A051577 -id:A051577 - cp's OEIS frontend

A136394 Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).

1, 1, 1, 1, 1, 5, 1, 20, 3, 1, 84, 35, 1, 409, 295, 15, 1, 2365, 2359, 315, 1, 16064, 19670, 4480, 105, 1, 125664, 177078, 56672, 3465, 1, 1112073, 1738326, 703430, 74025, 945, 1, 10976173, 18607446, 8941790, 1346345, 45045, 1, 119481284, 216400569, 118685336

Offset: 0

Vladeta Jovovic, May 03 2008

			Triangle (n,k) begins:
  1;
  1;
  1,    1;
  1,    5;
  1,   20,    3;
  1,   84,   35;
  1,  409,  295,  15;
  1, 2365, 2359, 315;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Jean-Luc Baril and Sergey Kirgizov, Transformation à la Foata for special kinds of descents and excedances, arXiv:2101.01928 [math.CO], 2021. See Theorem 2. p. 5.
FindStat - Combinatorial Statistic Finder, The number of nontrivial cycles of a permutation pi in its cycle decomposition
Bin Han, Jianxi Mao, and Jiang Zeng, Equidistributions around special kinds of descents and excedances, arXiv:2103.13092 [math.CO], 2021, see page 2.

Columns k=0-10 give: A000012, A006231, A289950, A289951, A289952, A289953, A289954, A289955, A289956, A289957, A289958.
Row sums give A000142.
Cf. A000276, A000483, A001147, A001705, A051577, A124324, A159964.

egf:= proc(k::nonnegint) option remember; x-> exp(x)* ((-x-ln(1-x))^k)/k! end; T:= (n,k)-> coeff(series(egf(k)(x), x=0, n+1), x, n) *n!; seq(seq(T(n,k), k=0..n/2), n=0..30); # Alois P. Heinz, Aug 14 2008
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-i)*
      `if`(i>1, x, 1)*binomial(n-1, i-1)*(i-1)!, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..15);  # Alois P. Heinz, Sep 25 2016
# third Maple program:
T:= proc(n, k) option remember; `if`(k<0 or k>2*n, 0,
      `if`(n=0, 1, add(T(n-i, k-`if`(i>1, 1, 0))*
       mul(n-j, j=1..i-1), i=1..n)))
    end:
seq(seq(T(n,k), k=0..n/2), n=0..15);  # Alois P. Heinz, Jul 16 2017

max = 12; egf = Exp[x*(1-y)]/(1-x)^y; s = Series[egf, {x, 0, max}, {y, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]*n!; t[0, 0] = t[1, 0] = 1; Table[t[n, k], {n, 0, max}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Jan 28 2014 *)

E.g.f.: exp(x*(1-y))/(1-x)^y. Binomial transform of triangle A008306. exp(x)*((-x-log(1-x))^k)/k! is e.g.f. of k-th column.
From Alois P. Heinz, Jul 13 2017: (Start)
T(2n,n) = A001147(n).
T(2n+1,n) = A051577(n) = (2*n+3)!!/3 = A001147(n+2)/3. (End)
From Alois P. Heinz, Aug 17 2023: (Start)
Sum_{k=0..floor(n/2)} k * T(n,k) = A001705(n-1) for n>=1.
Sum_{k=0..floor(n/2)} (-1)^k * T(n,k) = A159964(n-1) for n>=1. (End)

A051578 a(n) = (2*n+4)!!/4!!, related to A000165 (even double factorials).

Original entry on oeis.org

1, 6, 48, 480, 5760, 80640, 1290240, 23224320, 464486400, 10218700800, 245248819200, 6376469299200, 178541140377600, 5356234211328000, 171399494762496000, 5827582821924864000, 209792981589295104000, 7972133300393213952000, 318885332015728558080000

Offset: 0

Wolfdieter Lang

Row m=4 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..400
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 521.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1), A051577 (rows m=0..3), A051579, A051580, A051581, A051582, A051583.

Cf. A052587 (essentially the same).

List([0..20], n-> 2^(n-1)*Factorial(n+2) ); # G. C. Greubel, Nov 11 2019

[2^(n-1)*Factorial(n+2): n in [0..20]]; // G. C. Greubel, Nov 11 2019

a:= proc(n) option remember; `if`(n=0, 1, 2*(n+2)*a(n-1)) end:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 29 2019
seq(2^(n-1)*(n+2)!, n=0..20); # G. C. Greubel, Nov 11 2019

Table[2^(n-1)(n+2)!, {n,0,20}] (* Jean-François Alcover, Oct 05 2019 *)
Table[(2n+4)!!/8,{n,0,20}] (* Harvey P. Dale, Apr 06 2023 *)

vector(21, n, 2^(n-2)*(n+1)! ) \\ G. C. Greubel, Nov 11 2019

apply( {A051578(n)=(n+2)!<<(n-1)}, [0..18]) \\ M. F. Hasler, Nov 10 2024

[2^(n-1)*factorial(n+2) for n in (0..20)] # G. C. Greubel, Nov 11 2019

a(n) = (2*n+4)!!/4!!.
E.g.f.: 1/(1-2*x)^3.
a(n) ~ 2^(-1/2)*Pi^(1/2)*n^(5/2)*2^n*e^-n*n^n*{1 + 37/12*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
a(n) = (n+2)!*2^(n-1). - Zerinvary Lajos, Sep 23 2006. [corrected by Gary Detlefs, Apr 29 2019]
a(n) = 2^n*A001710(n+2). - R. J. Mathar, Feb 22 2008
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 6)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 6*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 6*x/(1 - 2*x/(1 - 8*x/(1 - 4*x/(1 - 10*x/(1 - 6*x/(1 - ... - (2*n + 4)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 6*x/(1 - 8*x/(1 - 2*x/(1 - 10*x/(1 - 4*x/(1 - 12*x/(1 - 6*x/(1 - ... - (2*n + 6)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 8*sqrt(e) - 12.
Sum_{n>=0} (-1)^n/a(n) = 8/sqrt(e) - 4. (End)
a(n) = A052587(n+2) for n > 0. - M. F. Hasler, Nov 10 2024

A051580 a(n) = (2*n+6)!!/6!!, related to A000165 (even double factorials).

Original entry on oeis.org

1, 8, 80, 960, 13440, 215040, 3870720, 77414400, 1703116800, 40874803200, 1062744883200, 29756856729600, 892705701888000, 28566582460416000, 971263803654144000, 34965496931549184000, 1328688883398868992000

Offset: 0

Wolfdieter Lang

Row m=6 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..400
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1), A051577, A051578, A051579 (rows m=0..5), A051581, A051582, A051583.

List([0..20], n-> Product([1..n], j-> 2*j+6) ); # G. C. Greubel, Nov 11 2019

[1] cat [(&*[2*j+6: j in [1..n]]): n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(2*j+6, j=1..n), n=0..20); # G. C. Greubel, Nov 11 2019

Table[2^n*Pochhammer[4, n], {n,0,20}] (* G. C. Greubel, Nov 11 2019 *)
Table[(2n+6)!!/6!!,{n,0,20}] (* Harvey P. Dale, Mar 03 2022 *)

vector(20, n, prod(j=1,n-1, 2*j+6) ) \\ G. C. Greubel, Nov 11 2019

[product( (2*j+6) for j in (1..n)) for n in (0..20)] # G. C. Greubel, Nov 11 2019

a(n) = (2*n+6)!!/6!!.
E.g.f.: 1/(1-2*x)^4.
a(n) = n!*2^(n-4)/3, n>=3. - Zerinvary Lajos, Sep 23 2006
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x/(x + 1/(2*k+8)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 8)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 8*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 8*x/(1 - 2*x/(1 - 10*x/(1 - 4*x/(1 - 12*x/(1 - 6*x/(1 - ... - (2*n + 6)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 8*x/(1 - 10*x/(1 - 2*x/(1 - 12*x/(1 - 4*x/(1 - 14*x/(1 - 6*x/(1 - ... - (2*n + 8)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 48*sqrt(e) - 78.
Sum_{n>=0} (-1)^n/a(n) = 30 - 48/sqrt(e). (End)

A051579 a(n) = (2*n+5)!!/5!!, related to A001147 (odd double factorials).

Original entry on oeis.org

1, 7, 63, 693, 9009, 135135, 2297295, 43648605, 916620705, 21082276215, 527056905375, 14230536445125, 412685556908625, 12793252264167375, 422177324717523375, 14776206365113318125, 546719635509192770625

Offset: 0

Wolfdieter Lang

Row m=5 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..400
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1), A051577, A051578 (rows m=0..4).

Cf. A051580, A051581, A051582, A051583, A178647.

List([0..20], n-> Product([0..n-1], j-> 2*j+7) ); # G. C. Greubel, Nov 12 2019

[1] cat [(&*[2*j+7: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 12 2019

df:=doublefactorial; seq(df(2*n+5)/df(5), n = 0..20); # G. C. Greubel, Nov 12 2019

Table[2^n*Pochhammer[7/2, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)

vector(20, n, prod(j=1,n-1, 2*j+5) ) \\ G. C. Greubel, Nov 12 2019

[product( (2*j+7) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = (2*n+5)!!/4!!.
E.g.f.: 1/(1-2*x)^(7/2).
a(n) ~ 8/15*sqrt(2)*n^3*2^n*e^-n*n^n*(1 + 107/24*n^-1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
G.f.: G(0)/(10*x) -1/(5*x), where G(k)= 1 + 1/(1 - x*(2*k+5)/(x*(2*k+5) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 13 2013
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 7)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 7*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 7*x/(1 - 2*x/(1 - 9*x/(1 - 4*x/(1 - 11*x/(1 - 6*x/(1 - ... - (2*n + 5)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 7*x/(1 - 9*x/(1 - 2*x/(1 - 11*x/(1 - 4*x/(1 - 13*x/(1 - 6*x/(1 - ... - (2*n + 7)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 15 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 20, where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 15 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)) - 10, where erfi is the imaginary error function. (End)

A051582 a(n) = (2*n+8)!!/8!!, related to A000165 (even double factorials).

Original entry on oeis.org

1, 10, 120, 1680, 26880, 483840, 9676800, 212889600, 5109350400, 132843110400, 3719607091200, 111588212736000, 3570822807552000, 121407975456768000, 4370687116443648000, 166086110424858624000, 6643444416994344960000

Offset: 0

Wolfdieter Lang

Row m=8 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..399
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1).

Cf. A051577, A051578, A051579, A051580, A051581 (rows m=0..7), A051583.

F:=Factorial;; List([0..20], n-> 2^n*F(n+4)/F(4) ); # G. C. Greubel, Nov 12 2019

F:=Factorial; [2^n*F(n+4)/F(4): n in [0..20]]; // G. C. Greubel, Nov 12 2019

seq(2^n*pochhammer(5, n), n=0..20); # G. C. Greubel, Nov 12 2019

(2Range[0,20]+8)!!/8!! (* Harvey P. Dale, Feb 03 2013 *)
Table[2^n*Pochhammer[5, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)

vector(20, n, n--; (n+4)!*2^(n-1)/12) \\ Michel Marcus, Feb 09 2015

f=factorial; [2^n*f(n+4)/f(4) for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = (2*n+8)!!/8!!.
E.g.f.: 1/(1-2*x)^5.
a(n) = (n+4)!*2^(n-1)/12. - Zerinvary Lajos, Sep 23 2006
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 10)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 10*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 10*x/(1 - 2*x/(1 - 12*x/(1 - 4*x/(1 - 14*x/(1 - 6*x/(1 - ... - (2*n + 8)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 10*x/(1 - 12*x/(1 - 2*x/(1 - 14*x/(1 - 4*x/(1 - 16*x/(1 - 6*x/(1 - ... - (2*n + 10)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 384*sqrt(e) - 632.
Sum_{n>=0} (-1)^n/a(n) = 384/sqrt(e) - 232. (End)

A051581 a(n) = (2*n+7)!!/7!!, related to A001147 (odd double factorials).

Original entry on oeis.org

1, 9, 99, 1287, 19305, 328185, 6235515, 130945815, 3011753745, 75293843625, 2032933777875, 58955079558375, 1827607466309625, 60311046388217625, 2110886623587616875, 78102805072741824375, 3046009397836931150625

Offset: 0

Wolfdieter Lang

Row m=7 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..400
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1).

Cf. A051577, A051578, A051579, A051580 (rows m=0..6), A051582, A051583.

List([0..20], n-> Product([0..n-1], j-> 2*j+9) ); # G. C. Greubel, Nov 12 2019

[1] cat [(&*[2*j+9: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 12 2019

df:=doublefactorial; seq(df(2*n+7)/df(7), n = 0..20); # G. C. Greubel, Nov 12 2019

Table[2^n*Pochhammer[9/2, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)

vector(20, n, prod(j=1,n-1, 2*j+7) ) \\ G. C. Greubel, Nov 12 2019

[product( (2*j+9) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = (2*n+7)!!/7!!.
E.g.f.: 1/(1-2*x)^(9/2).
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x/(x + 1/(2*k+9)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 9)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 9*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 9*x/(1 - 2*x/(1 - 11*x/(1 - 4*x/(1 - 13*x/(1 - 6*x/(1 - ... - (2*n + 7)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 9*x/(1 - 11*x/(1 - 2*x/(1 - 13*x/(1 - 4*x/(1 - 15*x/(1 - 6*x/(1 - ... - (2*n + 9)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 105 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 147, where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 77 - 105 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)), where erfi is the imaginary error function. (End)

A051583 a(n) = (2*n+9)!!/9!!, related to A001147 (odd double factorials).

Original entry on oeis.org

1, 11, 143, 2145, 36465, 692835, 14549535, 334639305, 8365982625, 225881530875, 6550564395375, 203067496256625, 6701227376468625, 234542958176401875, 8678089452526869375, 338445488648547905625, 13876265034590464130625

Offset: 0

Wolfdieter Lang

Row m=9 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..398
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1), A178647.

Cf. A051577, A051578, A051579, A051580, A051581, A051582 (rows m=0..8).

List([0..20], n-> Product([0..n-1], j-> 2*j+11) ); # G. C. Greubel, Nov 12 2019

[1] cat [(&*[2*j+11: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 12 2019

seq(2^n*pochhammer(11/2,n), n = 0..20); # G. C. Greubel, Nov 12 2019

(2*Range[0,20]+9)!!/945 (* Harvey P. Dale, Apr 10 2019 *)
Table[2^n*Pochhammer[11/2, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)

vector(20, n, prod(j=0,n-2, 2*j+11) ) \\ G. C. Greubel, Nov 12 2019

[product( (2*j+11) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = (2*n+9)!!/9!!.
E.g.f.: 1/(1-2*x)^(11/2).
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 11)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 11*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 11*x/(1 - 2*x/(1 - 13*x/(1 - 4*x/(1 - 15*x/(1 - 6*x/(1 - ... - (2*n + 9)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 11*x/(1 - 13*x/(1 - 2*x/(1 - 15*x/(1 - 4*x/(1 - 17*x/(1 - 6*x/(1 - ... - (2*n + 11)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 945 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 1332, where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 945 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)) - 684, where erfi is the imaginary error function. (End)

A000906 Exponential generating function: 2(1+3x)/(1-2*x)^(7/2).

Original entry on oeis.org

2, 20, 210, 2520, 34650, 540540, 9459450, 183783600, 3928374450, 91662070500, 2319050383650, 63246828645000, 1849969737866250, 57775977967207500, 1918987839625106250, 67548371954803740000, 2511955082069264081250

Offset: 0

N. J. A. Sloane

nonn

Ramanujan polynomials -psi_{n+2}(n+2,x) evaluated at 1.
With offset 2, second Eulerian transform of 0,1,2,3,4... - Ross La Haye, Mar 05 2005
With offset 1, a strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Dec 30 2016

			G.f. = 2 + 20*x + 210*x^2 + 2520*x^3 + 34650*x^4 + 540540*x^5 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.
C. Jordan, Calculus of Finite Differences. Budapest, 1939, p. 152.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
H. W. Gould, Harris Kwong, Jocelyn Quaintance, On Certain Sums of Stirling Numbers with Binomial Coefficients, J. Integer Sequences, 18 (2015), #15.9.6.
C. Jordan, On Stirling's Numbers, Tohoku Math. J., 37 (1933), 254-278.

a(n) = 2*A000457(n) = A051577(n+1) - A001147(n+2).

Negative coefficient of x of polynomials in A098503.

[Factorial(2*n+3)/(6*Factorial(n)*2^(n-1)): n in [0..30]]; // G. C. Greubel, May 15 2018

Table[(2 n + 5)!!/3 - (2 n + 3)!!, {n, 0, 20}] (* Vincenzo Librandi, Apr 11 2012 *)

a(n)=(2*n+6)!/(n+3)!/2^(n+3)/3-(2*n+4)!/(n+2)!/2^(n+2)

a(n) = (2n+5)!!/3 - (2n+3)!!.
a(n) -2*(n+4)*a(n-1) +3*(2*n+1)*a(n-2) = 0. - R. J. Mathar, Feb 20 2013
a(n) ~ 2^(n+7/2)*n^(n+3)/(3*exp(n)). - Ilya Gutkovskiy, Aug 17 2016
a(n) = (2n+3)!/( 3!*n!*2^(n-1) ). - G. C. Greubel, May 15 2018

A112292 An invertible triangle of ratios of double factorials.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 15, 15, 5, 1, 105, 105, 35, 7, 1, 945, 945, 315, 63, 9, 1, 10395, 10395, 3465, 693, 99, 11, 1, 135135, 135135, 45045, 9009, 1287, 143, 13, 1, 2027025, 2027025, 675675, 135135, 19305, 2145, 195, 15, 1, 34459425, 34459425, 11486475, 2297295, 328185, 36465, 3315, 255, 17, 1

Offset: 0

Paul Barry, Sep 01 2005

As a square array read by antidiagonals, column k has e.g.f. (1/(1-2x)^(1/2))*(1/(1-2x))^k. - Paul Barry, Sep 04 2005

Let G(m, k, p) = (-p)^k*Product_{j=0..k-1}(j - m - 1/p) and T(n, k, p) = G(n-1, n-k, p) then T(n, k, 1) = A094587(n, k), T(n, k, 2) is this sequence and T(n, k, 3) = A136214. - Peter Luschny, Jun 01 2009, revised Jun 18 2019

			Triangle begins
      1;
      1,     1;
      3,     3,    1;
     15,    15,    5,  1;
    105,   105,   35,  7,  1;
    945,   945,  315, 63,  9,  1;
  10395, 10395, 3465,693, 99, 11, 1;
Inverse is A112295, which begins
   1;
  -1,  1;
   0, -3,  1;
   0,  0, -5,  1;
   0,  0,  0, -7,  1;
   0,  0,  0,  0, -9,  1;
Similar results arise for higher factorials.

Columns include A001147, A051577, A051579.
Row sums are A112293.
Diagonal sums are A112294.
Cf. A094587 (p=1), this sequence (p=2), A136214 (p=3).

T[n_, k_] := If[k <= n, (2n-1)!!/(2k-1)!!, 0];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

T(n, k)=if(k<=n, (2n-1)!!/(2k-1)!!, 0);
T(n, k)=if(k<=n, n!*C(2n, n)2^(k-n)/(k!*C(2k, k)), 0);
T(n, k)=if(k<=n, 2^(n-k)(n-1/2)!/(k-1/2)!, 0);
T(n, k)=if(k<=n, (n+1)!*C(n)2^(k-n)/((k+1)!*C(k)), 0).

A288950 Number of relaxed compacted binary trees of right height at most one with empty initial and final sequence on level 0.

Original entry on oeis.org

1, 0, 1, 2, 15, 140, 1575, 20790, 315315, 5405400, 103378275, 2182430250, 50414138775, 1264936572900, 34258698849375, 996137551158750, 30951416768146875, 1023460181133390000, 35885072600989486875, 1329858572860198631250, 51938365373373313209375

Offset: 0

Michael Wallner, Jun 20 2017

nonn

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. The number of unbounded relaxed compacted binary trees of size n is A082161(n). The number of relaxed compacted binary trees of right height at most one of size n is A001147(n). See the Genitrini et al. and Wallner link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where node 3 is at depth 1 on the right of node 2 and where the node n+1 has a left sibling. See the Wallner link. - Michael Wallner, Apr 20 2017

			Denote by L the leaf and by o nodes. Every node has exactly two out-going edges or pointers. Internal edges are denoted by - or |. Pointers are omitted and may point to any node further right. The root is at level 0 at the very left.
The general structure is
  L-o-o-o-o-o-o-o-o-o
    |       |     | |
    o   o-o-o   o-o o.
For n=0 the a(0)=1 solution is L.
For n=1 we have a(1)=0 because we need nodes on level 0 and level 1.
For n=2 the a(2)=1 solution is
     L-o
       |
       o
and the pointers of the node on level 1 both point to the leaf.
For n=3 the a(3)=2 solutions have the structure
     L-o
       |
     o-o
where the pointers of the last node have to point to the leaf, but the pointer of the next node has 2 choices: the leaf of the previous node.

Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.

Cf. A001147 (relaxed compacted binary trees of right height at most one).
Cf. A082161 (relaxed compacted binary trees of unbounded right height).
Cf. A000032, A000246, A001879, A051577, A177145, A213527, A288950, A288952, A288953, A288954 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A001879.

terms = 21; (z + (1 - z)/3*(2 - z + (1 - 2z)^(-1/2)) + O[z]^terms // CoefficientList[#, z] &) Range[0, terms-1]! (* Jean-François Alcover, Dec 04 2018 *)

E.g.f.: z + (1-z)/3 * (2-z + (1-2*z)^(-1/2)).
From Seiichi Manyama, Apr 26 2025: (Start)
a(n) = (n-1)*(2*n-3)/(n-2) * a(n-1) for n > 3.
a(n) = A001879(n-2)/3 for n > 2. (End)

cp's OEIS Frontend

A136394 Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).

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A051578 a(n) = (2*n+4)!!/4!!, related to A000165 (even double factorials).

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A051580 a(n) = (2*n+6)!!/6!!, related to A000165 (even double factorials).

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A051579 a(n) = (2*n+5)!!/5!!, related to A001147 (odd double factorials).

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A051582 a(n) = (2*n+8)!!/8!!, related to A000165 (even double factorials).

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A051581 a(n) = (2*n+7)!!/7!!, related to A001147 (odd double factorials).

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A000906 Exponential generating function: 2(1+3x)/(1-2*x)^(7/2).