups of degree k for k >= 1, with order k! = A000142(k), which are k-transitive; 2) alternating groups of degree k for k >= 3, with order k!/2 = A001710(k), which are (k-2)-transitive; or 3) two exceptional groups, of orders 20 and 120.
The group of order 20 is AGL(1,5), which is 2-transitive on 5 letters.
The exceptional group of order 120 is PGL(2,5), which is 3-transitive on 6 letters, and contains AGL(1,5) as its one-point stabilizer. It is isomorphic as an abstract group, but not as a permutation group, to the symmetric group of degree 5. An outer automorphism of the symmetric group of degree 6 interchanges the two types of subgroup of order 120.

Hal M. Switkay, Table of n, a(n) for n = 1..48
Shreeram S. Abhyankar, Galois Theory on the Line in Non-Zero Characteristic, Bulletin of the AMS, 27 (1992), 68-133.

Cf. A000142, A001710, A187741.

A210443 G.f.: Sum_{n>=0} x^n * (1 + n^2x)^n / (1 + x + n^2x^2)^n.

Original entry on oeis.org

1, 1, 1, 6, 21, 150, 962, 8640, 80220, 884520, 10709520, 140873040, 2098741680, 32163828480, 568234774800, 9957054159360, 203333391011520, 4013297314266240, 92967912795139200, 2041979786688441600, 52890421861957680000, 1279950952105367942400, 36648398470742114918400

Offset: 0

Paul D. Hanna, Jan 20 2013

nonn

a(n) is divisible by ((n-1)/2)! for n>0.
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 + x^2 + 6*x^3 + 21*x^4 + 150*x^5 + 962*x^6 + 8640*x^7 +...
where
A(x) = 1 + (1+x)*x/(1+x+x^2) + (1+4*x)^2*x^2/(1+x+4*x^2)^2 + (1+9*x)^3*x^3/(1+x+9*x^2)^3 + (1+16*x)^4*x^4/(1+x+16*x^2)^4 + (1+25*x)^5*x^5/(1+x+25*x^2)^5 +...

Cf. A187741.

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

A222589 G.f. satisfies: A(x) = Sum_{n>=0} x^n(1 + nx)^n * A(x)^n / (1 + xA(x) + nx^2*A(x))^n.

Original entry on oeis.org

1, 1, 2, 5, 14, 41, 128, 409, 1355, 4564, 15728, 54904, 194740, 698042, 2532483, 9270351, 34268276, 127677731, 479723132, 1815553953, 6923744832, 26587139445, 102838915279, 400513959602, 1571152132075, 6206954038519, 24705172805012, 99071049959707, 400475021255313

Offset: 0

Paul D. Hanna, Feb 25 2013

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 41*x^5 + 128*x^6 + 409*x^7 +...
where:
A(x) = 1 + x*(1+x)*A(x)/(1+x*(1+x)*A(x)) + x^2*(1+2*x)^2*A(x)^2/(1+x*(1+2*x)*A(x))^2 + x^3*(1+3*x)^3*A(x)^3/(1+x*(1+3*x)*A(x))^3 + x^4*(1+4*x)^4*A(x)^4/(1+x*(1+4*x)*A(x))^4 +...
Also,
A(x) = 1/2 + (1 + 2*x*A(x))/2 * (1 + 2*x^2*A(x) + 6*x^4*A(x)^2 + 24*x^6*A(x)^3 + 120*x^8*A(x)^4 + 720*x^10*A(x)^5 + 5040*x^12*A(x)^6 +...).

Cf. A187741.

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0, n, (x+m*x^2)^m*A^m / (1 + x*A+m*x^2*A +x*O(x^n))^m));polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

{a(n)=local(A=1+x);for(i=1,n,A=1/2+(1+2*x*A)*sum(k=0,n,(k+1)!/2*x^(2*k)*(A+x*O(x^n))^k));polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

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

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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A187742 G.f.: Sum_{n>=0} (n+x)^n * x^n / (1 + n*x + 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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A202365 G.f.: Sum_{n>=0} (n-x)^n * x^n / (1 + n*x - x^2)^n.

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A208236 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + n*k*x) / (1 + x + n*k*x^2).

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A361382 The orders, with repetition, of subset-transitive permutation groups.

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A210443 G.f.: Sum_{n>=0} x^n * (1 + n^2*x)^n / (1 + x + n^2*x^2)^n.

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A222589 G.f. satisfies: A(x) = Sum_{n>=0} x^n*(1 + n*x)^n * A(x)^n / (1 + x*A(x) + n*x^2*A(x))^n.

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

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

A208236 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + nkx) / (1 + x + nkx^2).

A210443 G.f.: Sum_{n>=0} x^n * (1 + n^2x)^n / (1 + x + n^2x^2)^n.

A222589 G.f. satisfies: A(x) = Sum_{n>=0} x^n(1 + nx)^n * A(x)^n / (1 + xA(x) + nx^2*A(x))^n.