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

%I A090219 #14 Apr 08 2025 08:45:32
%S A090219 1,-1,4,1,-8,10,-1,12,-30,20,1,-64,600,-1600,1225,-1,80,-1000,4000,
%T A090219 -6125,3136,1,-96,1500,-8000,18375,-18816,7056,-1,448,-21000,280000,
%U A090219 -1500625,3687936,-4148928,1728000,1,-512,28000,-448000,3001250,-9834496,16595712,-13824000,4492125,-1
%N A090219 Signed triangle used to compute column sequences of array A078741 ((3,3)-Stirling2).
%C A090219 The formula for the column no. 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.
%H A090219 Wolfdieter Lang, <a href="/A090219/a090219.txt">First 10 rows</a>.
%F A090219 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.
%e A090219 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.
%e A090219 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.
%e A090219 Triangle begins:
%e A090219   [1];
%e A090219   [-1,4];
%e A090219   [1,-8,10];
%e A090219   [-1,12,-30,20];
%e A090219   [1,-64,600,-1600,1225];
%e A090219   ...
%Y A090219 Companion sequence A090220 for denominators D(m).
%K A090219 sign,easy,tabl
%O A090219 3,3
%A A090219 _Wolfdieter Lang_, Dec 01 2003