This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A238385 #42 Jul 27 2017 09:09:48 %S A238385 1,1,1,-1,2,1,2,-3,3,1,-6,8,-6,4,1,24,-30,20,-10,5,1,-120,144,-90,40, %T A238385 -15,6,1,720,-840,504,-210,70,-21,7,1,-5040,5760,-3360,1344,-420,112, %U A238385 -28,8,1,40320,-45360,25920,-10080,3024,-756,168,-36,9,1,-362880,403200,-226800,86400,-25200,6048,-1260,240,-45,10,1 %N A238385 Shifted lower triangular matrix A238363 with a main diagonal of ones. %C A238385 Shift A238363 and add a main diagonal of ones to obtain this array. The row polynomials form a special Sheffer sequence of polynomials, an Appell sequence. %F A238385 a(n,k) = (-1)^(n+k-1)*n!/((n-k)*k!) for k<n and a(n,n)=1. %F A238385 Along the n-th diagonal (n>0) Diag(n,k) = a(n+k,k) = (-1)^(n-1)(n-1)! * A007318(n+k,k). %F A238385 E.g.f.: (log(1+t)+1)*exp(x*t). %F A238385 E.g.f. for inverse: exp(x*t)/(log(1+t)+1). %F A238385 The lowering/annihilation and raising/creation operators for the row polynomials are L=D=d/dx and R=x+1/[(1+D)(1+log(1+D))], i.e., L p(n,x)= n*p(n-1,x) and R p(n,x)= p(n+1,x). %F A238385 E.g.f. of row sums: (log(1+t)+1)*exp(t). Cf. |row sums-1|=|A002741|. %F A238385 E.g.f. of unsigned row sums: (-log(1-t)+1)*exp(t). Cf. A002104 + 1. %F A238385 Let dP = A132440, the infinitesimal generator for the Pascal matrix, I, the identity matrix, and T, this entry's lower triangular matrix, then exp(T-I)=I+dP, i.e., T=I+log(I+dP). Also, ((T-I)_n)^n=0, where (T-I)_n denotes the n X n submatrix, i.e., (T-I)_n is nilpotent of order n. - _Tom Copeland_, Mar 02 2014 %F A238385 Dividing each subdiagonal by its first element (-1)^(n-1)*(n-1)! yields Pascal's triangle A007318. This is equivalent to multiplying the e.g.f. by exp(t)/(log(1+t)+1). - _Tom Copeland_, Apr 16 2014 %F A238385 From _Tom Copeland_, Apr 25 2014: (Start) %F A238385 A) T = [St1]*[dP]*[St2] + I = [padded A008275]*A132440*A048993 + I %F A238385 B) = [St1]*[dP]*[St1]^(-1) + I %F A238385 C) = [St2]^(-1)*[dP]*[St2] + I %F A238385 D) = [St2]^(-1)*[dP]*[St1]^(-1) + I, %F A238385 where [St1]=padded A008275 just as [St2]=A048993=padded A008277 and I=identity matrix. Cf. A074909. (End) %F A238385 From _Tom Copeland_, Jul 26 2017: (Start) %F A238385 p_n(x) = (1 + log(1+D)) x^n = (1 + D - D^2/2 + D^3/3- ...) x^n = x^n + n! * Sum_(k=1,..,n) (-1)^(k+1) (1/k) x^(n-k)/(n-k)!. %F A238385 Unsigned T with the first two diagonals nulled gives an exponential infinitesimal generator M (infinigen) for the rencontres numbers A008290, and negated M gives the infinigen for A055137; i.e., with M = |T| - I - dP = -log(I-dP)-dP, then e^M = e^(-dP) / (I-dP) = lower triangular A008290, and e^(-M) = e^dP (I-dP) = A007318 * (I-dP) = lower triangular A055137. The matrix formulation is consistent with the operator relations e^(-D) / (1-D) x^n = n-th row polynomial of A008290 and e^D (1-D) x^n = n-th row polynomial of A055137. (End) %e A238385 The triangle a(n,k) begins: %e A238385 n\k 0 1 2 3 4 5 6 7 8 9 10 ... %e A238385 0: 1 %e A238385 1: 1 1 %e A238385 2: -1 2 1 %e A238385 3: 2 -3 3 1 %e A238385 4: -6 8 -6 4 1 %e A238385 5: 24 -30 20 -10 5 1 %e A238385 6: -120 144 -90 40 -15 6 1 %e A238385 7: 720 -840 504 -210 70 -21 7 1 %e A238385 8: -5040 5760 -3360 1344 -420 112 -28 8 1 %e A238385 9: 40320 -45360 25920 -10080 3024 -756 168 -36 9 1 %e A238385 10: -362880 403200 -226800 86400 -25200 6048 -1260 240 -45 10 1 %e A238385 ... formatted by _Wolfdieter Lang_, Mar 09 2014 %e A238385 ---------------------------------------------------------------------------- %Y A238385 Cf. A002741, A007318, A008275, A008277, A008290, A048993, A055137, A074909, A132440, A238363. %K A238385 sign,tabl,easy %O A238385 0,5 %A A238385 _Tom Copeland_, Feb 25 2014