A021009 -id:A021009 - cp's OEIS frontend

A201199 Triangle version of the array w(N,L) of the total number of round trips of length L on closed Laguerre graphs Lc_N.

1, 1, 2, 1, 4, 3, 1, 18, 9, 4, 1, 76, 53, 16, 5, 1, 322, 357, 120, 25, 6, 1, 1364, 2489, 1024, 233, 36, 7, 1, 5778, 17509, 9424, 2545, 404, 49, 8, 1, 24476, 123449, 89536, 29985, 5400, 645, 64, 9, 1, 103682, 870893, 862560, 367505, 78392, 10213, 968, 81, 10

Offset: 0

Wolfdieter Lang, Nov 30 2011

For Laguerre graphs (open and closed ones) see the W. Lang link on Jacobi graphs under A201198. There one also finds a sketch of the closed Laguerre graph Lc_4 as Fig.4.

The total number of round trips on the closed Laguerre graph Lc_N, for N>=3, with N vertices N^2 loops, binomial(N,2) lines between neighboring vertices and two lines between the first and the last vertex (in total (3*N-1)*N/2+2 = (3*N^2-N+4)/2 lines) is w(N,L) = sum(w(N,L;p_n->p_n),n=1..N) = Trace((L_N)^L) = sum((x_n^{(N)})^L,n=1..N), with the N x N symmetric adjacency matrix, also called Lc_N, having non-vanishing elements (Lc_N)[n,n] = 2*n-1, n=1..N, (Lc_N)[n,n+1] = (Lc_N)[n+1,n] = n, n=1..N-1, and (Lc_N)[1,N]= 2=(Lc_N)[N,1]. The eigenvalues of Lc_N are x_n^{(N)}. They are the zeros of the characteristic polynomial Lac_N(x):=Det(x*1_N -Lc_N) with the N x N unit matrix 1_N. These are the polynomials Lac_N(x) = La(N,x) - 4*La1(N-2,x) - 4*(N-1)!, with the ordinary monic Laguerre polynomials La(N,x) with coefficient array given by A021009(n,m)*(-1)^n and the first associated monic Laguerre polynomials La1(N-2,x) with coefficient array given by A199577(n,m). For N=1 one has Lc_1=L_1 (Laguerre graph with one vertex and one loop) with L_1(x)=x-1, and for N=2 one has a graph where one vertex has one loop, the other three, and there are two lines joining these vertices, hence Lc_2(x)= x^2-4*x-1.

			The array w(N,L) starts:
N\L 0   1    2     3      4        5         6  ...
1:  1   1    1     1      1        1         1
2:  2   4   12    40    136      464      1584
3:  3   9   53   357   2489    17509    123449
4:  4  16  120  1024   9424    89536    862560
5:  5  25  233  2545  29985   367505   4599521
6:  6  36  404  5400  78392  1188336  18460016
7:  7  49  645 10213 176473  3195829  59473593
8:  8  64  968 17728 355536  7493504 162671840
9:  9  81 1385 28809 657953 15826041 392792273
...The triangle a(K,N) = w(N,K-N+1) starts:
K\N 1      2       3      4      5     6     7   8  9..
0:  1
1:  1      2
2:  1      4       3
3:  1     18       9      4
4:  1     76      53     16      5
5:  1    322     357    120     25     6
6:  1   1364    2489   1024    233    36     7
7:  1   5778   17509   9424   2545   404    49   8
8:  1  24476  123449  89536  29985  5400   645  64  9
...
For the graph Lc_4, shown in the W. Lang link as Figure 4, the counting for round trips of length L=2 for each of the four vertices V_i, i=1..4, read from left to right, is as follows.
V_1: 1+1+(1+1+2*1), V_2: 3+2*binomial(3,2)+1+(1+1+2*1),
V_3: 5+2*binomial(5,2)+(1+1+2*1)+(3+2*binomial(3,2)),
V_4: 7+2*binomial(7,2)+(3+2*binomial(3,2))+(1+1+2*1),
this sums to the total number  w(4,2)= 120  =  a(5,4).
Compared to the open L_4 graph (see the corresponding A201198 entry 4*28 = 112) one has to add 2*(1+1+2*1)=8 from the new two lines joining V_1 and V_4.

Wolfdieter Lang, Counting walks on Jacobi graphs: an application of orthogonal polynomials.

Cf. A201198 (open Laguerre graphs).

a(K,N) =  w(N,K-N+1),K>=0, N=1,...,K+1, with w(N,L) the total number of round trips of length L on the closed Laguerre graph Lc_N described above in the comment section.
The o.g.f. of w(N,L) is: G(N,x)=y*(d/dx)Lac_N(x)/Lac_N(x) with y=1/x.
  The characteristic polynomial Lac_N(x) has also been given in the comment section above.

A218234 Infinitesimal generator for padded Pascal matrix A097805 (as lower triangular matrices).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0

Offset: 0

Tom Copeland, Oct 24 2012

Matrix T begins
0;
0,0;
0,1,0;
0,0,2,0;
0,0,0,3,0;
0,0,0,0,4,0;
Let M(t) = exp(t*T) = limit [1 + t*T/n]^n as n tends to infinity.
Then M(1) = the lower triangular padded Pascal matrix A097805, with inverse M(-1).
Given a polynomial sequence p_n(x) with p_0(x)=1 and the lowering and raising operators L and R defined by L P_n(x) = n * P_(n-1)(x) and R P_n(x) = P_(n+1)(x), the matrix T represents the action of R^2*L in the p_n(x) basis. For p_n(x) = x^n, L = D = d/dx and R = x. For p_n(x) = x^n/n!, L = DxD and R = D^(-1).
See A132440 for an analog and more general discussion.

P. Blasiak and P. Flajolet, Combinatorial models of creation-annihilation
T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
T. Copeland, Mathemagical Forests
T. Copeland, Addendum to Mathemagical Forests
G. Dattoli, B. Germano, M. Martinelli, and P. Ricci, Touchard like polynomials and generalized Stirling polynomials
W. Lang, Combinatorial interpretation of generalized Stirling numbers

Table[PadLeft[{n-1, 0}, n+1], {n, 0, 12}]  // Flatten (* Jean-François Alcover, Apr 30 2014 *)

The matrix operation b = T*a can be characterized in several ways in terms of the coefficients a(n) and b(n), their o.g.f.s A(x) and B(x), or e.g.f.s EA(x) and EB(x):
  1) b(0) = 0, b(1) = 0, b(n) = (n-1) * a(n-1),
  2) B(x) = x^2D A(x)= x (xDx)(1/x)A(x) = x^2 * Lag(1,-:xD:) A(x)/x , or
  3) EB(x) = D^(-1)xD  EA(x),
  where D is the derivative w.r.t. x, (D^(-1)x^j/j!) = x^(j+1)/(j+1)!, (:xD:)^j = x^j*D^j, and Lag(n,x) are the Laguerre polynomials A021009.
So the exponentiated operator can be characterized as
  4) exp(t*T) A(x) = exp(t*x^2D) A(x) = x exp(t*xDx)(1/x)A(x)
     = x [sum(n=0,1,...) (t*x)^n * Lag(n,-:xD:)] A(x)/x
     = x [exp{[t*u/(1-t*u)]*:xD:} / (1-t*u) ] A(x)/x (eval. at u=x)
     = A[x/(1-t*x)], a special Moebius or linear fractional trf.,
  5) exp(t*T) EA(x) =  D^(-1) exp(t*x)D EA(x), a shifted Euler trf.
     for an e.g.f., or
  6) [exp(t*T) * a]_n = [M(t) * a]_n
     = [sum(k=0,...,n-1) binomial(n-1,k)*  t^(n-1-k) * a(k+1)] with [M(t) * a]_0 = a_0
For generalizations and more on the operator x^2D, see A132440 and the references therein and above, and A094638.

A305465 a(n) = Sum_{k=0..floor(n/2)} ((n-k)!/k!)binomial(n-k,k)n^(n-2*k).

Original entry on oeis.org

1, 1, 9, 174, 6433, 387045, 34372513, 4223468872, 685727920641, 142133068151865, 36615156774045001, 11474421446955693006, 4298048476279871328289, 1896322606147540294800349, 973319784969445114237699713, 575000041101937659730069884960

Offset: 0

Seiichi Manyama, Jun 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Main diagonal of A305401.

Cf. A021009, A305467.

Join[{1},Table[Sum[(n-k)!/k! Binomial[n-k,k]n^(n-2k),{k,0,Floor[n/2]}],{n,20}]] (* Harvey P. Dale, Sep 22 2019 *)

{a(n) = sum(k=0, n/2, ((n-k)!/k!)*binomial(n-k, k)*n^(n-2*k))}

a(n) ~ n! * n^n. - Vaclav Kotesovec, Jun 03 2018

A331333 Interpolating the factorial and the powers of 2. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 2, 2, 8, 4, 6, 36, 36, 8, 24, 192, 288, 128, 16, 120, 1200, 2400, 1600, 400, 32, 720, 8640, 21600, 19200, 7200, 1152, 64, 5040, 70560, 211680, 235200, 117600, 28224, 3136, 128, 40320, 645120, 2257920, 3010560, 1881600, 602112, 100352, 8192, 256

Offset: 0

Peter Luschny, Jan 19 2020

			Triangle starts:
  [0] 1
  [1] 1,     2
  [2] 2,     8,      4
  [3] 6,     36,     36,      8
  [4] 24,    192,    288,     128,     16
  [5] 120,   1200,   2400,    1600,    400,     32
  [6] 720,   8640,   21600,   19200,   7200,    1152,   64
  [7] 5040,  70560,  211680,  235200,  117600,  28224,  3136,   128
  [8] 40320, 645120, 2257920, 3010560, 1881600, 602112, 100352, 8192, 256

T(n, 0) = A000142(n), T(n, n) = A000079(n).
Row sums: A087912, alternating row sums: A295382, antidiagonal sums: A222467, positive half sums: A129683, negative half sums: A331334.
Cf. A021009.

A331333 := proc(n, k) local S; S := proc(n, k) option remember;
`if`(k = 0, 1, `if`(k > n, 0, 2*S(n-1, k-1)/k + S(n-1, k))) end: n!*S(n, k) end:
seq(seq(A331333(n, k), k=0..n), n=0..8);

T(n, k) = n!*S(n, k) where S(n, k) is recursively defined by:
if k = 0 then 1 else if k > n then 0 else 2*S(n-1, k-1)/k + S(n-1, k).
From Peter Bala, Jan 19 2020: (Start)
T(n,k) = 2^k*(n!/k!)*binomial(n,k).
E.g.f.: exp((2*x*t)/(1 - x))/(1 - x) = 1 + (1 + 2*t)*x + (2 + 8*t + 4*t^2)*x^2/2! + .... Cf. A021009. (End)

A104558 Triangle, read by rows, equal to the matrix inverse of A104557 and related to Laguerre polynomials.

Original entry on oeis.org

1, -1, 1, 0, -2, 1, 0, 2, -4, 1, 0, 0, 6, -6, 1, 0, 0, -6, 18, -9, 1, 0, 0, 0, -24, 36, -12, 1, 0, 0, 0, 24, -96, 72, -16, 1, 0, 0, 0, 0, 120, -240, 120, -20, 1, 0, 0, 0, 0, -120, 600, -600, 200, -25, 1, 0, 0, 0, 0, 0, -720, 1800, -1200, 300, -30, 1, 0, 0, 0, 0, 0, 720, -4320, 5400, -2400, 450, -36, 1, 0, 0, 0, 0, 0, 0, 5040, -15120, 12600, -4200, 630, -42, 1

Offset: 0

Paul D. Hanna, Mar 16 2005

Even-indexed rows are found in A066667 (generalized Laguerre polynomials). Odd-indexed rows are found in A021009 (Laguerre polynomials L_n(x)). Row sums equal A056920 (offset 1). Absolute row sums equal A056953 (offset 1).

			Rows begin:
  1;
  -1,1;
  0,-2,1;
  0,2,-4,1;
  0,0,6,-6,1;
  0,0,-6,18,-9,1;
  0,0,0,-24,36,-12,1;
  0,0,0,24,-96,72,-16,1;
  0,0,0,0,120,-240,120,-20,1;
  0,0,0,0,-120,600,-600,200,-25,1;
  ...
Unsigned columns read downwards equals rows of matrix inverse A104557 read backwards:
  1;
  1,1;
  2,2,1;
  6,6,4,1;
  24,24,18,6,1;
  120,120,96,36,9,1;
  ...

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

Cf. A104557, A066667, A021009, A056920, A056953.

/* As triangle */ [[(-1)^(n-k)*Factorial(n-k)*Binomial(1+ Floor(n/2), k +1 -Floor((n+1)/2))*Binomial(Floor((n+1)/2), k - Floor(n/2)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 14 2018

T[n_, k_] := (-1)^(n - k)*(n - k)!*Binomial[1 + Floor[n/2], k + 1 - Floor[(n + 1)/2]]*Binomial[Floor[(n+1)/2], k -Floor[n/2]]; Table[T[n, k], {n, 0, 20}, {k, 0, n}]//Flatten (* G. C. Greubel, May 14 2018 *)

{T(n,k)=(-1)^(n-k)*(n-k)!*binomial(1+n\2,k+1-(n+1)\2)* binomial( (n+1)\2,k-n\2)};

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

A130562 Triangular table of denominators of the coefficients of Laguerre-Sonin polynomials L(1/2,n,x).

Original entry on oeis.org

1, 2, 1, 8, 2, 2, 16, 8, 4, 6, 128, 16, 16, 4, 24, 256, 128, 32, 16, 48, 120, 1024, 256, 256, 32, 192, 240, 720, 2048, 1024, 512, 256, 384, 64, 96, 5040, 32768, 2048, 2048, 512, 3072, 384, 384, 10080, 40320, 65536, 32768, 4096, 2048, 6144, 3072, 2304, 40320

Offset: 0

Wolfdieter Lang, Jul 13 2007

The corresponding numerator table is given in A131440.

			Triangle begins:
    1;
    2,   1;
    8,   2,  2;
   16,   8,  4,  6;
  128,  16, 16,  4, 24;
  256, 128, 32, 16, 48, 120;
  ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 775, 22.3.9.
Wolfdieter Lang, Rational coefficients and more

Cf. A021009 (Coefficient table of n!*L(n, 0, x)).

from sympy import binomial, factorial, Integer
def a(n, m): return ((-1)**m * binomial(n + 1/Integer(2), n -m) / factorial(m)).denominator
for n in range(21): print([a(n, m) for m in range(n + 1)]) # Indranil Ghosh, Jun 29 2017

a(n,m) = denom(L(1/2,n,m)) with L(1/2,n,m)=((-1)^m)*binomial(n+1/2,n-m)/m!, n>=m>=0, else 0 (taken in lowest terms).

A216313 Total number of cycles in all partial permutations of {1,2,...,n}.

Original entry on oeis.org

0, 1, 5, 29, 200, 1609, 14809, 153453, 1767240, 22383681, 309123733, 4621295117, 74331184256, 1279614456041, 23470211031097, 456836915073277, 9403557603534960, 204061142480099649, 4655419598313230277, 111378768040665868093, 2788108620329147151896

Offset: 0

Geoffrey Critzer, Sep 04 2012

nonn

Cf. A002720, A021009.

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[D[Exp[ x/(1-x)]/(1-x)^y,y]/.y->1,{x,0,nn}],x]

E.g.f.: exp(x/(1-x))*log(1/(1-x))/(1-x).
a(n) = sum(k=0..n, A216294(n,k)*k ).
a(n) = (4*n-3)*a(n-1) - (6*n^2 - 17*n + 13)*a(n-2) + (n-2)^2*(4*n-9)*a(n-3) - (n-3)^3*(n-2)*a(n-4). - Vaclav Kotesovec, Sep 24 2013
a(n) ~ sqrt(2)/4 * n^(n+1/4) * exp(2*sqrt(n)-n-1/2) * (log(n)*(1 + 31/(48*sqrt(n)) + 553/(4608*n)) + 1/sqrt(n) + 43/(48*n)). - Vaclav Kotesovec, Sep 24 2013

A265649 Triangle of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..n} T(n,k)x^k where T(0,0) = 1, and T(n,k) = 0 for k < 0 or k > n, and T(n,k) = T(n-1,k-1) + (2n-1+k)*T(n-1,k) for n > 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 3, 5, 1, 15, 33, 12, 1, 105, 279, 141, 22, 1, 945, 2895, 1830, 405, 35, 1, 10395, 35685, 26685, 7500, 930, 51, 1, 135135, 509985, 435960, 146685, 23310, 1848, 70, 1, 2027025, 8294895, 7921305, 3076290, 589575, 60270, 3318, 92, 1, 34459425, 151335135, 158799690, 69447105, 15457365, 1915515, 136584, 5526, 117, 1

Offset: 0

Werner Schulte, Dec 11 2015

The polynomials p(n,x) satisfy the differential equation: x*y''' + (3*x+1)*y'' + (2*x+2)*y' - 2*n*y = 0 where y' = dy/dx (first derivative).

Appears to be the exponential Riordan array [1/sqrt(1 - 2x), 1/(sqrt(1 - 2x) - 1)]. [Barry, Example 1] - Eric M. Schmidt, Sep 23 2017

			The triangle T(n,k) begins:
n\k:        0        1        2        3       4      5     6   7  8
  0:        1
  1:        1        1
  2:        3        5        1
  3:       15       33       12        1
  4:      105      279      141       22       1
  5:      945     2895     1830      405      35      1
  6:    10395    35685    26685     7500     930     51     1
  7:   135135   509985   435960   146685   23310   1848    70   1
  8:  2027025  8294895  7921305  3076290  589575  60270  3318  92  1
  etc.
The polynomial corresponding to row 3 is p(3,x) = 15 + 33*x + 12*x^2 + x^3.

Paul Barry, A Note on d-Hankel Transforms, Continued Fractions, and Riordan Arrays, arXiv:1702.04011 [math.CO], 2017.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See pp. 21, 28.

Cf. A000007, A000326, A001147, A001497, A021009, A129890, A035342.

T := (n, k) -> local j; 2^n*add((-1)^(k-j)*binomial(k, j)*pochhammer((j+1)/2, n), j=0..k) / k!: for n from 0 to 6 do seq(T(n, k), k=0..n) od;  # Peter Luschny, Mar 04 2024

(* The function RiordanArray is defined in A256893. *)
rows = 10;
R = RiordanArray[1/Sqrt[1 - 2 #]&, 1/Sqrt[1 - 2 #] - 1&, rows, True];
R // Flatten (* Jean-François Alcover, Jul 20 2019 *)

Recurrence: p(0,x) = 1 and p(n+1,x) = (2*n+1+x)*p(n,x) + x*p'(n,x).
T(n,0) = A001147(n), T(n+1,1) = A129890(n), T(n+1,n) = A000326(n+1), and Sum_{k=0..n} (-1)^k*k!*T(n,k) = A000007(n).
Recurrence: k^2*(k+1)*T(n,k+1) = (2*n+2-2*k)*T(n,k-1)-k*(3*k-1)*T(n,k).
Conjecture: T(n,k) = 2^(n-k)*(n-k)!*binomial(n,k)*(Sum_{j=0..n-k} (-1/4)^j* binomial(2*j+k,j)*binomial(n,j+k)).
Conjecture: T(n,k) = (-1)^k*Sum_{j=0..n-1} A001497(n-1,j)*A021009(j+1,k).
T(n,k) = (Sum_{i=0..k} (-1)^(k-i) * binomial(k, i)*Product_{j=1..n} (2*j+i-1))/k!. - Werner Schulte, Mar 03 2024
T(n,k) = (2^n/k!)*(Sum_{j=0..k}(-1)^(k-j)*binomial(k,j)*Pochhammer((j + 1)/2, n)). - Peter Luschny, Mar 04 2024

A331325 a(n) = n!*[x^n] cosh(x/(1-x))/(1-x).

Original entry on oeis.org

1, 1, 3, 15, 97, 745, 6571, 65359, 723969, 8842257, 118091251, 1712261551, 26786070433, 449634481465, 8059974923547, 153634497337455, 3102367733191681, 66145005096272929, 1484586887025099619, 34983117545622446287, 863397428225495045601, 22269844592814969946761

Offset: 0

Peter Luschny, Jan 21 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..443

Cf. A002720, A009940, A021009, A331326.

gf := cosh(x/(1 - x))/(1 - x): ser := series(gf, x, 22):
seq(n!*coeff(ser, x, n), n=0..21);
# Alternative: seq(add(abs(A021009(n, 2*k)), k=0..n/2), n=0..21);
A331325 := proc(n) local S; S := proc(n, k) option remember; `if`(k = 0, 1,
`if`(k > n, 0, S(n-1, k-1)/k + S(n-1, k))) end: n!*add(S(n, 2*k), k=0..n) end:
seq(A331325(n), n=0..21);

a[n_] := n! HypergeometricPFQ[{1/2 - n/2, -n/2}, {1, 1/2, 1/2}, 1/4];
Array[a, 22, 0]

x='x+O('x^22); Vec(serlaplace(cosh(x/(1-x))/(1-x)))

def A331325():
    sa, sb, ta, tb, n = 1, 2, 1, 0, 2
    yield sa
    yield ta
    while(True):
        s = 2*n*sb - ((n-1)**2)*sa
        t = 2*(n-1)*tb - ((n-1)**2)*ta
        sa, sb, ta, tb = sb, s, tb, t
        n += 1
        yield (s + t)//2
a = A331325(); print([next(a) for _ in range(22)])

a(n) + A331326(n) = A002720(n).
a(n) - A331326(n) = A009940(n).
a(n) = Sum_{k=0..n/2} |A021009(n, 2*k)|.
a(n) = Sum_{k=0..n} binomial(n, 2*k)*n!/(2*k)!.
a(n) = n!*hypergeom([1/2 - n/2, -n/2], [1/2, 1/2, 1], 1/4).
(n+1)^2*(n+2)^2*a(n) - 4*(n+2)^3*a(n+1) + (6*n^2+30*n+37)*a(n+2) - 4*(n+3)*a(n+3)+a(n+4)=0. - Robert Israel, Jan 23 2020
Sum_{n>=0} a(n) * x^n / (n!)^2 = (1/2) * exp(x) * (BesselI(0,2*sqrt(x)) + BesselJ(0,2*sqrt(x))). - Ilya Gutkovskiy, Jul 18 2020
a(n) ~ 2^(-3/2) * exp(2*sqrt(n)-n-1/2) * n^(n+1/4) * (1 + 31/(48*sqrt(n))). - Vaclav Kotesovec, Feb 17 2024

A331326 a(n) = n!*[x^n] sinh(x/(1 - x))/(1 - x).

Original entry on oeis.org

0, 1, 4, 19, 112, 801, 6756, 65563, 717760, 8729857, 116570980, 1693096131, 26548383984, 446689827169, 8023582921732, 153192673528651, 3097301219335936, 66095983547942913, 1484384376886189380, 34991710162280602867, 863797053818651591920, 22282392569877969167521

Offset: 0

Peter Luschny, Jan 21 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..443

Cf. A002720, A009940, A021009, A331325.

gf := sinh(x/(1 - x))/(1 - x): ser := series(gf, x, 22):
seq(n!*coeff(ser, x, n), n=0..20);
# Alternative: seq(add(abs(A021009(n, 2*k+1)), k=0..n/2), n=0..21);
A331326 := proc(n) local S; S := proc(n, k) option remember; `if`(k = 0, 1,
`if`(k > n, 0, S(n-1, k-1)/k + S(n-1, k))) end: n!*add(S(n, 2*k+1), k=0..n) end:
seq(A331326(n), n=0..21);

a[n_] := n n! HypergeometricPFQ[{1/2 - n/2, 1 - n/2}, {1, 3/2, 3/2}, 1/4];
Array[a, 22, 0]

x='x+O('x^22); concat(0,Vec(serlaplace(sinh(x/(1-x))/(1-x))))

def A331326():
    sa, sb, ta, tb, n = 1, 2, 1, 0, 2
    yield 0
    yield ta
    while(True):
        s = 2*n*sb - ((n-1)**2)*sa
        t = 2*(n-1)*tb - ((n-1)**2)*ta
        sa, sb, ta, tb = sb, s, tb, t
        n += 1
        yield (s - t)//2
a = A331326(); print([next(a) for _ in range(22)])

a(n) + A331325(n) = A002720(n).
A331325(n) - a(n) = A009940(n).
a(n) = Sum_{k=0..n/2} |A021009(n, 2*k+1)|.
a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*n!/(2*k+1)!.
a(n) = n*n!*hypergeom([1/2 - n/2, 1 - n/2], [1, 3/2, 3/2], 1/4).
(n+1)^2*(n+2)^2*a(n) - 4*(n+2)^3*a(n+1) + (6*n^2+30*n+37)*a(n+2) - 4*(n+3)*a(n+3)+a(n+4) = 0. - Robert Israel, Jan 22 2020
Sum_{n>=0} a(n) * x^n / (n!)^2 = (1/2) * exp(x) * (BesselI(0,2*sqrt(x)) - BesselJ(0,2*sqrt(x))). - Ilya Gutkovskiy, Jul 17 2020
a(n) ~ 2^(-3/2) * exp(2*sqrt(n)-n-1/2) * n^(n+1/4) * (1 + 31/(48*sqrt(n))). - Vaclav Kotesovec, Feb 17 2024

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