A090219 Signed triangle used to compute column sequences of array A078741 ((3,3)-Stirling2).

1, -1, 4, 1, -8, 10, -1, 12, -30, 20, 1, -64, 600, -1600, 1225, -1, 80, -1000, 4000, -6125, 3136, 1, -96, 1500, -8000, 18375, -18816, 7056, -1, 448, -21000, 280000, -1500625, 3687936, -4148928, 1728000, 1, -512, 28000, -448000, 3001250, -9834496, 16595712, -13824000, 4492125, -1

Offset

3,3

Comments

The formula for the column nr. k sequence of array A078741 is c(k;n) = b(k-2)*sum(a(k,m)*fallfac(m+2,3)^n,m=1..k-2),n>=0, k>=3 and fallfac(p,3) and b(n) are defined in the formula below.

Links

Table of n, a(n) for n=3..48.

W. Lang, First 10 rows.

Formula

a(n, m)= A089505(n-2, m)*(sum(A089517(n, p)/fallfac(m+2, 3)^p, p=0..floor(2*(n-3)/3)))/b(n-2), n>=3, 1<= m<= n-2, else 0; with fallfac(q, 3)=A008279(q, 3)=q*(q-1)*(q-2) and b(n)=N(n)/D(n) where D(n) := A090220(n) and N(n) is given in A090220 for n=1..26.

Example

The third (k=5) column sequence of array A078741 is A078741(n+3,5)=c(5; n)= b(3)*(1*(3*2*1)^n -8*(4*3*2)^n +10*(5*4*3)^n), with b(3)= N(3)/A090220(3)=3/1=3, n>=0. This is 9*A089518.
The fifth (k=7) column sequence of array A078741 is A078741(n+3,7)=c(7; n)= b(5)*(1*(3*2*1)^n -64*(4*3*2)^n +600*(5*4*3)^n -1600*(6*5*4)^n +1225*(7*6*5)^n), with b(5)= N(5)/A090220(5)=3/2, n>=0. This is the sequence [243, 149580, 49658508, 13062960720,... ] which has a factor of 27.
[1]; [ -1,4]; [1,-8,10]; [ -1,12,-30,20]; [1,-64,600,-1600,1225]; ...

Crossrefs

Companion sequence A090220 for denominators D(m).

Sequence in context: A105533 A179193 A124848 * A264285 A125129 A324780

Adjacent sequences: A090216 A090217 A090218 * A090220 A090221 A090222

Keyword

sign,easy,tabl

Author

Wolfdieter Lang, Dec 01 2003

Status

approved