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A208896 Rectangular table where the g.f. of row n satisfies: R(n,x) = 1 + x*R(n,x)^n * [d/dx x/R(n,x)] for n>=0, as read by antidiagonals.

%I A208896 #13 Mar 30 2012 18:37:37
%S A208896 1,1,1,1,1,-2,1,1,-1,9,1,1,0,3,-56,1,1,1,0,-13,425,1,1,2,0,0,71,-3726,
%T A208896 1,1,3,3,-1,0,-461,36652,1,1,4,9,0,1,0,3447,-397440,1,1,5,18,19,-12,0,
%U A208896 0,-29093,4695489,1,1,6,30,72,0,-14,0,0,273343,-59941550
%N A208896 Rectangular table where the g.f. of row n satisfies: R(n,x) = 1 + x*R(n,x)^n * [d/dx x/R(n,x)] for n>=0, as read by antidiagonals.
%C A208896 The following property accounts for the zeros along the main diagonal.
%C A208896 The row g.f.s satisfy: [x^k] R(n,x)^(k-n+1) = [x^k] R(n,x)^(k-n) for k>=2
%C A208896 and thus when k=n we have [x^n] R(n,x) = [x^n] R(n,x)^0 = 0 for n>=2.
%H A208896 Paul D. Hanna, <a href="/A208896/b208896.txt">Rows n = 0..46, flattened.</a>
%e A208896 Coefficients in the n-th row g.f., R(n,x), of this table begins:
%e A208896 n=0: [1, 1,-2,  9, -56,  425, -3726,  36652, -397440, 4695489, ...];
%e A208896 n=1: [1, 1,-1,  3, -13,   71,  -461,   3447,  -29093,  273343, ...];
%e A208896 n=2: [1, 1, 0,  0,   0,    0,     0,      0,       0,       0, ...];
%e A208896 n=3: [1, 1, 1,  0,  -1,    1,     0,      0,      -5,      27, ...];
%e A208896 n=4: [1, 1, 2,  3,   0,  -12,   -14,     43,      96,     -50, ...];
%e A208896 n=5: [1, 1, 3,  9,  19,    0,  -195,   -732,    -453,    6495, ...];
%e A208896 n=6: [1, 1, 4, 18,  72,  201,     0,  -4200,  -27984,  -91044, ...];
%e A208896 n=7: [1, 1, 5, 30, 175,  880,  3106,      0, -114485,-1124735, ...];
%e A208896 n=8: [1, 1, 6, 45, 344, 2451, 14946,  64522,       0,-3805692, ...];
%e A208896 n=9: [1, 1, 7, 63, 595, 5453, 45927, 331177, 1704795,       0, ...];
%e A208896 n=10:[1, 1, 8, 84, 944,10550,112336,1094604, 9157984,55095601, 0,...]; ...
%e A208896 in which the main diagonal is zeros for n>=2.
%e A208896 Initial row g.f.s are illustrated by the following.
%e A208896 R(0,x) = 1 + x*[d/dx x/R(0,x)] begins:
%e A208896 R(0,x) = 1 + x - 2*x^2 + 9*x^3 - 56*x^4 + 425*x^5 - 3726*x^6 +...
%e A208896 which satisfies: [x^k] R(0,x)^(k+1) = [x^k] R(0,x)^k for k>=2.
%e A208896 ...
%e A208896 R(1,x) = 1 + x*R(1,x)*[d/dx x/R(1,x)] begins:
%e A208896 R(1,x) = 1 + x - x^2 + 3*x^3 - 13*x^4 + 71*x^5 - 461*x^6 + 3447*x^7 +...
%e A208896 which satisfies: [x^k] R(1,x)^k = [x^k] R(1,x)^(k-1) for k>=2.
%e A208896 ...
%e A208896 R(2,x) = 1 + x*R(2,x)^2*[d/dx x/R(2,x)] is satisfied by:
%e A208896 R(2,x) = 1 + x,
%e A208896 which satisfies: [x^k] R(2,x)^(k-1) = [x^k] R(2,x)^(k-2) = 0 for k>=2.
%e A208896 ...
%e A208896 R(3,x) = 1 + x*R(3,x)^3*[d/dx x/R(3,x)] begins:
%e A208896 R(3,x) = 1 + x + x^2 - x^4 + x^5 - 5*x^8 + 27*x^9 - 147*x^10 + 996*x^11 +...
%e A208896 which satisfies: [x^k] R(3,x)^(k-2) = [x^k] R(3,x)^(k-3) for k>=2.
%e A208896 ...
%o A208896 (PARI) {T(n,k)=local(ROWn=1+x+x*O(x^k));for(i=0,k,ROWn=1+x*ROWn^n*deriv(x/ROWn));polcoeff(ROWn,k)}
%o A208896 for(n=0,12,for(k=0,12,print1(T(n,k),","));print(""))
%Y A208896 Cf. A158883 (row 0), A158882 (row 1), A208897, A208898.
%K A208896 sign,tabl
%O A208896 0,6
%A A208896 _Paul D. Hanna_, Mar 03 2012