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 A238385 Shifted lower triangular matrix A238363 with a main diagonal of ones. 16
 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, -15, 6, 1, 720, -840, 504, -210, 70, -21, 7, 1, -5040, 5760, -3360, 1344, -420, 112, -28, 8, 1, 40320, -45360, 25920, -10080, 3024, -756, 168, -36, 9, 1, -362880, 403200, -226800, 86400, -25200, 6048, -1260, 240, -45, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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. LINKS FORMULA a(n,k) = (-1)^(n+k-1)*n!/((n-k)*k!) for k0) Diag(n,k) = a(n+k,k) = (-1)^(n-1)(n-1)! * A007318(n+k,k). E.g.f.: (log(1+t)+1)*exp(x*t). E.g.f. for inverse: exp(x*t)/(log(1+t)+1). 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). E.g.f. of row sums: (log(1+t)+1)*exp(t). Cf. |row sums-1|=|A002741|. E.g.f. of unsigned row sums: (-log(1-t)+1)*exp(t). Cf. A002104 + 1. 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 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 From Tom Copeland, Apr 25 2014: (Start) A) T = [St1]*[dP]*[St2] + I = [padded A008275]*A132440*A048993 + I B) = [St1]*[dP]*[St1]^(-1) + I C) = [St2]^(-1)*[dP]*[St2] + I D) = [St2]^(-1)*[dP]*[St1]^(-1) + I,   where [St1]=padded A008275 just as [St2]=A048993=padded A008277 and I=identity matrix. Cf. A074909. (End) From Tom Copeland, Jul 26 2017: (Start) 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)!. 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) EXAMPLE The triangle a(n,k) begins: n\k       0       1        2      3       4     5      6    7   8   9 10 ... 0:        1 1:        1       1 2:       -1       2        1 3:        2      -3        3      1 4:       -6       8       -6      4       1 5:       24     -30       20    -10       5     1 6:     -120     144      -90     40     -15     6      1 7:      720    -840      504   -210      70   -21      7    1 8:    -5040    5760    -3360   1344    -420   112    -28    8   1 9:    40320  -45360    25920 -10080    3024  -756    168  -36   9   1 10: -362880  403200  -226800  86400  -25200  6048  -1260  240 -45  10  1 ... formatted by Wolfdieter Lang, Mar 09 2014 ---------------------------------------------------------------------------- CROSSREFS Cf. A002741, A007318, A008275, A008277, A008290, A048993, A055137, A074909, A132440, A238363. Sequence in context: A247453 A109449 A129570 * A215652 A305715 A165014 Adjacent sequences:  A238382 A238383 A238384 * A238386 A238387 A238388 KEYWORD sign,tabl,easy AUTHOR Tom Copeland, Feb 25 2014 STATUS approved

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Last modified February 18 03:44 EST 2020. Contains 332006 sequences. (Running on oeis4.)