A345747 -id:A345747 - cp's OEIS frontend

A358080 Expansion of e.g.f. 1/(1 - x^2 * exp(x)).

1, 0, 2, 6, 36, 260, 2190, 21882, 248696, 3181320, 45229050, 707208590, 12063902532, 222939837276, 4436813677478, 94605994108290, 2151763873634160, 51999544476324752, 1330540380342907506, 35936656483848501654, 1021700660649312689660

Offset: 0

Seiichi Manyama, Oct 30 2022

Seiichi Manyama, Table of n, a(n) for n = 0..425

Cf. A006153, A358081.

Cf. A216507, A345747, A358064.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^2*exp(x))))

a(n) = n!*sum(k=0, n\2, k^(n-2*k)/(n-2*k)!);

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

a(n) ~ n! / ((1 + LambertW(1/2)) * 2^(n+1) * LambertW(1/2)^n). - Vaclav Kotesovec, Oct 30 2022

A355575 a(n) = n! * Sum_{k=0..floor(n/3)} k^(n - 3*k)/k!.

Original entry on oeis.org

1, 0, 0, 6, 24, 120, 1080, 10080, 120960, 1874880, 34473600, 738460800, 17982518400, 489858969600, 14834839219200, 498452777222400, 18583796335104000, 768773914900992000, 35220800475250790400, 1779227869201400217600, 98469904378626772992000

Offset: 0

Seiichi Manyama, Sep 17 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..323

Cf. A345747, A354436.

Cf. A292889, A352945.

Join[{1}, Table[n!*Sum[k^(n - 3*k)/k!, {k, 0, n/3}], {n, 1, 20}]] (* Vaclav Kotesovec, Oct 30 2022 *)

PARI
```
a(n) = n!*sum(k=0, n\3, k^(n-3*k)/k!);
```

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^(3*k)/(k!*(1-k*x)))))

E.g.f.: Sum_{k>=0} x^(3*k) / (k! * (1 - k * x)).

a(n) ~ sqrt(Pi) * exp((n - 1/2)/LambertW(exp(3/4)*(2*n - 1)/8) - 2*n) * n^(2*n + 1/2) / (sqrt(1 + LambertW(exp(3/4)*(2*n - 1)/8)) * 2^(2*n + 1/2) * LambertW(exp(3/4)*(2*n - 1)/8)^n). - Vaclav Kotesovec, Oct 30 2022

A347433 Irregular triangle read by rows: T(n,k) is the difference between the total arch lengths of a semi-meander multiplied by its number of exterior arches and total arch lengths of the semi-meanders with n + 1 top arches generated by the exterior arch splitting algorithm on the given semi-meander.

Original entry on oeis.org

4, 4, 4, 10, 4, 11, 4, 12, 20, 4, 13, 22, 4, 14, 24, 34, 4, 15, 26, 37, 4, 16, 28, 40, 52, 4, 17, 30, 43, 56, 4, 18, 32, 46, 60, 74, 4, 19, 34, 49, 64, 79, 4, 20, 36, 52, 68, 84, 100, 4, 21, 38, 55, 72, 89, 106, 4, 22, 40, 58, 76, 94, 112, 130, 4, 23, 42, 61, 80, 99

Offset: 2

Roger Ford, Sep 01 2021

			n = number of top arches, k = number of exterior top arches:
n\k  2   3   4   5   6
2:   4
3:   4
4:   4   10
5:   4   11
6:   4   12  20
7:   4   13  22
8:   4   14  24  34
9:   4   15  26  37
10:  4   16  28  40  52
Length of each arch = 1 + number of arches covered:
Top arches of a given semi-meander:       Arch splitting generated
n = 5, k = 2                              semi-meanders (6 top arches):
     1     1    = 2 exterior arches                /\
           /\                                     //\\
     /\   //\\                                   ///\\\
    //\\ ///\\\                           /\ /\ ////\\\\
    21   321    = 9 length of top arches  1  1  4321     = 12 length of top arches
                                            /\
                                           //\\   /\
                                          ///\\\ //\\ /\
                                          321    21   1  = 10 length of top arches
    T(5,2) = 4 + (5+2)(2-2) = 4 --------------------------- 4 = (12+10) - (2 * 9);
Top arches of given semi meander:
n = 5, k = 3                                    /\
    1   1    1   = 3 exterior arches           /  \
        /\   /\                               /    \
    /\ //\\ //\\                             //\  /\\
    1  21   21   = 7 length top arches   /\ ///\\//\\\
                                         1  521  21     = 12 length of top arches
                                                   /\
                                          /\      //\\
                                         //\\ /\ ///\\\
                                         21   1  321    = 10 length of top arches
                                            /\
                                           /  \
                                          /  /\\
                                         //\//\\\ /\ /\
                                         41 21    1  1  = 10 length of top arches
    T(5,3) = 4 + (5+2)(3-2) = 11 --------------------- 11 = (12+10+10) - (3 * 7).

Cf. A345747.

For n >= 2 and k = 2..floor((n+2)/2), T(n,k) = 4 + (n+2)*(k-2).

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A358080 Expansion of e.g.f. 1/(1 - x^2 * exp(x)).

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A355575 a(n) = n! * Sum_{k=0..floor(n/3)} k^(n - 3*k)/k!.

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