A001147 -id:A001147 - cp's OEIS frontend

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

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)

A133221 A001147 with each term repeated.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 15, 15, 105, 105, 945, 945, 10395, 10395, 135135, 135135, 2027025, 2027025, 34459425, 34459425, 654729075, 654729075, 13749310575, 13749310575, 316234143225, 316234143225, 7905853580625, 7905853580625, 213458046676875, 213458046676875

Offset: 0

N. J. A. Sloane, Oct 13 2007

nonn

Normally such sequences are excluded from the OEIS, but I have made an exception for this one because so many variants of it have occurred in recent submissions.
For n>=2, a(n) = product of odd positive integers <=(n-1). - Jaroslav Krizek, Mar 21 2009
a(n) is, for n>=3, the number of way to choose floor((n-1)/2) disjoint pairs of items from n-1 items. It is then a fortiori the size of the conjugacy class of the reversal permutation [n-1,n-2,n-3,...,3,2,1]=(1 n-1)(2 n-2)(3 n-3)... in the symmetric group on n-1 elements. - Karl-Dieter Crisman, Nov 03 2009

G. C. Greubel, Table of n, a(n) for n = 0..800

Cf. A006882, A055634.

Appears in A161736. - Johannes W. Meijer, Jun 18 2009

f[x_] := E^(x^2/2) + Sqrt[Pi/2]*Erfi[x/Sqrt[2]]; CoefficientList[ Series[f[x], {x, 0, 29}], x]*Range[0, 29]! (* Jean-François Alcover, Sep 25 2012, after Sergei N. Gladkovskii *)
Table[(n - 1 - Mod[n, 2])!!, {n, 0, 20}] (* Eric W. Weisstein, Dec 31 2017 *)
Table[((2 n + (-1)^n - 3)/2)!!, {n, 0, 20}] (* Eric W. Weisstein, Dec 31 2017 *)

a(n) = my(k = (2*n + (-1)^n - 3)/2); prod(i=0, (k-1)\2, k - 2*i) \\ Iain Fox, Dec 31 2017

def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
def A133221(n): return Gauss_factorial(n-1, 2)
[A133221(n) for n in (0..29)]  # Peter Luschny, Oct 01 2012

E.g.f.: x*U(0)  where U(k)= 1 + (2*k+1)/(x - x^4/(x^3 + (2*k+2)*(2*k+3)/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 25 2012
G.f.: 1+x*G(0), where G(k)= 1 + x*(2*k+1)/(1 - x/(x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013
a(n) = (2*floor(n/2)-1)!! = (n-1-(n mod 2))!!. - Alois P. Heinz, Sep 24 2024

A185411 A triangular decomposition of the double factorial numbers A001147.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 10, 1, 0, 8, 60, 36, 1, 0, 16, 296, 516, 116, 1, 0, 32, 1328, 5168, 3508, 358, 1, 0, 64, 5664, 42960, 64240, 21120, 1086, 1, 0, 128, 23488, 320064, 900560, 660880, 118632, 3272, 1, 0, 256, 95872, 2225728, 10725184, 14713840, 6049744, 638968, 9832, 1

Offset: 0

Paul Barry, Jan 26 2011

Row sums are A001147. Reversal of A185410. Contains A156919 as submatrix.
Row n counts perfect matchings of [2n] by number of matches in which the smaller entry is odd. For example, T(2,1)=2 counts 13/24, 14/23, in each of which only the first matching pair has an odd smaller entry. Outline proof. Consider the map on perfect matchings of [2n] given by "delete the entries n and n-1 and, if they were not originally matched to each other, match up their now-unmatched partners". Consideration of this map and its effect on the statistic "number of matches in which the smaller entry is odd" yields the Mathematica recurrence below. - David Callan, Dec 13 2011
Triangle T(n,k), 0 <= k <= n, given by (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, ...) DELTA (1, 0, 3, 0, 5, 0, 7, 0, 9, 11, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 12 2013
T(n,k), 0 <= k <= n, is the number of signed permutations of [n] that are products of balanced cycles (i.e., cuspidal elements of the type B Coxeter group) and have excedance number of type B equal to k. - Jose Bastidas, Jul 05 2023

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,     1;
  0,   4,    10,       1;
  0,   8,    60,      36,        1;
  0,  16,   296,     516,      116,        1;
  0,  32,  1328,    5168,     3508,      358,       1;
  0,  64,  5664,   42960,    64240,    21120,    1086,      1;
  0, 128, 23488,  320064,   900560,   660880,  118632,   3272,    1;
  0, 256, 95872, 2225728, 10725184, 14713840, 6049744, 638968, 9832, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Jose Bastidas, Christophe Hohlweg, and Franco Saliola, The Primitive Eulerian polynomial, arXiv:2306.15556 [math.CO], 2023. See Table 2 p. 18.
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, arXiv: 1204.4963v3 [math.CO], 2012.
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, Elect. J. Combinat. 20 (1) (2013) #P11.
Shi-Mei Ma, T. Mansour, and D. Callan, Some combinatorial arrays related to the Lotka-Volterra system, arXiv:1404.0731 [math.CO], 2014.
Shi-Mei Ma, T. Mansour and D. G. L. Wang, Combinatorics of Dumont differential system on the Jacobi elliptic functions, arXiv:1403.0233 [math.CO], 2014.
Shi-Mei Ma, Toufik Mansour, David G.L. Wang, and Yeong-Nan Yeh, Several variants of the Dumont differential system and permutation statistics, Science China Mathematics 60 (2018).
Shi-Mei Ma and Y.-N. Yeh, Stirling permutations, cycle structures of permutations and perfect matchings, arXiv:1503.06601 [math.CO], 2015.

Columns 0-1 give: A000007, A131577.

Cf. A001147, A185410, A156919 (another version).

u[n_, 0] := If[n==0, 1, 0]; u[n_, m_] /; m==1 := 2^(n - 1); u[n_, m_] /; m==n>=1 := 1; u[n_, m_] /; 1David Callan, Dec 13 2011 *)

G.f.: 1/(1-xy/(1-2x/(1-3xy/(1-4x/(1-5xy/(1-6x/(1-7xy/(1- ... (continued fraction).
T(n,k) = (2n-2k+1)*T(n-1,k-1) + 2k*T(n-1,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n. - Philippe Deléham, Feb 12 2013
T(n,k) = 2^(n-k)*A211399(n,k). - Philippe Deléham, Feb 12 2013

Sequence terms corrected by Paul Barry, Jan 27 2011

A099020 Euler-Seidel matrix T(k,n) with start sequence A001147, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 2, 1, 1, 4, 2, 1, 0, 10, 6, 4, 3, 3, 26, 16, 10, 6, 3, 0, 76, 50, 34, 24, 18, 15, 15, 232, 156, 106, 72, 48, 30, 15, 0, 764, 532, 376, 270, 198, 150, 120, 105, 105, 2620, 1856, 1324, 948, 678, 480, 330, 210, 105, 0, 9496, 6876, 5020, 3696, 2748, 2070, 1590, 1260, 1050, 945, 945

Offset: 0

Ralf Stephan, Sep 23 2004

In an Euler-Seidel matrix, the rows are consecutive pairwise sums and the columns consecutive differences, with the first column the inverse binomial transform of the start sequence.

			1,   0,  1,  0,   3,   0,   15, ...
1,   1,  1,  3,   3,  15,   15, ...
2,   2,  4,  6,  18,  30,  120, ...
4,   6, 10, 24,  48, 150,  330, ...
10, 16, 34, 72, 198, 480, 1590, ...

Alois P. Heinz, Rows n = 0..140, flattened
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

First column is A000085, 2nd A013989, main diagonal is in A099021.

T:= proc(k, n) option remember; `if`(k=0, `if`(irem(n, 2)=0,
      doublefactorial(n-1), 0), T(k-1, n) +T(k-1, n+1))
    end:
seq(seq(T(d-n, n), n=0..d), d=0..14);  # Alois P. Heinz, Oct 14 2012

t[0, n_?EvenQ] := (n-1)!!; t[0, n_?OddQ] := 0; t[k_, n_] := t[k, n] = t[k-1, n] + t[k-1, n+1]; Table[t[k-n, n], {k, 0, 10}, {n, 0, k}] // Flatten (* Jean-François Alcover, Dec 10 2012 *)

Recurrence: T(0, 2n) = (2n-1)!!, T(0, 2n+1) = 0, T(k, n) = T(k-1, n) + T(k-1, n+1).

A305868 Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A001147 (double factorial of odd numbers).

Original entry on oeis.org

1, 2, 12, 87, 816, 9194, 122028, 1859460, 32002076, 613890984, 12989299596, 300556859080, 7550646317520, 204687481289946, 5955892982437120, 185158929516065160, 6125200081143892800, 214837724609502834082, 7963817560398871790604, 311101285877489780292000, 12773912991134665452205048

Offset: 1

Ilya Gutkovskiy, Jun 12 2018

nonn

Inverse Euler transform of A001147.

			1/((1 - x) * (1 - x^2)^2 * (1 - x^3)^12 * (1 - x^4)^87 * (1 - x^5)^816 * ... * (1 - x^n)^a(n) * ...) = 1 + 1*x + 1*3*x^2 + 1*3*5*x^3 + 1*3*5*7*x^4 + ... + A001147(k)*x^k + ...

Seiichi Manyama, Table of n, a(n) for n = 1..404
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Double Factorial
Index entries for sequences related to factorial numbers

Cf. A001147, A112354, A305867, A305870.

nn = 21; f[x_] := Product[1/(1 - x^n)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-k x, 1, {k, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
nmax = 20; s = ConstantArray[0, nmax]; Do[s[[j]] = j*(2*j - 1)!! - Sum[s[[d]]*(2*j - 2*d - 1)!!, {d, 1, j - 1}], {j, 1, nmax}]; Table[Sum[MoebiusMu[k/d]*s[[d]], {d, Divisors[k]}]/k, {k, 1, nmax}] (* Vaclav Kotesovec, Aug 09 2019 *)

Product_{n>=1} 1/(1 - x^n)^a(n) = 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))).

a(n) ~ 2^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Aug 09 2019

A305870 Product_{n>=1} (1 + x^n)^a(n) = g.f. of A001147 (double factorial of odd numbers).

Original entry on oeis.org

1, 3, 12, 90, 816, 9206, 122028, 1859550, 32002076, 613891800, 12989299596, 300556868286, 7550646317520, 204687481411974, 5955892982437120, 185158929517924710, 6125200081143892800, 214837724609534836158, 7963817560398871790604, 311101285877490394183800, 12773912991134665452205048

Offset: 1

Ilya Gutkovskiy, Jun 12 2018

nonn

Inverse weigh transform of A001147.

			(1 + x) * (1 + x^2)^3 * (1 + x^3)^12 * (1 + x^4)^90 * (1 + x^5)^816 * ... * (1 + x^n)^a(n) * ... = 1 + 1*x + 1*3*x^2 + 1*3*5*x^3 + 1*3*5*7*x^4 + ... + A001147(k)*x^k + ...

Alois P. Heinz, Table of n, a(n) for n = 1..404
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Double Factorial
Index entries for sequences related to factorial numbers

Cf. A001147, A168246, A305868, A305869.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; doublefactorial(2*n-1)-b(n, n-1) end:
seq(a(n), n=1..23);  # Alois P. Heinz, Jun 13 2018

nn = 21; f[x_] := Product[(1 + x^n)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-k x, 1, {k, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

Product_{n>=1} (1 + x^n)^a(n) = 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))).

A111106 Riordan array (1, x*g(x)) where g(x) is g.f. of double factorials A001147.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 15, 7, 3, 1, 0, 105, 36, 12, 4, 1, 0, 945, 249, 64, 18, 5, 1, 0, 10395, 2190, 441, 100, 25, 6, 1, 0, 135135, 23535, 3807, 691, 145, 33, 7, 1, 0, 2027025, 299880, 40032, 5880, 1010, 200, 42, 8, 1

Offset: 0

Philippe Deléham, Oct 13 2005, Dec 20 2008

Triangle T(n,k), 0 <= k <= n, given by [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

			Rows begin:
  1;
  0,       1;
  0,       1,      1;
  0,       3,      2,     1;
  0,      15,      7,     3,    1;
  0,     105,     36,    12,    4,    1;
  0,     945,    249,    64,   18,    5,   1;
  0,   10395,   2190,   441,  100,   25,   6,  1:
  0,  135135,  23535,  3807,  691,  145,  33,  7, 1;
  0, 2027025, 299880, 40032, 5880, 1010, 200, 42, 8, 1;

Columns: A000007, A001147, A034430; diagonals: A000012, A001477, A055998.

Cf. A084938, A111088, A112934, A168441.

# Uses function PMatrix from A357368.
PMatrix(10, n -> doublefactorial(2*n-3)); # Peter Luschny, Oct 19 2022

T(n, k) = Sum_{j=0..n-k} T(n-1, k-1+j)*A111088(j).
Sum_{k=0..n} T(n, k) = A112934(n).
G.f.: 1/(1-xy/(1-x/(1-2x/(1-3x/(1-4x/(1-... (continued fraction). - Paul Barry, Jan 29 2009
Sum_{k=0..n} T(n,k)*2^(n-k) = A168441(n). - Philippe Deléham, Nov 28 2009

A185410 A decomposition of the double factorials A001147.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 10, 4, 0, 1, 36, 60, 8, 0, 1, 116, 516, 296, 16, 0, 1, 358, 3508, 5168, 1328, 32, 0, 1, 1086, 21120, 64240, 42960, 5664, 64, 0, 1, 3272, 118632, 660880, 900560, 320064, 23488, 128, 0, 1, 9832, 638968, 6049744, 14713840, 10725184, 2225728, 95872, 256, 0

Offset: 0

Paul Barry, Jan 26 2011

Row sums are A001147. Reversal of A185411.
From Peter Bala, Jul 24 2012: (Start)
This is the case k = 2 of the 1/k—Eulerian polynomials introduced by Savage and Viswanathan. They give a combinatorial interpretation of the triangle in terms of an ascent statistic on sets of inversion sequences and a geometric interpretation in terms of lecture hall polytopes.
Row reverse of A156919.
(End)
Triangle T(n,k), 0<=k<=n, given by (1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...) DELTA (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 12 2013

			Triangle begins:
  1,
  1,    0,
  1,    2,      0,
  1,   10,      4,       0,
  1,   36,     60,       8,        0,
  1,  116,    516,     296,       16,        0,
  1,  358,   3508,    5168,     1328,       32,       0,
  1, 1086,  21120,   64240,    42960,     5664,      64,     0,
  1, 3272, 118632,  660880,   900560,   320064,   23488,   128,   0,
  1, 9832, 638968, 6049744, 14713840, 10725184, 2225728, 95872, 256, 0,
  ...
In the Savage-Viswanathan paper, the coefficients appear as
  1
  1    2
  1   10     4
  1   36    60     8
  1  116   516   296    16
  1  358  3508  5168  1328   32
  1 1086 21120 64240 42960 5664 64
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
S.-M. Ma and T. Mansour, The 1/k-Eulerian polynomials and k-Stirling permutations, arXiv preprint arXiv:1409.6525 [math.CO], 2014.
C. D. Savage and G. Viswanathan, The 1/k-Eulerian polynomials, Elec. J. of Comb., Vol. 19, Issue 1, #P9 (2012).

Cf. A156919, A001147 (row sums), A112857, A173018, A186695, A202038 (alt. row sums).

T[0, 0] := 1;  T[n_, -1] := 0;  T[n_, n_] := 0; T[n_, k_] := T[n, k] = (n - k)*T[n - 1, k - 1] + (2*k + 1)*T[n - 1, k]; Join[{1}, Table[If[k < 0, 0, If[k >= n, 0, 2^k*T[n, k]]], {n, 1, 5}, {k, 0, n}] // Flatten] (* G. C. Greubel, Jun 30 2017 *)

G.f.: 1/(1-x/(1-2xy/(1-3x/(1-4xy/(1-5x/(1-6xy/(1-7x/(1-8xy/(1- .... (continued fraction).
From Peter Bala, Jul 24 2012: (Start)
T(n,k) = sum {j=0..k}(-1)^(k-j)/4^j*C(n+1/2,k-j)*C(2*j,j)*(2*j+1)^n.
Recurrence equation: T(n+1,k) = (2*k+1)*T(n,k) + 2*(n-k+1)*T(n,k-1).
E.g.f.: sqrt(E(x,2*z)) = 1 + z + (1+2*x)*z^2/2! + (1+10*x+4*x^2)*z^3/3! + ..., where E(x,z) = (1-x)/(exp(z*(x-1)) - x) is the e.g.f. for the Eulerian numbers (version A173018). Cf. A156919.
Row polynomial R(n,x) = sum {k = 1..n} 2^(n-2*k)*C(2*k,k)*k!*Stirling2(n,k)*(x-1)^(n-k). R(n,4*x)/(1-4*x)^(n+1/2) = sum {k>=0} C(2*k,k)*(2*k+1)^n*x^k. The sequence of rational functions x*R(n,x)/(1-x)^(n+1) conjecturally occurs in the first column of (I - x*A112857)^(-1). (1+x)^(n-1)*R(n,x/(x+1)) gives the n-th row polynomial of A186695.
Row sums A001147. Alt. row sums A202038. (End)
T(n,k) = 2^k*A102365(n,k). - Philippe Deléham, Feb 12 2013

A185971 Convolution inverse of A001147.

Original entry on oeis.org

1, -1, -2, -10, -74, -706, -8162, -110410, -1708394, -29752066, -576037442, -12277827850, -285764591114, -7213364729026, -196316804255522, -5731249477826890, -178676789473121834, -5925085744543837186

Offset: 0

Michael Somos, Feb 08 2011

sign

			1 - x - 2*x^2 - 10*x^3 - 74*x^4 - 706*x^5 - 8162*x^6 - 110410*x^7 - ...
From _Paul D. Hanna_, Mar 03 2012: (Start)
The coefficients in A(x)^n begin:
n=1: [1,  -1, -2, -10,  -74,  -706,  -8162, -110410, ...];
n=2: [1,  -2,(-3),-16, -124, -1224, -14516, -200192, ...];
n=3: [1,  -3,(-3),-19, -156, -1596, -19412, -272772, ...];
n=4: [1,  -4, -2,(-20),-175, -1856, -23136, -331008, ...];
n=5: [1,  -5,  0,(-20),-185, -2031, -25920, -377280, ...];
n=6: [1,  -6,  3, -20,(-189),-2142, -27951, -413568, ...];
n=7: [1,  -7,  7, -21,(-189),-2205, -29379, -441519, ...];
n=8: [1,  -8, 12, -24, -186,(-2232),-30324, -462504, ...];
n=9: [1,  -9, 18, -30, -180,(-2232),-30882, -477666, ...];
n=10:[1, -10, 25, -40, -170, -2212,(-31130),-487960, ...];
n=11:[1, -11, 33, -55, -154, -2178,(-31130),-494186, ...]; ...
where the coefficients in parenthesis demonstrate the properties:
(2) [x^n] A(x)^(2*n-2) = [x^n] A(x)^(2*n-1) for n>=2,
(3) [x^n] A(x)^(2*n-1) = -(2*n-1)*A000699(n) for n>=1:
A000699 = [1/1, 3/3, 20/5, 189/7, 2232/9, 31130/11, ...].
Note: g.f. of A000699, G(x), satisfies: G(x) = x + x^2*[d/dx G(x)^2/x].
(End)

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A001147, A000699, A185971, A208975, A208961.

a[n_] := If[ n < 0, 0, SeriesCoefficient[ 1 / (Sum[ (2 k - 1)!! x^k, {k, 0, n}] + O[x]^(n + 1)), n]];

{a(n) = if( n<0, 0, polcoeff( 1 / sum( k=0, n, x^k * (2*k)! / (2^k * k!), x * O(x^n)), n))}

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1-x*A^2*deriv(x/A^2));polcoeff(A,n)} /* Paul D. Hanna, Mar 03 2012 */

def A185971_list(len): # len >= 1
    if len == 1: return [1]
    T = [0]*(2*len-1); T[1] = 1; R = [1,-1]
    for n in (1..2*len-3):
        a,b,c = 1,0,0
        for k in range(n,-1,-1):
            r = a-(k+2)*c
            if k < n : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u;
        if is_even(n): R.append(-abs(u))
    return R
A185971_list(18)  # Peter Luschny, Nov 01 2012

G.f.: 1 / ( Sum_{k>=0} (2*k-1)!! * x^k ).
a(n) = -A000698(n) if n > 0.
G.f. A(x) = 1 - x * B(x) * C(x) where B = g.f. for A001147 and C = g.f. for A005416.
G.f.: A(x) = 1 - x/W(0); W(k) = 1 + x + x*2k - x*(2k+3)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2011
From Paul D. Hanna, Mar 03 2012: (Start)
G.f. A(x) satisfies:
(1) A(x) = 1 - x*A(x)^2 * [d/dx x/A(x)^2].
(2) [x^n] A(x)^(2*n-2) = [x^n] A(x)^(2*n-1) for n>=2.
(3) [x^n] A(x)^(2*n-1) = -(2*n-1)*A000699(n) for n>=1.  (End)
G.f. A(x) = G(0) where G(k)=  1 - x*(2*k+1)/(1 - (2*k+2)*x/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 11 2012
G.f. A(x)=1-x/Q(0) where Q(k)= 1 - (k+2)*x/Q(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Aug 20 2012
G.f. A(x) = G(0) where G(k)= 1 - x*(k+1)/G(k+1); (continued fraction, 1-step).- Sergei N. Gladkovskii, Oct 28 2012
G.f.: 1/(1 + x*(Q(0) - 1)/(x+1)) where Q(k)= 1 + (2*k+1)/(1-x/(x + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 11 2013
G.f.: Q(0), where Q(k)= 1 + (k+1)*sqrt(x) - sqrt(x)/(1-sqrt(x)*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 18 2013
G.f.: Q(0), where Q(k)= 1 + (2*k+1)*x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 02 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - 1 + 2*x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
a(n) ~ -n^n * 2^(n+1/2) / exp(n). - Vaclav Kotesovec, Feb 23 2014

A085881 Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).

Original entry on oeis.org

1, 1, 1, 3, 6, 3, 15, 45, 45, 15, 105, 420, 630, 420, 105, 945, 4725, 9450, 9450, 4725, 945, 10395, 62370, 155925, 207900, 155925, 62370, 10395, 135135, 945945, 2837835, 4729725, 4729725, 2837835, 945945, 135135, 2027025, 16216200, 56756700, 113513400, 141891750, 113513400, 56756700, 16216200, 2027025

Offset: 0

N. J. A. Sloane, Aug 17 2003

			Triangle starts:
       1;
       1,      1;
       3,      6,       3;
      15,     45,      45,      15;
     105,    420,     630,     420,     105;
     945,   4725,    9450,    9450,    4725,     945;
   10395,  62370,  155925,  207900,  155925,   62370,  10395;
  135135, 945945, 2837835, 4729725, 4729725, 2837835, 945945, 135135;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A001147, A007318, A084938, A164961.

Flat(List([0..10], n-> List([0..n], k-> Binomial(n,k)*Binomial(2*n,n) *Factorial(n)/2^n ))); # G. C. Greubel, Feb 07 2020

[Binomial(n,k)*Binomial(2*n,n)*Factorial(n)/2^n: k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 07 2020

T:= (n, k)-> n!/2^n*binomial(n, k)*binomial(2*n, n):
seq(seq(T(n, k), k=0..n), n=0..10); # Yu-Sheng Chang, Jan 16 2020

Table[Binomial[n, k]*(2*n-1)!!, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 07 2020 *)

T(n,k) = binomial(n,k)*binomial(2*n,n)*n!/2^n;
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 07 2020

[[binomial(n,k)*(2*n-1).multifactorial(2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 07 2020

Triangle given by [1, 2, 3, 4, 5, 6, ...] DELTA [1, 2, 3, 4, 5, 6, ...] where DELTA is Deléham's operator defined in A084938.
T(n,k) = A164961(n,k)/2^n. - Philippe Deléham, Jan 07 2012
Recurrence equation: T(n+1,k) = (2*n+1)*(T(n,k) + T(n,k-1)). - Peter Bala, Jul 15 2012
E.g.f.: 1/sqrt(1-2*x-2*x*y). - Peter Bala, Jul 15 2012
G.f.: W(0), where W(k) = 1 - (k+1)*x*(1+y)/( (k+1)*x*(1+y) - 1/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 03 2013

cp's OEIS Frontend

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

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A133221 A001147 with each term repeated.

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A185411 A triangular decomposition of the double factorial numbers A001147.

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A099020 Euler-Seidel matrix T(k,n) with start sequence A001147, read by antidiagonals.

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A305868 Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A001147 (double factorial of odd numbers).

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A305870 Product_{n>=1} (1 + x^n)^a(n) = g.f. of A001147 (double factorial of odd numbers).

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A111106 Riordan array (1, x*g(x)) where g(x) is g.f. of double factorials A001147.

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