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COMMENT
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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 1-step 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 1-step 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 2-step 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 2-step 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 3-step 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 3-step 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)! ] = (1-m*x)^(-1/m) .
Also with P(n,t) = sum(k=0,...,n-1) T(n,k+1) * t^k = 1*(1+t)*(1+2t)...(1+(n-1)*t) and P(0,t)=1, exp[P(.,t)*x] = (1-tx)^(-1/t) .
T(n,k+1) = (1/k!) (D_t)^k (D_x)^n [ (1-tx)^(-1/t) - 1 ] eval. at t=x=0 .
And, (1-tx)^(-1/t) - 1 is the e.g.f. for plane increasing m-ary trees when t = (m-1), discussed by Bergeron et al. in "Varieties of Increasing Trees" and the book Combinatorial Species and Tree-Like 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.
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