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 A106208 Triangular matrix T, read by rows, that satisfies: [T^-1](n,k) = -(k+1)*T(n-1,k) when (n-1)>=k>=0, with T(n,n) = 1 and T(n+1,n) = (n+1) for n>=0. 1
 1, 1, 1, 3, 2, 1, 16, 10, 3, 1, 127, 78, 21, 4, 1, 1363, 832, 216, 36, 5, 1, 18628, 11342, 2901, 460, 55, 6, 1, 311250, 189286, 48081, 7456, 840, 78, 7, 1, 6173791, 3752320, 949800, 145660, 15955, 1386, 105, 8, 1, 142190703, 86392756, 21826470, 3327340 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Column 0 is A082161 (offset 1). Column 1 is (1/2)*A102087. Row sums form A106209. LINKS FORMULA T(n, k) = A102086(n, k)/(k+1) for n>=0, k>=0. T(n, 0) = A082161(n) for n>0, with T(0, 0) = 1. G.f. for column k: 1 = Sum_{n>=0} T(n+k, k)*x^n*prod_{j=1, n+1} (1-(j+k)*x). EXAMPLE Triangle T begins: 1; 1,1; 3,2,1; 16,10,3,1; 127,78,21,4,1; 1363,832,216,36,5,1; 18628,11342,2901,460,55,6,1; 311250,189286,48081,7456,840,78,7,1; 6173791,3752320,949800,145660,15955,1386,105,8,1; ... Matrix inverse T^-1 begins: 1; -1,1; -1,-2,1; -3,-4,-3,1; -16,-20,-9,-4,1; -127,-156,-63,-16,-5,1; -1363,-1664,-648,-144,-25,-6,1; -18628,-22684,-8703,-1840,-275,-36,-7,1; ... where [T^-1](n,k) = -(k+1)*T(n-1,k) when (n-1)>=k>=0. G.f. for column 0: 1 = 1(1-x) + 1*x*(1-x)(1-2x) + 3*x^2*(1-x)(1-2x)(1-3x) + ... + T(n,0)*x^n*(1-x)(1-2x)(1-3x)*..*(1-(n+1)*x) + ... G.f. for column 1: 1 = 1(1-2x) + 2*x*(1-2x)(1-3x) + 10*x^2*(1-2x)(1-3x)(1-4x) + ... + T(n+1,1)*x^n*(1-2x)(1-3x)(1-4x)*..*(1-(n+2)*x) + ... G.f. for column 2: 1 = 1(1-3x) + 3*x*(1-3x)(1-4x) + 21*x^2*(1-3x)(1-4x)(1-5x) + ... + T(n+2,2)*x^n*(1-3x)(1-4x)(1-5x)*..*(1-(n+3)*x) + ... PROG (PARI) T(n, k)=if(n

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Last modified April 10 02:39 EDT 2020. Contains 333392 sequences. (Running on oeis4.)