A053817 -id:A053817 - cp's OEIS frontend

A344229 a(n) = n*a(n-1) + n^signum(n mod 4), a(0) = 1.

1, 2, 6, 21, 85, 430, 2586, 18109, 144873, 1303866, 13038670, 143425381, 1721104573, 22374359462, 313241032482, 4698615487245, 75177847795921, 1278023412530674, 23004421425552150, 437084007085490869, 8741680141709817381, 183575282975906165022

Offset: 0

Alois P. Heinz, May 12 2021

nonn

This sequence is one of many possible solutions to puzzle 16 on the Meerdaelquiz puzzle page, cf. the Delestinne link and A090805.

Alois P. Heinz, Table of n, a(n) for n = 0..450
Hugo Delestinne, Meerdaelquiz [broken link]
N. J. A. Sloane and Brady Haran, A Sequence with a Mistake, Numberphile video (2021)

Cf. A000142, A000522, A010873, A033540, A053817, A057427, A090805, A166486, A277506.

a:= proc(n) a(n):= n*a(n-1) + n^signum(n mod 4) end: a(0):= 1:
seq(a(n), n=0..23);

A368759 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * (1 + Sum_{j=0..n} j^k/j!).

Original entry on oeis.org

2, 1, 3, 1, 2, 7, 1, 2, 6, 22, 1, 2, 8, 21, 89, 1, 2, 12, 33, 88, 446, 1, 2, 20, 63, 148, 445, 2677, 1, 2, 36, 141, 316, 765, 2676, 18740, 1, 2, 68, 351, 820, 1705, 4626, 18739, 149921, 1, 2, 132, 933, 2428, 4725, 10446, 32431, 149920, 1349290, 1, 2, 260, 2583, 7828, 15265, 29646, 73465, 259512, 1349289, 13492901

Offset: 0

Seiichi Manyama, Jan 04 2024

			Square array begins:
     2,    1,    1,     1,     1,     1,      1, ...
     3,    2,    2,     2,     2,     2,      2, ...
     7,    6,    8,    12,    20,    36,     68, ...
    22,   21,   33,    63,   141,   351,    933, ...
    89,   88,  148,   316,   820,  2428,   7828, ...
   446,  445,  765,  1705,  4725, 15265,  54765, ...
  2677, 2676, 4626, 10446, 29646, 99366, 375246, ...

Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials.

Columns k=0..3 give A038159, A033540(n+1), A053817, A368760.

Cf. A337085.

PARI
```
T(n, k) = n!*(1+sum(j=0, n, j^k/j!));
```

T(0,k) = 1 + 0^k and T(n,k) = n^k + n * T(n-1,k) for n>0.
T(n,k) = n! + A337085(n,k).
E.g.f. of column k: (1+ B_k(x) * exp(x)) / (1-x), where B_n(x) = Bell polynomials.

A348311 a(n) = n! * Sum_{k=1..n} (-1)^k * (k-2) / (k-1)!.

Original entry on oeis.org

0, 1, 2, 3, 20, 85, 534, 3703, 29672, 266985, 2669930, 29369131, 352429692, 4581585853, 64142202110, 962133031455, 15394128503504, 261700184559313, 4710603322067922, 89501463119290195, 1790029262385804260, 37590614510101889061, 826993519222241559782

Offset: 0

Ilya Gutkovskiy, Oct 11 2021

nonn

Cf. A000166, A000240, A030297, A053817, A180189, A212291.

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

a(n) = n!*sum(k=1, n, (-1)^k * (k-2) / (k-1)!); \\ Michel Marcus, Oct 20 2021

E.g.f.: x * (1 + x) * exp(-x) / (1 - x).
a(0) = 0; a(n) = n * (a(n-1) + (-1)^n * (n-2)).
a(n) = n * (2 * A000166(n-1) + (-1)^n).

cp's OEIS Frontend

A344229 a(n) = n*a(n-1) + n^signum(n mod 4), a(0) = 1.

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A368759 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * (1 + Sum_{j=0..n} j^k/j!).

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A348311 a(n) = n! * Sum_{k=1..n} (-1)^k * (k-2) / (k-1)!.

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