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A371762 Triangle read by rows: the polynomial coefficients of the numerator of the rational solution of the linear recurrence equations of the rows of A371761.

%I A371762 #7 Apr 08 2024 18:59:39
%S A371762 1,0,1,0,1,2,0,1,7,6,0,1,19,46,24,0,1,46,251,326,120,0,1,104,1163,
%T A371762 3016,2556,720,0,1,225,4831,23283,35848,22212,5040,0,1,473,18523,
%U A371762 158531,417148,437228,212976,40320,0,1,976,66886,976636,4285549,7084804,5586444,2239344,362880
%N A371762 Triangle read by rows: the polynomial coefficients of the numerator of the rational solution of the linear recurrence equations of the rows of A371761.
%C A371762 Let R(n) = N(n)/D(n) denote the ordinary rational generating function of row n of A371761 as given by its linear recurrence equation. N(n) is the row polynomial Sum_{k=0..n} T(n, k)*x^k and D(n) = Sum_{k=0..n} Stirling1(n+1, n+1-k)*x^k. Thus A371761(n, k) = [x^k] N(n)/D(n).
%e A371762 Triangle starts:
%e A371762   [0] 1;
%e A371762   [1] 0, 1;
%e A371762   [2] 0, 1,   2;
%e A371762   [3] 0, 1,   7,     6;
%e A371762   [4] 0, 1,  19,    46,     24;
%e A371762   [5] 0, 1,  46,   251,    326,     120;
%e A371762   [6] 0, 1, 104,  1163,   3016,    2556,     720;
%e A371762   [7] 0, 1, 225,  4831,  23283,   35848,   22212,    5040;
%e A371762   [8] 0, 1, 473, 18523, 158531,  417148,  437228,  212976,   40320;
%e A371762   [9] 0, 1, 976, 66886, 976636, 4285549, 7084804, 5586444, 2239344, 362880;
%e A371762 .
%e A371762 The rational generating function for row 3 of A371761 is:
%e A371762 gf = (6*x^3 + 7*x^2 + x)/(-6*x^3 + 11*x^2 - 6*x + 1).
%Y A371762 Cf. A029767 (row sums), A000142 (main diagonal), A067318 (subdiagonal).
%Y A371762 Cf. A371761.
%K A371762 nonn,tabl
%O A371762 0,6
%A A371762 _Peter Luschny_, Apr 06 2024