A034430 -id:A034430 - cp's OEIS frontend

A082590 Expansion of 1/((1 - 2x)sqrt(1 - 4*x)).

1, 4, 14, 48, 166, 584, 2092, 7616, 28102, 104824, 394404, 1494240, 5692636, 21785872, 83688344, 322494208, 1246068806, 4825743832, 18726622964, 72798509728, 283443548276, 1105144970992, 4314388905704, 16862208539008, 65972020761116, 258354647959984, 1012627828868072

Offset: 0

Vladeta Jovovic, May 13 2003

nonn

Row sums of A068555 and A112336. - Paul Barry, Sep 04 2005
Hankel transform is 2^n*(-1)^C(n+1,2) (A120617). - Paul Barry, Apr 26 2009
Number of n-lettered words in the alphabet {1, 2, 3, 4} with as many occurrences of the substring (consecutive subword) [1, 2] as of [1, 3]. - N. J. A. Sloane, Apr 08 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Shalosh B. Ekhad and Doron Zeilberger, Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type, arXiv:1112.6207 [math.CO], 2011. See subpages for rigorous derivations of the g.f., the recurrence, asymptotics for this sequence.
Alejandro Erickson and Frank Ruskey, Enumerating maximal tatami mat coverings of square grids with v vertical dominoes, arXiv:1304.0070 [math.CO], 2013.
Y. Kamiyama, On the middle dimensional homology classes of equilateral polygon spaces, arXiv:1507.03161 [math.AT], 2015.

Bisection of A226302.

Cf. A034430, A068555, A076729, A112336.

A082590 := proc(n)
    coeftayl( 1/(1-2*x)/sqrt(1-4*x),x=0,n) ;
end proc: # R. J. Mathar, Nov 06 2013
A082590 := n -> 2^n*JacobiP(n, 1/2, -1 - n, 3):
seq(simplify(A082590(n)), n = 0..26);  # Peter Luschny, Jan 22 2025

CoefficientList[ Series[ 1/((1 - 2*x)*Sqrt[1 - 4*x]), {x, 0, 25}], x] (* Jean-François Alcover, Mar 26 2013 *)
Table[2^(n) JacobiP[n, 1/2, -1-n, 3], {n, 0, 30}] (* Vincenzo Librandi, May 26 2013 *)

a(n) = 2^n*JacobiP(n, 1/2, -1-n, 3).
A034430(n) = (n!/2^n)*a(n). A076729(n) = n!*a(n).
a(n) = Sum_{k=0..n+1} binomial(2*n+2, k) * sin((n - k + 1)*Pi/2). - Paul Barry, Nov 02 2004
From Paul Barry, Sep 04 2005: (Start)
a(n) = Sum_{k=0..n} 2^(n-k)*binomial(2*k, k).
a(n) = Sum_{k=0..n} (2*k)! * (2*(n-k))!/(n!*k!*(n-k)!). (End)
a(n) = Sum_{k=0..n} C(2*n, n)*C(n, k)/C(2*n, 2*k). - Paul Barry, Mar 18 2007
G.f.: 1/(1 - 4*x + 2*x^2/(1 + x^2/(1 - 4*x + x^2/(1 + x^2/(1 - 4*x + x^2/(1 + ... (continued fraction). - Paul Barry, Apr 26 2009
D-finite with recurrence: n*a(n) + 2*(-3*n+1)*a(n-1) + 4*(2*n-1)*a(n-2) = 0. - R. J. Mathar, Dec 03 2012
a(n) ~ 2^(2*n + 1)/sqrt(Pi*n). - Vaclav Kotesovec, Aug 15 2013
a(n) = 2^(n + 1)*Pochhammer(1/2, n+1)*hyper2F1([1/2,-n], [3/2], -1)/n!. - Peter Luschny, Aug 02 2014
a(n) - 2*a(n-1) = A000984(n). - R. J. Mathar, Apr 24 2024
a(n) = 2^n*JacobiP(n, 1/2, -1 - n, 3). - Peter Luschny, Jan 22 2025

A076729 a(n) = A001147(n+1) * Integral_{x=0..1} (1 + x^2)^n dx.

Original entry on oeis.org

1, 4, 28, 288, 3984, 70080, 1506240, 38384640, 1133072640, 38038533120, 1431213235200, 59645279232000, 2726781752217600, 135661078090137600, 7295806823277772800, 421717409630060544000, 26071235813929033728000, 1716456412254215503872000, 119894838461795743137792000

Offset: 0

Al Hakanson (hawku(AT)hotmail.com), Oct 28 2002

nonn

Numerator of the integral where denominator is equal to (2n+1)!! = A001147(n+1).
Also numerator of the integral (1-x^2)^-(n+1/2) for x from 0 to sqrt(1/2). Here the sequence starts at n=1; at n=2 the function is 4.
a(n) = Integral_{x=0..log(1+sqrt(2))} cosh(x)^(2*n-1) dx where the denominators are b(n) = (2*n)!/(n!*2^n). E.g., a(3)=28 and b(3)=15; both offsets are 1. - Al Hakanson (hawkuu(AT)excite.com), Mar 02 2004
Self-convolution of A001813. - Vladimir Reshetnikov, Oct 11 2016

			For n = 3, (2n+1)!! = 105 and the integral is 96/35 = 288/105, so a(3) = 288.

Cf. A001147, A077595, A077745, A086891, A034430, A001813.

seq((doublefactorial(2*n+1))*add((binomial(n, i))/(2*i+1), i=0..n), n=0..20) ; # John M. Campbell, Feb 06 2016
A076729 := n -> 2^n*n!*JacobiP(n, 1/2, -1 - n, 3):
seq(simplify(A076729(n)), n = 0..18);  # Peter Luschny, Jan 22 2025

a[n_] := (2n + 1)!!*Integrate[(1 + x^2)^n, {x, 0, 1}]; Table[ a[n], {n, 0, 16}] (* Robert G. Wilson v, Feb 27 2004 *)
Round@Table[-(2 n + 1)!! Im[Beta[2, n + 1, 1/2]]/2, {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Oct 08 2016 *)
nxt[{n_,a_}]:={n+1,2a(n+1)+(2(n+1))!/(n+1)!}; NestList[nxt,{0,1},20][[All,2]] (* Harvey P. Dale, Feb 04 2023 *)

a(n)=if(n<0,0,subst(intformal((1+x^2)^n),x,1)*(2*n+1)!/2^n/n!)

a(n) = 2*n*a(n-1) + (2*n)!/n!.
a(n) = 2^n*Sum_{k=0..n} A001147(k)*A001147(n-k).
a(n) = (2*n+1)!*Sum_{k=0..n} k!*(-2)^k/((2*k+1)!*(n-k)!).
a(n) = (2*n+1)!!*hypergeom([1/2, -n], [3/2], -1). - Vladeta Jovovic, Dec 05 2002
E.g.f.: 1/((1-2*x)*sqrt(1-4*x)). - Vladeta Jovovic, May 11 2003
G.f.: hypergeom([1,1/2],[],4*x)^2  - Mark van Hoeij, May 16 2013
a(n) ~ 2^(2*n+3/2)*n^n/exp(n). - Vaclav Kotesovec, Oct 05 2013
a(n) = (2n+1)!!*Sum_{i=0..n} binomial(n,i)/(2i+1). - John M. Campbell, Feb 06 2016
From Vladimir Reshetnikov, Oct 08 2016: (Start)
a(n) = 2^n*A034430(n) = -(2*n+1)!! * Im(Beta(2, n+1, 1/2))/2.
Recurrence: 2*(3*n+2)*a(n) = a(n+1) + 4*n*(2*n+1)*a(n-1). (End)
Expansion of square of continued fraction 1/(1 - 2*x/(1 - 4*x/(1 - 6*x/(1 - 8*x/(1 - 10*x/(1 - ...)))))). - Ilya Gutkovskiy, Apr 19 2017
a(n) = 2^n*n!*JacobiP(n, 1/2, -1 - n, 3). - Peter Luschny, Jan 22 2025

A336601 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords are excluded by (i.e., are outside and do not contain) the marked chord.

Original entry on oeis.org

1, 4, 2, 22, 16, 7, 160, 136, 88, 36, 1464, 1344, 1044, 624, 249, 16224, 15504, 13344, 9624, 5484, 2190, 211632, 206592, 188952, 152832, 104322, 58080, 23535, 3179520, 3139200, 2977920, 2594880, 1990080, 1309680, 725040, 299880, 54092160, 53729280, 52096320, 47681280, 39652560, 29174400, 18809640, 10473120, 4426065

Offset: 1

Donovan Young, Jul 31 2020

			Triangle begins:
     1;
     4,    2;
    22,   16,    7;
   160,  136,   88,  36;
  1464, 1344, 1044, 624, 249;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can either be (1,2) and it excludes one other chord, namely (3,4), or vice-versa, hence T(2,1) = 2.

Donovan Young, Table of n, a(n) for n = 1..9870
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Row sums are n*A001147(n) for n > 0.
The first column is A087547(n) for n > 0.
Leading diagonal is A034430(n-1) for n > 0.
Cf. A336598, A336599, A336600.

CoefficientList[Normal[Series[1/(1-y)/Sqrt[1-2*x]*ArcTan[(x*(1-y))/Sqrt[(1-2*x)]/Sqrt[1-2*y*x]],{x,0,10}]]/.{x^n_.->x^n*n!},{x,y}]

E.g.f.: arctan(x*(1 - y)/sqrt((1 - 2*x)*(1 - 2*x*y)))/(1 - y)/sqrt(1 - 2*x).

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

A233481 Number of singletons (strong fixed points) in pair-partitions.

Original entry on oeis.org

0, 1, 4, 21, 144, 1245, 13140, 164745, 2399040, 39834585, 742940100, 15374360925, 349484058000, 8654336615925, 231842662751700, 6679510641428625, 205916703920928000, 6762863294018456625, 235719416966063530500, 8689887736412502745125

Offset: 0

Wojciech Bozejko, Dec 11 2013

nonn

For h(V) = number of singletons (non-crossing chords) in the pair-partition of 2n-elementary set P_2(2n), let T(2n) = sum_{V in P_2(2n)} h(V).
Elements of the sequence a(n) = T(2n).
a(n) is the number of linear chord diagrams on 2n vertices with one marked chord such that none of the remaining n-1 chords cross the marked chord, see [Young]. - Donovan Young, Aug 11 2020

G. C. Greubel, Table of n, a(n) for n = 0..400
Marek Bozejko and Wojciech Bozejko, Generalized Gaussian processes and relations with random matrices and positive definite functions on permutation groups, arXiv:1301.2502 [math.PR], 2013.
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Cf. A001147, A034430, A082590, A076729, A336598.

A081054 counts pair-partitions of a fixed size without singletons, i.e., linear chord diagrams with 2n nodes and n arcs in which each arc crosses another arc.

a := n -> 2*n*GAMMA(1/2+n)*hypergeom([1/2,-n+1],[3/2],-1)/sqrt(Pi);
seq(simplify(a(n)), n = 0..19); # Peter Luschny, Dec 16 2013
# Alternative:
u := (z/2)^2: egf := 2*u*exp(u)*hypergeom([1/2], [3/2], u): ser := series(egf, z, 40): seq((2*n)!*coeff(ser, z, 2*n), n = 0..19); # Peter Luschny, Mar 14 2023

Table[Sum[(2 k - 1)!! (2 n - 2 k - 1)!!, {k, 0, n - 1}], {n,0,30}] (* T. D. Noe, Dec 13 2013 *)

def A233481():
    a, b, n = 0, 1, 1
    while True:
        yield a
        n += 1
        a, b = b, n*((3*n-4)*b/(n-1)-(2*n-3)*a)
a = A233481(); [next(a) for i in range(17)]  # Peter Luschny, Dec 14 2013

a(n) = T_{2n} = n*sum_{k=0..(n-1)} (2k-1)!!*(2n-2k-1)!!, where (2n-1)!! = 1*3*5*...*(2n-1).
From Peter Luschny, Dec 16 2013: (Start)
E.g.f.: x/((1-x)*sqrt(1-2*x)).
a(n) = 2*n*Gamma(1/2+n)*2_F_1([1/2,-n+1],[3/2],-1)/sqrt(Pi), where 2_F_1 is the hypergeometric function.
a(n) = n*((3*n-4)*a(n-1)/(n-1)-(2*n-3)*a(n-2)) for n>1.
a(n) = n*A034430(n-1) for n>=1.
a(n+1)/(n+1)! = JacobiP(n, 1/2, -n-1, 3).
2^n*a(n+1)/(n+1)! = A082590(n).
2^n*a(n+1)/(n+1) = A076729(n). (End)
a(n) ~ 2^(n+1/2) * n^n / exp(n). - Vaclav Kotesovec, Dec 20 2013
a(n) = (2*n)! * [z^(2*n)] 2*u*exp(u)*hypergeom([1/2], [3/2], u), where u = (z/2)^2. - Peter Luschny, Mar 14 2023

A331817 a(n) = (n!)^2 * Sum_{k=0..n} (2k)! / (2^k (k!)^3 * (n - k)!).

Original entry on oeis.org

1, 2, 9, 66, 681, 9090, 148905, 2889810, 64805265, 1648535490, 46896669225, 1475099460450, 50831084252025, 1904311245686850, 77061447551313225, 3349828945512299250, 155672917524626126625, 7701743926471878533250, 404153655359180645543625

Offset: 0

Ilya Gutkovskiy, Jan 27 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..380

Cf. A000165, A001147, A032031, A034430, A052577.

[(Factorial(n))^2*&+[Factorial(2*k)/(2^k*(Factorial(k))^3*Factorial(n-k)):k in [0..n]]:n in [0..18]]; // Marius A. Burtea, Jan 27 2020

f:= gfun:-rectoproc({a(n + 2) = 2*(3 + 2*n)*a(n + 1) - 3*(n + 1)^2*a(n), a(0)=1, a(1)=2},a(n), remember):
map(f, [$0..30]); # Robert Israel, Feb 17 2020

Table[n!^2 Sum[(2 k)!/(2^k k!^3 (n - k)!), {k, 0, n}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/Sqrt[1 - 4 x + 3 x^2], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Hypergeometric2F1[1/2, -n, 1, -2], {n, 0, 18}]

seq(n) = {Vec(serlaplace(1/(sqrt(1 - 4*x + 3*x^2 + O(x*x^n)))))} \\ Andrew Howroyd, Jan 27 2020

E.g.f.: 1 / sqrt(1 - 4*x + 3*x^2).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * (2*k - 1)!! * (n - k)!.
a(n) = n! * 2F1(1/2, -n; 1; -2).
a(n) ~ 3^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Jan 28 2020
D-finite with recurrence a(n + 2) = 2*(3 + 2*n)*a(n + 1) - 3*(n + 1)^2*a(n). - Robert Israel, Feb 17 2020

A305577 a(n) = Sum_{k=0..n} k!!*(n - k)!!.

Original entry on oeis.org

1, 2, 5, 10, 26, 58, 167, 414, 1324, 3606, 12729, 37674, 145578, 463770, 1944879, 6614190, 29852856, 107616150, 518782545, 1970493210, 10077228270, 40125873690, 216425656215, 899557170750, 5091758227620, 22011865939350, 130202223160905, 583641857191050, 3594820517111250

Offset: 0

Ilya Gutkovskiy, Jun 05 2018

nonn

Convolution of A006882 with itself.

Alois P. Heinz, Table of n, a(n) for n = 0..807
Poloni, Federico; Del Corso, Gianna M. Counting Fiedler pencils with repetitions. Linear Algebra Appl. 532, 463-499 (2017), corollary 24.
Eric Weisstein's World of Mathematics, Double Factorial
Index entries for sequences related to factorial numbers

Cf. A003149, A006882, A034430, A059371, A111308, A126674, A305578.

a:= proc(n) option remember; `if`(n<4, n^2+1,
      ((3*n^2-4*n-2)*a(n-2) +(n+1)*a(n-3)
       -2*a(n-1) -(n-1)^2*n*a(n-4))/(2*n-4))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 14 2018

Table[Sum[k!! (n - k)!!, {k, 0, n}], {n, 0, 28}]
nmax = 28; CoefficientList[Series[Sum[k!! x^k, {k, 0, nmax}]^2, {x, 0, nmax}], x]

G.f.: (Sum_{k>=0} k!!*x^k)^2.

A173424 Triangle read by rows: T(n, k) = (2n - 2k)!(2k)!/(2^n(n - k)!k!).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 15, 3, 3, 15, 105, 15, 9, 15, 105, 945, 105, 45, 45, 105, 945, 10395, 945, 315, 225, 315, 945, 10395, 135135, 10395, 2835, 1575, 1575, 2835, 10395, 135135, 2027025, 135135, 31185, 14175, 11025, 14175, 31185, 135135, 2027025, 34459425

Offset: 0

Roger L. Bagula, Feb 18 2010

			Triangle T(n, k) starts:
  [0]       1;
  [1]       1,      1;
  [2]       3,      1,     3;
  [3]      15,      3,     3,    15;
  [4]     105,     15,     9,    15,   105;
  [5]     945,    105,    45,    45,   105,   945;
  [6]   10395,    945,   315,   225,   315,   945, 10395;
  [7]  135135,  10395,  2835,  1575,  1575,  2835, 10395, 135135;
  [8] 2027025, 135135, 31185, 14175, 11025, 14175, 31185, 135135, 2027025;

Cf. A034430 (row sums), A006882, A001147.

T := (n, k) -> doublefactorial(2*n-1) * binomial(n, k) / binomial(2*n, 2*k):
for n from 0 to 8 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Apr 15 2023

t[n_, k_] = (2*n - 2*k)!*(2*k)!/(2^n*(n - k)!*k!);
Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}]; Flatten[%]

T(n, k) = A006882(2*n-2*k-1) * A006882(2*k-1).
T(n, k) = A001147(n-k) * A001147(k).
From Peter Luschny, Apr 15 2023: (Start)
T(n, k) = (1/Pi) * 2^n * Gamma(k + 1/2) * Gamma(n - k + 1/2).
T(n, k) = (2*n-1)!! * binomial(n, k) / binomial(2*n, 2*k). (End)

Formula added by the Assoc. Editors of the OEIS, Feb 24 2010

A188287 Convolution of A000085 with itself.

Original entry on oeis.org

1, 2, 5, 12, 32, 88, 260, 800, 2604, 8824, 31340, 115568, 443760, 1763456, 7260256, 30835712, 135124496, 609027360, 2822461648, 13417923008, 65401203584, 326242088064, 1664539966400, 8674167861760, 46140838036160, 250248380068736, 1383064482739392, 7782094359642880

Offset: 0

Groux Roland, Mar 26 2011

nonn

a(n) is also the moment of order n for the measure of density: x*exp(-(x-1)^2)*erfi((x-1)/sqrt(2)) over the interval -infinity..infinity, with erfi the Imaginary Error Function.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000085, A034430.

seq(n)={Vec(serlaplace(exp(x + x^2/2 + O(x*x^n)))^2)} \\ Andrew Howroyd, Nov 04 2019

a(n) = Sum_{k=0..floor(n/2)} binomial(n+1,2*k+1)*A034430(k).

Terms a(20) and beyond from Andrew Howroyd, Nov 04 2019

A293470 a(n) = [x^n] (1/(1 - x/(1 - 2x/(1 - 3x/(1 - 4x/(1 - 5x/(1 - 6*x/(1 - ...))))))))^n, a continued fraction.

Original entry on oeis.org

1, 1, 7, 64, 691, 8506, 117586, 1811902, 30977059, 585159526, 12157511122, 276365651992, 6835179127294, 182885413524568, 5265255383238592, 162296482607602714, 5332203008816278819, 185989603728568482598, 6863252473075010369626, 267102762222709967674384, 10932746393513621360731066

Offset: 0

Ilya Gutkovskiy, Oct 09 2017

nonn

Index entries for sequences related to factorial numbers

Cf. A001147, A034430, A113551, A287899, A293471.

Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-k x, 1, {k, 1, n}])^n, {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[Sum[(2 k - 1)!! x^k, {k, 0, n}]^n, {x, 0, n}], {n, 0, 20}]

a(n) ~ 2^(n + 1/2) * n^(n+1) / exp(n - 1/2). - Vaclav Kotesovec, Sep 16 2021

cp's OEIS Frontend

A082590 Expansion of 1/((1 - 2*x)*sqrt(1 - 4*x)).

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A076729 a(n) = A001147(n+1) * Integral_{x=0..1} (1 + x^2)^n dx.

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A336601 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords are excluded by (i.e., are outside and do not contain) the marked chord.

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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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A233481 Number of singletons (strong fixed points) in pair-partitions.

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A331817 a(n) = (n!)^2 * Sum_{k=0..n} (2*k)! / (2^k * (k!)^3 * (n - k)!).

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Magma

Maple

Mathematica

PARI

Formula

A305577 a(n) = Sum_{k=0..n} k!!*(n - k)!!.

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Maple

Mathematica

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A173424 Triangle read by rows: T(n, k) = (2*n - 2*k)!*(2*k)!/(2^n*(n - k)!*k!).

A082590 Expansion of 1/((1 - 2x)sqrt(1 - 4*x)).

A331817 a(n) = (n!)^2 * Sum_{k=0..n} (2k)! / (2^k (k!)^3 * (n - k)!).

A173424 Triangle read by rows: T(n, k) = (2n - 2k)!(2k)!/(2^n(n - k)!k!).

A293470 a(n) = [x^n] (1/(1 - x/(1 - 2x/(1 - 3x/(1 - 4x/(1 - 5x/(1 - 6*x/(1 - ...))))))))^n, a continued fraction.