A084261 -id:A084261 - cp's OEIS frontend

A002896 Number of 2n-step polygons on cubic lattice.

1, 6, 90, 1860, 44730, 1172556, 32496156, 936369720, 27770358330, 842090474940, 25989269017140, 813689707488840, 25780447171287900, 825043888527957000, 26630804377937061000, 865978374333905289360, 28342398385058078078010, 932905175625150142902300

Offset: 0

N. J. A. Sloane

Number of walks with 2n steps on the cubic lattice Z^3 beginning and ending at (0,0,0).
If A is a random matrix in USp(6) (6 X 6 complex matrices that are unitary and symplectic) then a(n) is the 2n-th moment of tr(A^k) for all k >= 7. - Andrew V. Sutherland, Mar 24 2008
Diagonal of the rational function R(x,y,z,w) = 1/(1 - (w*x*y + w*x*z + w*y + x*z + y + z)). - Gheorghe Coserea, Jul 14 2016
Constant term in the expansion of (x + 1/x + y + 1/y + z + 1/z)^(2n). - Harry Richman, Apr 29 2020

			1 + 6*x + 90*x^2 + 1860*x^3 + 44730*x^4 + 1172556*x^5 + 32496156*x^6 + ...

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).

Alois P. Heinz, Table of n, a(n) for n = 0..645
David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008.
Nikolai Beluhov, Powers of 2 in Balanced Grid Colourings, arXiv:2504.21451 [math.CO], 2025.
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024. See p. 5.
Li Gan, On the algebraic area of cubic lattice walks, arXiv:2307.08732 [math-ph], 2023.
S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 23.
W. K. Hayman, A Generalisation of Stirling's Formula, Journal für die reine und angewandte Mathematik, (1956), vol. 196, pp. 67-95.
John A. Hendrickson, Jr., On the enumeration of rectangular (0,1)-matrices, Journal of Statistical Computation and Simulation, 51 (1995), 291-313.
Heng Huat Chan, Yoshio Tanigawa, Yifan Yang, and Wadim Zudilin, New analogues of Clausen's identities arising from the theory of modular forms, Advances in Mathematics, 228:2 (2011), pp. 1294-1314; doi:10.1016/j.aim.2011.06.011.
G. S. Joyce, The simple cubic lattice Green function, Phil. Trans. Roy. Soc., 273 (1972), 583-610.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017.
Nobu C. Shirai and Naoyuki Sakumichi, Universal Negative Energetic Elasticity in Polymer Chains: Crossovers among Random, Self-Avoiding, and Neighbor-Avoiding Walks, arXiv:2408.14992 [cond-mat.soft], 2024. See p. 5.
Chunwei Song and Bowen Yao, On Combinatorial Rectangles with Minimum oo-Discrepancy, arXiv:1909.05648 [math.CO], 2019. See p. 7 for another interpretation.
Ralph A. Raimi and Jet Wimp, Review of book "A=B" by M. Petkovsek et al., Mathematical Intelligencer, 23 (No. 4, 2001), pp. 72-77.

C(2n, n) times A002893.
Cf. A049020, A049037, A084261, A138540, A174516.
Related to diagonal of rational functions: A268545-A268555.
Row k=3 of A287318.

a := proc(n) local k; binomial(2*n,n)*add(binomial(n,k)^2 *binomial(2*k,k), k=0..n); end;
# second Maple program
a:= proc(n) option remember; `if`(n<2, 5*n+1,
      (2*(2*n-1)*(10*n^2-10*n+3) *a(n-1)
       -36*(n-1)*(2*n-1)*(2*n-3) *a(n-2)) /n^3)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 02 2012
A002896 := n -> binomial(2*n,n)*hypergeom([1/2, -n, -n], [1, 1], 4):
seq(simplify(A002896(n)), n=0..16); # Peter Luschny, May 23 2017

Table[Binomial[2n,n] Sum[Binomial[n,k]^2 Binomial[2k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jan 24 2012 *)
a[ n_] := If[ n < 0, 0, HypergeometricPFQ[ {-n, -n, 1/2}, {1, 1}, 4] Binomial[ 2 n, n]] (* Michael Somos, May 21 2013 *)

a(n)=binomial(2*n,n)*sum(k=0,n,binomial(n, k)^2*binomial(2*k, k)) \\ Charles R Greathouse IV, Oct 31 2011

def A002896():
    x, y, n = 1, 6, 1
    while True:
        yield x
        n += 1
        x, y = y, ((4*n-2)*((10*(n-1)*n+3)*y-18*(n-1)*(2*n-3)*x))//n^3
a = A002896()
[next(a) for i in range(17)]  # Peter Luschny, Oct 09 2013

a(n) = C(2*n, n)*Sum_{k=0..n} C(n, k)^2*C(2*k, k).
a(n) = (4^n*p(1/2, n)/n!)*hypergeom([-n, -n, 1/2], [1, 1], 4), where p(a, k) = Product_{i=0..k-1} (a+i).
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2*x)^3. - Corrected by Christopher J. Smyth, Oct 29 2012
D-finite with recurrence: n^3*a(n) = 2*(2*n-1)*(10*n^2-10*n+3)*a(n-1) - 36*(n-1)*(2*n-1)*(2*n-3)*a(n-2). - Vladeta Jovovic, Jul 16 2004
An asymptotic formula follows immediately from an observation of Bruce Richmond and me in SIAM Review - 31 (1989, 122-125). We use Hayman's method to find the asymptotic behavior of the sum of squares of the multinomial coefficients multi(n, k_1, k_2, ..., k_m) with m fixed. From this one gets a_n ~ (3/4)*sqrt(3)*6^(2*n)/(Pi*n)^(3/2). - Cecil C Rousseau (ccrousse(AT)memphis.edu), Mar 14 2006
G.f.: (1/sqrt(1+12*z)) * hypergeom([1/8,3/8],[1],64/81*z*(1+sqrt(1-36*z))^2*(2+sqrt(1-36*z))^4/(1+12*z)^4) * hypergeom([1/8, 3/8],[1],64/81*z*(1-sqrt(1-36*z))^2*(2-sqrt(1-36*z))^4/(1+12*z)^4). - Sergey Perepechko, Jan 26 2011
a(n) = binomial(2*n,n)*A002893(n). - Mark van Hoeij, Oct 29 2011
G.f.: (1/2)*(10-72*x-6*(144*x^2-40*x+1)^(1/2))^(1/2)*hypergeom([1/6, 1/3],[1],54*x*(108*x^2-27*x+1+(9*x-1)*(144*x^2-40*x+1)^(1/2)))^2. - Mark van Hoeij, Nov 12 2011
PSUM transform is A174516. - Michael Somos, May 21 2013
0 = (-x^2+40*x^3-144*x^4)*y''' + (-3*x+180*x^2-864*x^3)*y'' + (-1+132*x-972*x^2)*y' + (6-108*x)*y, where y is the g.f. - Gheorghe Coserea, Jul 14 2016
a(n) = [(x y z)^0] (x + 1/x + y + 1/y + z + 1/z)^(2*n). - Christopher J. Smyth, Sep 25 2018
a(n) = (1/Pi)^3*Integral_{0 <= x, y, z <= Pi} (2*cos(x) + 2*cos(y) + 2*cos(z))^(2*n) dx dy dz. - Peter Bala, Feb 10 2022
a(n) = Sum_{i+j+k=n, 0<=i,j,k<=n} multinomial(2n [i,i,j,j,k,k]). - Shel Kaphan, Jan 16 2023
Sum_{k>=0} a(k)/36^k = A086231 = (sqrt(3)-1) * (Gamma(1/24) * Gamma(11/24))^2 / (32*Pi^3). - Vaclav Kotesovec, Apr 23 2023
G.f.: HeunG(1/9,1/12,1/4,3/4,1,1/2,4*x)^2 (see Hassani et al.). - Stefano Spezia, Feb 16 2025

A081124 Binomial transform of floor(n/2)!.

Original entry on oeis.org

1, 2, 4, 8, 17, 38, 90, 224, 585, 1594, 4520, 13288, 40409, 126782, 409646, 1360512, 4637681, 16202034, 57941164, 211860488, 791272129, 3015807254, 11719800674, 46401584096, 187039192185, 767058993386, 3198568491792, 13553864902504

Offset: 0

Paul Barry, Mar 07 2003

G. C. Greubel, Table of n, a(n) for n = 0..880
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012

Cf. A081123, A004526.

Cf. A084261.

Table[Sum[Binomial[n,k]*Floor[k/2]!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 15 2013 *)

for(n=0,50, print1(sum(k=0,n, binomial(n,k)*(floor(k/2))!), ", ")) \\ G. C. Greubel, Feb 02 2017

a(n) = Sum_{k=0..n} C(n, k)*floor(k/2)!.
E.g.f.: exp(x)*(1+sqrt(Pi)/2*(x+2)*exp(x^2/4)*erf(x/2)). - Vladeta Jovovic, Sep 25 2003
Conjecture: 2*a(n) -4*a(n-1) +(-n+2)*a(n-2) +(n-1)*a(n-3)=0. - R. J. Mathar, Nov 24 2012
a(n) ~ sqrt(Pi*n)/2 * exp(sqrt(2*n)-n/2-1/2)*(n/2)^(n/2) * (1+5/(3*sqrt(2*n))). - Vaclav Kotesovec, Aug 15 2013

A161556 Exponential Riordan array [1 + (sqrt(Pi)/2)xexp(x^2/4)*erf(x/2), x].

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 2, 0, 6, 0, 1, 0, 10, 0, 10, 0, 1, 6, 0, 30, 0, 15, 0, 1, 0, 42, 0, 70, 0, 21, 0, 1, 24, 0, 168, 0, 140, 0, 28, 0, 1, 0, 216, 0, 504, 0, 252, 0, 36, 0, 1, 120, 0, 1080, 0, 1260, 0, 420, 0, 45, 0, 1

Offset: 0

Paul Barry, Jun 13 2009

Row sums are A084261.

			Triangle begins
   1;
   0,   1;
   1,   0,   1;
   0,   3,   0,   1;
   2,   0,   6,   0,   1;
   0,  10,   0,  10,   0,   1;
   6,   0,  30,   0,  15,   0,   1;
   0,  42,   0,  70,   0,  21,   0,   1;
  24,   0, 168,   0, 140,   0,  28,   0,   1;
Production matrix begins
   0,   1;
   1,   0,   1;
   0,   2,   0,   1;
  -1,   0,   3,   0,   1;
   0,  -4,   0,   4,   0,   1;
   6,   0, -10,   0,   5,   0,   1;
   0,  36,   0, -20,   0,   6,   0,   1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A155856, A156367.

T[n_, k_] := Boole[k <= n] Binomial[n, k] ((n-k)/2)! (1 + (-1)^(n-k))/2; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 30 2016 *)

T(n,k) = [k<=n]*binomial(n,k)*((n-k)/2)!*(1+(-1)^(n-k))/2.

G.f.: 1/(1-x*y-x^2/(1-x*y-x^2/(1-x*y-2x^2/(1-x*y-2x^2/(1-x*y-3x^2/(1-... (continued fraction).

A137704 Hankel transform of aerated factorial numbers.

Original entry on oeis.org

1, 1, 1, 2, 8, 96, 3456, 497664, 286654464, 825564856320, 11888133931008000, 1027134771639091200000, 532466665617704878080000000, 1932215036193527461576704000000000

Offset: 0

Paul Barry, Feb 07 2008

Hankel transform of A084261. Hankel transform of A000142 (n!) with interpolated zeros.

a(n+1) is the Hankel transform of A003319 aerated. [From Paul Barry, Oct 07 2008]

a(n):=Product{k=0..n, floor((k+2)/2)^(n-k)};

a(n) ~ n^(n^2/2 + 3*n/2 + 7/6) * Pi^(n + 3/2) / (A^4 * 2^(n^2/2 + n/2 - 1/3) * exp(3*n^2/4 + 3*n/2 - 1/3)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jan 20 2024

A156367 Triangle T(n, k) = binomial(n+k, 2k)k!, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 30, 42, 24, 1, 15, 70, 168, 216, 120, 1, 21, 140, 504, 1080, 1320, 720, 1, 28, 252, 1260, 3960, 7920, 9360, 5040, 1, 36, 420, 2772, 11880, 34320, 65520, 75600, 40320, 1, 45, 660, 5544, 30888, 120120, 327600, 604800, 685440, 362880

Offset: 0

Paul Barry, Feb 08 2009

			Triangle begins
  1;
  1,  1;
  1,  3,   2;
  1,  6,  10,    6;
  1, 10,  30,   42,    24;
  1, 15,  70,  168,   216,   120;
  1, 21, 140,  504,  1080,  1320,   720;
  1, 28, 252, 1260,  3960,  7920,  9360,  5040;
  1, 36, 420, 2772, 11880, 34320, 65520, 75600, 40320;

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A084261 (diagonal sums), A155856 (row reversal), A155857 (row sums)

Flatten[Table[Binomial[n+k,2k]k!,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jun 17 2015 *)

flatten([[factorial(k)*binomial(n+k, 2*k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 05 2021

G.f.: 1/(1 -x -x*y/(1 -x -x*y/(1 -x -2*x*y/(1 -x -2*x*y/(1 -x -3*x*y/(1 -x -3*x*y/(1 - ... (continued fraction).
T(n, k) = binomial(n+k, 2*k)*k!
T(n, k) = A155856(n, n-k).
Sum_{k=0..n} T(n, k) = A155857(n).
sum_{k=0..floor(n/2)} T(n, k) = A084261(n).

A162748 Row sums of factorial-Pascal matrix A162747.

Original entry on oeis.org

1, 2, 5, 14, 42, 132, 430, 1444, 4984, 17648, 64024, 237712, 902416, 3499680, 13853424, 55931168, 230142848, 964460288, 4113656704, 17846729984, 78708574976, 352678567424, 1604739694848, 7411167960576, 34723660917760

Offset: 0

Paul Barry, Jul 12 2009

Second binomial transform of aerated factorial numbers. Binomial transform of A084261. Hankel transform is A137704.

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

Table[Sum[Binomial[n,k]*2^(n-k)*(k/2)!*(1+(-1)^k)/2,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 15 2013 *)

G.f.: 1/(1-2x-x^2/(1-2x-x^2/(1-2x-2x^2/(1-2x-2x^2/(1-2x-3x^2/(1-2x-3x^2/(1-2x-4x^2/(1-2x-... (continued fraction);
a(n)=sum{k=0..floor(n/2), C(n,2k)*2^(n-2k)*F(k+1)}=sum{k=0..n, C(n,k)*2^(n-k)*(k/2)!*(1+(-1)^k)/2}.
a(n)=sum{k=0..n, A161556(n,k)*2^k}. - Paul Barry, Apr 11 2010
E.g.f.: exp(2x)*(1+(sqrt(Pi)/2)*x*exp(x^2/4)*erf(x/2)). - Paul Barry, Sep 17 2010
Apparently -2*a(n) +8*a(n-1) +(n-8)*a(n-2) +2*(2-n)*a(n-3)=0. - R. J. Mathar, Oct 25 2012
a(n) ~ 1/2 * sqrt(Pi*n) * exp(2*sqrt(2*n)-n/2-2) * (n/2)^(n/2) * (1 + 1/(3*sqrt(2*n))). - Vaclav Kotesovec, Aug 15 2013

Minor edits by Vaclav Kotesovec, Jul 22 2015

A093190 Array t read by antidiagonals: number of {112,212}-avoiding words.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 9, 4, 1, 8, 21, 16, 5, 1, 10, 39, 52, 25, 6, 1, 12, 63, 136, 105, 36, 7, 1, 14, 93, 292, 365, 186, 49, 8, 1, 16, 129, 544, 1045, 816, 301, 64, 9, 1, 18, 171, 916, 2505, 3006, 1603, 456, 81, 10, 1, 20, 219, 1432, 5225, 9276, 7315, 2864, 657, 100, 11

Offset: 1

Ralf Stephan, Apr 20 2004

t(k,n) = number of n-long k-ary words that simultaneously avoid the patterns 112 and 212.

			Square array begins as:
  1  1   1   1    1    1 ... 1*A000012;
  2  4   6   8   10   12 ... 2*A000027;
  3  9  21  39   63   93 ... 3*A002061;
  4 16  52 136  292  544 ... 4*A135859;
  5 25 105 365 1045 2505 ... ;
Antidiagonal rows begins as:
  1;
  1,  2;
  1,  4,  3;
  1,  6,  9,   4;
  1,  8, 21,  16,   5;
  1, 10, 39,  52,  25,  6;
  1, 12, 63, 136, 105, 36, 7;

G. C. Greubel, Antidiagonal rows n = 1..50, flattened
A. Burstein and T. Mansour, Words restricted by patterns with at most 2 distinct letters, arXiv:math/0110056 [math.CO], 2001.

Main diagonal is A052852.

Antidiagonal sums are in A084261 - 1.

[(&+[Factorial(j)*Binomial(k,j)*Binomial(n-k,j-1): j in [0..n-k+1]]): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 09 2021

T[n_, k_]:= Sum[j!*Binomial[k, j]*Binomial[n-k, j-1], {j,0,n-k+1}];
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 09 2021 *)

t(n,k)=sum(j=0,k,j!*binomial(k,j)*binomial(n-1,j-1))

flatten([[ sum(factorial(j)*binomial(k,j)*binomial(n-k,j-1) for j in (0..n-k+1)) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 09 2021

t(n, k) = Sum{j=0..n} j!*C(n, j)*C(k-1, j-1). (square array)

T(n, k) = Sum_{j=0..n-k+1} j!*binomial(k,j)*binomial(n-k,j-1). (number triangle) - G. C. Greubel, Mar 09 2021

A113485 Number of partitions of [n] avoiding the pattern 12/34.

Original entry on oeis.org

1, 2, 5, 14, 41, 122, 367, 1114, 3423, 10670, 33841, 109398, 361045, 1217346, 4195267, 14775986, 53172411, 195396310, 732806677, 2802898190, 10926431393, 43381582538, 175311002903, 720640632074, 3011495745175, 12786738800254

Offset: 1

Adam Goyt, Jan 09 2006

nonn

The first sum in the formula counts those partitions with a single block of size at least 3. The second sum counts those partitions with blocks of size at most 2. It's easy to see that to avoid 12/34 a partition cannot contain more than one block of size at least 3.

The elements shown satisfy the hypergeometric recurrence 2*a(n) -10*a(n-1) +(-n+13)*a(n-2) +2*(2*n+1)*a(n-3) +3*(-n-5)*a(n-4) +4*(-n+6)*a(n-5) +4*(n-5)*a(n-6)=0. - R. J. Mathar, Jan 25 2013

			For n=1,2,3 a(n)=B_n, where B_n is the n-th Bell number, since there aren't enough distinct elements for such a partition to contain a copy of 12/34. By a similar argument a(4)=B_4-1=14.

M. Klazar, Counting Pattern-free Set Partitions I: A Generalization of Stirling Numbers of the Second Kind, Europ. J. Combinatorics, Vol. 21 (2000), pp. 367-378.

G. C. Greubel, Table of n, a(n) for n = 1..880

Cf. A084261.

Table[Sum[Sum[(k+1)^2 Binomial[n, 2k+p] k!, {k, 0, Floor[(n-p)/2]}], {p, 3, n}]+Sum[Binomial[n, 2k] k!, {k, 0, Floor[n/2]}], {n, 1, 31}]

a(n)=sum(p=3,n, sum(k=0,(n-p)\2, binomial(n,2*k+p)*(k+1)^2*k!)) + sum(k=0,n\2, binomial(n,2*k)) \\ Charles R Greathouse IV, Mar 12 2017

a(n) = Sum[Sum[(k+1)^2 binomial[n, 2k+p] k!, {k, 0, Floor[(n-p)/2]}], {p, 3, n}] + Sum[binomial[n, 2k] k!, {k, 0, Floor[n/2]}].
From Vaclav Kotesovec, Jun 10 2019: (Start)
Recurrence: 2*(n^2-8*n+13)*a(n) = 2*(4*n^2-31*n+43)*a(n-1) + (n^3-17*n^2+87*n-91)*a(n-2) - (3*n^3-27*n^2+78*n-64)*a(n-3) + 2*(n-3)*(n^2-6*n+6)*a(n-4).
a(n) ~ sqrt(Pi) * exp(sqrt(2*n) - n/2 - 1/2) * n^(n/2 + 1) / 2^(n/2 + 3/2) * (1 + 4*sqrt(2)/(3*sqrt(n))). (End)

A174516 Partial sums of A002896.

Original entry on oeis.org

1, 7, 97, 1957, 46687, 1219243, 33715399, 970085119, 28740443449, 870830918389, 26860099935529, 840549807424369, 26620996978712269, 851664885506669269, 27482469263443730269, 893460843597349019629, 29235859228655427097639

Offset: 0

Jonathan Vos Post, Mar 20 2010

nonn

			a(4) = 1 + 6 + 90 + 1860 + 44730 = 46687.

Cf. A002896, A049020, A049037, A084261, A138540, A140476.

b[n_] := b[n] = (* A002896 *) Binomial[2*n, n]*HypergeometricPFQ[{1/2, -n, -n}, {1, 1}, 4]; a[n_] := Sum[b[k], {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Dec 20 2011 *)

a(n) = Sum_{i=0..n} A002896(i).
G.f.: g/(1-x) where g is the o.g.f. of A002896. - Mark van Hoeij, Nov 12 2011
a(n) ~ 2^(2*n) * 3^(2*n + 7/2) / (35 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Feb 17 2024

A372829 a(n) = n! * Sum_{k=0..floor(n/2)} k! / (2*k)!.

Original entry on oeis.org

1, 1, 3, 9, 38, 190, 1146, 8022, 64200, 577800, 5778120, 63559320, 762712560, 9915263280, 138813690960, 2082205364400, 33315285870720, 566359859802240, 10194477476803200, 193695072059260800, 3873901441188844800, 81351930264965740800, 1789742465829286214400

Offset: 0

Ilya Gutkovskiy, May 14 2024

nonn

Cf. A001813, A009179, A084261.

A372829 := proc(n)
    add( k!/(2*k)!,k=0..floor(n/2)) ;
    %*n! ;
end proc:
seq(A372829(n),n=0..70) ; # R. J. Mathar, Sep 27 2024

Table[n! Sum[k!/(2 k)!, {k, 0, Floor[n/2]}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(1 + Sqrt[Pi] x Exp[x^2/4] Erf[x/2]/2)/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = n! * sum(k=0, n\2,  k! / (2*k)!); \\ Michel Marcus, May 14 2024

E.g.f.: (1 + sqrt(Pi) * x * exp(x^2/4) * erf(x/2) / 2) / (1 - x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * k! * (n-2*k)!.
a(n) ~ n! * (1 + exp(1/4)*sqrt(Pi)*erf(1/2)/2). - Vaclav Kotesovec, May 14 2024
D-finite with recurrence 2*a(n) -2*n*a(n-1) -n*a(n-2) +n*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 27 2024

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