A202365 -id:A202365 - cp's OEIS frontend

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

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).

A355570 Regular triangle of certain polynomial expansion coefficients for the n-th power series.

Original entry on oeis.org

1, 0, 1, 1, -2, 2, 0, 5, -10, 6, 1, -10, 40, -54, 24, 0, 21, -140, 336, -336, 120, 1, -42, 462, -1764, 3024, -2400, 720, 0, 85, -1470, 8442, -22176, 29520, -19440, 5040, 1, -170, 4580, -38178, 144648, -288000, 313200, -176400, 40320, 0, 341, -14080, 166452, -875952, 2451240, -3920400, 3603600, -1774080, 362880

Offset: 2

Michel Marcus, Jul 07 2022

See the paper by Muschielok for precise definition.

			Triangle begins:
  1;
  0,   1;
  1,  -2,    2;
  0,   5,  -10,   6;
  1, -10,   40, -54,   24;
  0,  21, -140, 336, -336, 120;
  ...

Michel Marcus, Table of n, a(n) for n = 2..4951 (Rows 2..100)
Christoph Muschielok, Another Approach on Power Sums, arXiv:2207.01935 [math.CO], 2022.
Christoph Muschielok, Further Identities for cmk- and amk-Weighted Sums and a Remark on a Representation of Pythagoras' Equation, arXiv:2208.11032 [math.NT], 2022.

Cf. A000142 (right diagonal), A202365 (subdiagonal).

mat(n) = my(M = matrix(n, n)); M[1, 1] = 1; for (i=2, n, my(p=x + prod(k=-1, i-2, x+k)/(i-2)!); for (j=1, i, M[i, j] = polcoef(p, j, x));); my(iM = 1/M); matrix(n-1, n-1, i, j, iM[i+1, j+1]);
tabl(nn) = {my(m = mat(nn)); for (n=1, nn-1, row = vector(n, k, m[n, k]); print(row, ", "););}

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

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

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A355570 Regular triangle of certain polynomial expansion coefficients for the n-th power series.

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

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cp's OEIS Frontend

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

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A355570 Regular triangle of certain polynomial expansion coefficients for the n-th power series.

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PARI

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

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