A130915 -id:A130915 - cp's OEIS frontend

A365974 Expansion of e.g.f. exp( Sum_{k>=0} x^(5k+3) / (5k+3) ).

1, 0, 0, 2, 0, 0, 40, 0, 5040, 2240, 0, 1663200, 246400, 479001600, 605404800, 44844800, 699941088000, 274450176000, 355699625881600, 836634972096000, 156436600320000, 1437392253237248000, 1021561084051200000, 1124111547465274368000

Offset: 0

Seiichi Manyama, Sep 23 2023

nonn

Cf. A038205, A130915, A365973.

Cf. A365982.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=0, N\5, x^(5*k+3)/(5*k+3)))))

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-3)/5)} a(n-5*k-3)/(n-5*k-3)!.

A054479 Number of sets of cycle graphs of 2n nodes where the 2-colored edges alternate colors.

Original entry on oeis.org

1, 0, 6, 120, 6300, 514080, 62785800, 10676746080, 2413521910800, 700039083744000, 253445583029839200, 112033456760809584000, 59382041886244720843200, 37175286835046004765120000, 27139206193305890195912400000, 22852066417535931447551359680000

Offset: 0

Christian G. Bower, Mar 29 2000

nonn

Also number of permutations in the symmetric group S_2n in which cycle lengths are even and greater than 2, cf. A130915. - Vladeta Jovovic, Aug 25 2007

a(n) is also the number of ordered pairs of disjoint perfect matchings in the complete graph on 2n vertices. The sequence A006712 is the number of ordered triples of perfect matchings. - Matt Larson, Jul 23 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001147, A001818, A053871, A006712.

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-2*j)*binomial(n-1, 2*j-1)*(2*j-1)!, j=2..n/2))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..15);  # Alois P. Heinz, Mar 06 2023

Table[(n-1)*(2*n)!^2 * HypergeometricPFQ[{2-n},{3/2-n},-1/2] / (4^n*(n-1/2)*(n!)^2), {n, 0, 20}] (* Vaclav Kotesovec, Mar 29 2014 after Mark van Hoeij *)

x='x+O('x^66); v=Vec(serlaplace(1/(sqrt(exp(x^2)*(1-x^2))))); vector(#v\2,n,v[2*n-1]) \\ Joerg Arndt, May 13 2013

If b(2n)=a(n) then e.g.f. of b is 1/(sqrt(exp(x^2)*(1-x^2))).
a(n) = 4^n*(n-1)*gamma(n+1/2)^2*hypergeom([2-n],[3/2-n],-1/2)/(Pi*(n-1/2)). - Mark van Hoeij, May 13 2013
a(n) ~ 2^(2*n+1) * n^(2*n) / exp(2*n+1/2). - Vaclav Kotesovec, Mar 29 2014

A365980 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(2k+3) / (2k+3) ).

Original entry on oeis.org

1, 0, 0, 2, 0, 24, 80, 720, 5376, 53760, 490752, 6289920, 68766720, 1024607232, 13520332800, 226177695744, 3498759290880, 65257155624960, 1153246338220032, 23793010526453760, 472374431008948224, 10686755493583257600, 235406405307208826880

Offset: 0

Seiichi Manyama, Sep 23 2023

nonn

Cf. A355284, A365981, A365982.

Cf. A130915, A296676.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x-atanh(x))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-3)/2)} (2*k+2)! * binomial(n,2*k+3) * a(n-2*k-3).

E.g.f.: 1 / ( 1 + x - arctanh(x) ).

A138022 Triangular array read by rows: e.g.f. sqrt(1-z^2)exp(xz)/(1+z).

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -3, 3, -3, 1, 9, -12, 6, -4, 1, -45, 45, -30, 10, -5, 1, 225, -270, 135, -60, 15, -6, 1, -1575, 1575, -945, 315, -105, 21, -7, 1, 11025, -12600, 6300, -2520, 630, -168, 28, -8, 1, -99225, 99225, -56700, 18900, -5670, 1134, -252, 36, -9, 1, 893025, -992250, 496125, -189000, 47250, -11340, 1890, -360, 45, -10, 1

Offset: 1

Roger L. Bagula, May 01 2008

			Triangle starts:
       1;
      -1,      1;
       1,     -2,      1;
      -3,      3,     -3,     1;
       9,    -12,      6,    -4,     1;
     -45,     45,    -30,    10,    -5,    1;
     225,   -270,    135,   -60,    15,   -6,    1;
   -1575,   1575,   -945,   315,  -105,   21,   -7,  1;
   11025, -12600,   6300, -2520,   630, -168,   28, -8,  1;
  -99225,  99225, -56700, 18900, -5670, 1134, -252, 36, -9, 1;
  ...

Cf. A130915 (row sums).

g := sqrt(1-z^2)*exp(x*z)/(1+z); gser := n -> series(g, z, n+2):
seq(print(seq(coeff(n!*coeff(gser(n),z,n),x,i),i=0..n)),n=0..10); # Peter Luschny, Aug 21 2014

max=10; g=Exp[x*z]*Sqrt[(1-z)/(1+z)]; gser=Series[g,{z,0,max}]; p[n_]:=n!*Coefficient[gser,z,n]; T[n_,k_]:=Coefficient[p[n],x,k]; Flatten[Table[T[n,k],{n,0,max},{k,0,n}]]
T[n_, k_] := If[n==k, 1, (-1)^(n + k)*(n - k)!*Sum[Sum[2^(j - i)*StirlingS1[j, i]*Binomial[n - k - 1, j - 1]/j!, {j, i, n - k}], {i, 1, n - k}]*Binomial[n, k]];Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Detlef Meya, Jan 16 2024 *)

The unsigned version has the e.g.f. exp(x*z)/sqrt((1-z)/(1+z)). - Peter Luschny, Aug 21 2014
T(n+3,k+1) = T(n+2,k) - T(n+2,k+1) + (n+1)*(n+2)*(T(n+1,k+1)-T(n,k)) with T(n,n) = 1, T(n,n-1) = -n, T(n+2,0) = T(n+1,0) + (n^2+n)*T(n,0). - Robert Israel, Aug 21 2014
T(n, k) = 1 if n = k, otherwise (-1)^(n+k)*(n-k)!*Sum_{i = 1..n-k} (Sum_{j = i..n-k} 2^(j-i)*Stirling1(j, i)*binomial(n-k-1, j-1)/j!)*binomial(n, k). - Detlef Meya, Jan 16 2024

Edited by Peter Luschny and Joerg Arndt, Aug 21 2014

A365973 Expansion of e.g.f. exp( Sum_{k>=0} x^(4k+3) / (4k+3) ).

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 40, 720, 0, 2240, 172800, 3628800, 246400, 49420800, 3531340800, 87223136000, 18450432000, 3006222336000, 225434879488000, 6411312940032000, 2744461025280000, 435228787435520000, 35074217524469760000, 1126838040745697280000

Offset: 0

Seiichi Manyama, Sep 23 2023

nonn

Cf. A038205, A130915, A365974.

Cf. A365981.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=0, N\4, x^(4*k+3)/(4*k+3)))))

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-3)/4)} a(n-4*k-3)/(n-4*k-3)!.

cp's OEIS Frontend

A365974 Expansion of e.g.f. exp( Sum_{k>=0} x^(5*k+3) / (5*k+3) ).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A054479 Number of sets of cycle graphs of 2n nodes where the 2-colored edges alternate colors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A365980 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(2*k+3) / (2*k+3) ).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A138022 Triangular array read by rows: e.g.f. sqrt(1-z^2)*exp(x*z)/(1+z).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A365973 Expansion of e.g.f. exp( Sum_{k>=0} x^(4*k+3) / (4*k+3) ).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A365974 Expansion of e.g.f. exp( Sum_{k>=0} x^(5k+3) / (5k+3) ).

A365980 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(2k+3) / (2k+3) ).

A138022 Triangular array read by rows: e.g.f. sqrt(1-z^2)exp(xz)/(1+z).

A365973 Expansion of e.g.f. exp( Sum_{k>=0} x^(4k+3) / (4k+3) ).