
COMMENTS

To motivate the definition, consider c(t) = column vector(1, t, t^2, t^3, t^4, t^5,...), T = A094638 and the list of integers.
Starting at 1 and sampling every integer to the right, we obtain (1,2,3,4,5,...) from which factorials may be formed. It's true that
T * c(1) = (1, 1*2, 1*2*3, 1*2*3*4,...), giving n! for n>0. Call this sequence the right 1step factorial (n,+1)! .
Starting at 1 and sampling every integer to the left, we obtain (1,0,1,2,3,4,5,...). And,
T * c(1) = (1, 1*0, 1*0*1, 1*0*1*2,...) = (1,0,0,0,...) . Call this the left 1step factorial (n,1)! .
Sampling every other integer to the right, we obtain (1,3,5,7,9,...).
T * c(2) = (1, 1*3, 1*3*5, ...) = (1,3,15,105,945,...), giving A001147 for n>0, the right 2step factorial, (n,+2)! .
Sampling every other integer to the left, we obtain (1,1,3,5,7...).
T * c(2) = (1, 1*1, 1*1*3, 1*1*3*5,...) = (1,1,3,15,105,945,...) = signed A001147, the left 2step factorial, (n,2)! .
Sampling every 3 steps to the right, we obtain (1,4,7,10,...).
T * c(3) = (1, 1*4, 1*4*7,...) = (1,4,28,280,...), giving A007559 for n>0, the right 3step factorial, (n,+3)! .
Sampling every 3 steps to the left, we obtain (1,2,5,8,11,...), giving
T * c(3) = (1, 1*2, 1*2*5, 1*2*5*8,...) = (1,2,10,80,880,...) = signed A008544 = the left 3step factorial, (n,3)! .
The list partition transform A133314 of [1,T * c(t)] gives signed [1,T *c(t)] . For example:
LPT[1,T*c(1)] = LPT[1,(n,+1)! ] = LPT[A000142] = (1,1,0,0,0,...) = signed [1,(n,1)! ]
LPT[1,T*c(2)] = LPT[1,(n,+2)! ] = LPT[A001147] = (1,1,1,3,15,105,945,...) = (1,A001147) = signed [1,(n,2)! ]
LPT[1,T*c(3)] = LPT[1,(n,+3)! ] = LPT[A007559] = (1,1,2,10,80,880,...) = (1,A008544) = signed [1,(n,3)! ]
LPT[1,T*c(3)] = LPT[1,(n,3)! ] = LPT[A000001] = signed A007559 = signed [1,(n,+3)! ] .
And, e.g.f.[1,T * c(m)] = e.g.f.[1,(n,m)! ] = (1m*x)^(1/m) .
Also with P(n,t) = sum(k=0,...,n1) T(n,k+1) * t^k = 1*(1+t)*(1+2t)...(1+(n1)*t) and P(0,t)=1, exp[P(.,t)*x] = (1tx)^(1/t) .
T(n,k+1) = (1/k!) (D_t)^k (D_x)^n [ (1tx)^(1/t)  1 ] eval. at t=x=0 .
And, (1tx)^(1/t)  1 is the e.g.f. for plane increasing mary trees when t = (m1), discussed by Bergeron et al. in "Varieties of Increasing Trees" and the book Combinatorial Species and TreeLike Structures, cited in the OEIS.
The above relations reveal the intimate connections, through T or LPT or sampling, between the right and left step factorials, (n,+m)! and (n,m)!. The pairs have conjugate interpretations as trees, ignoring signs, which Callan and Lang have noted in several of the OEIS entries above. Also note unsigned (n,2)! is the diagonal of A001498 and (n,+2)!, the first subdiagonal.
