A102409 Even triangle n!. This table read by rows gives the coefficients of sum formulas of n-th factorials (A000142). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+3, where k=[2*n+1+(-1)^(n-1)]/4 and T(i,k) satisfies n! = Sum_{i=1..k+3} T(i,k) * n^(i-1) / (2*k-2)!.

0, 1, 0, 0, 0, -20, 8, 0, 0, 20280, -6530, -1275, 362, 3, 0, -8749440, 21627600, -4871940, -66510, 48300, 390, 0, -261763004160, 72965762016, 13117344800, -3757930680, 72406040, 13101144, 90440, 0, -974260634054400, -1140185248443360, 353509119454680, -8136128999880, -3234018579750

Offset

1,6

Comments

Incidentally, the sum of signed coefficients for each k-th row is divisible by (2*k-2)!. Moreover, another variant (but an incomplete one, and sorted differently) of the above sequence is presented in A101751.

Links

Table of n, a(n) for n=1..35.

Example

Triangle starts:
0, 1, 0, 0;
0, -20, 8, 0, 0;
20280, -6530, -1275, 362, 3, 0;
-8749440, 21627600, -4871940, -66510, 48300, 390, 0;
-261763004160, 72965762016, 13117344800, -3757930680, 72406040, 13101144, 90440, 0;
...
11!=39916800; substituting n=11 in the formula of the k-th row we obtain k=6 and the coefficients T(i,6) are those needed for computing 11!.
=> 11! = [ -974260634054400 -1140185248443360*11 +353509119454680*11^2 -8136128999880*11^3 -3234018579750*11^4 +109743298560*11^5 +6053880420*11^6 +34067880*11^7 +9450*11^8 ]/10! = 39916800.

Crossrefs

Cf. A102410, A008276, A094216, A000142, A094638, A101751, A102411, A102412, A101752, A003422, A101559, A101032, A099731.

Sequence in context: A370499 A196102 A196099 * A334691 A097395 A108967

Adjacent sequences: A102406 A102407 A102408 * A102410 A102411 A102412

Keyword

sign,tabf,uned

Author

André F. Labossière, Jan 07 2005

Status

approved