A024420 -id:A024420 - cp's OEIS frontend

A024421 a(n) = n!*(1/C(n,0) - 1/C(n,1) - ... - 1/C(n,[ n/2 ])).

1, 1, 1, 4, 14, 84, 516, 3936, 32544, 305280, 3123360, 35112960, 427559040, 5629236480, 79568697600, 1203310080000, 19381043404800, 331357630464000, 5993331073228800, 114354145640448000, 2295517848809472000, 48363337839624192000

Offset: 0

Clark Kimberling

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..449

Cf. A024420.

Table[n!(1/Binomial[n,0]-Sum[1/Binomial[n,d],{d,Floor[n/2]}]),{n,0,30}] (* Harvey P. Dale, Feb 03 2024 *)

a(n) = n!*(1 - sum(k=1, floor(n/2), 1/binomial(n,k))); \\ Michel Marcus, Jul 10 2019

More terms from Sean A. Irvine, Jul 10 2019

A370418 Triangle read by rows. T(n, k) = (n - k)! * (n + k)!.

Original entry on oeis.org

1, 1, 2, 4, 6, 24, 36, 48, 120, 720, 576, 720, 1440, 5040, 40320, 14400, 17280, 30240, 80640, 362880, 3628800, 518400, 604800, 967680, 2177280, 7257600, 39916800, 479001600, 25401600, 29030400, 43545600, 87091200, 239500800, 958003200, 6227020800, 87178291200

Offset: 0

Peter Luschny, Feb 27 2024

			Triangle starts:
[0]      1;
[1]      1,      2;
[2]      4,      6,     24;
[3]     36,     48,    120,     720;
[4]    576,    720,   1440,    5040,   40320;
[5]  14400,  17280,  30240,   80640,  362880,  3628800;
[6] 518400, 604800, 967680, 2177280, 7257600, 39916800, 479001600;

Cf. A010050 (main diagonal), A009445 (subdiagonal), A001044 (column 0), A175430 (column 1), A024420 (bisection is alternating sum).

Cf. A119502, A143084.

T := (n, k) -> (n - k)! * (n + k)!:
seq(seq(T(n, k), k = 0..n), n = 0..7);

Table[(n - k)!*(n + k)!, {n, 0, 7}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 05 2024 *)

Sum_{k=0..n} (-1)^k*T(n, k) = n!^2 / 2 + (-1)^n * (2*n + 2)! / (2*n + 2)^2.

cp's OEIS Frontend

A024421 a(n) = n!*(1/C(n,0) - 1/C(n,1) - ... - 1/C(n,[ n/2 ])).

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