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A358735 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.

%I A358735 #37 Mar 17 2023 10:56:43
%S A358735 1,1,1,1,4,2,1,10,16,6,1,20,70,76,24,1,35,224,496,428,120,1,56,588,
%T A358735 2260,3808,2808,720,1,84,1344,8140,23008,32152,21096,5040,1,120,2772,
%U A358735 24772,107328,245560,298688,178848,40320
%N A358735 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.
%C A358735 This sequence is essentially A204024 except for extra row, alternating signs and reversed rows.
%C A358735 The sequence of polynomials a(m) satisfies a(m)*a(m-2) = a(m-1) * (a(m-1) + x*a(m-2) + a(m-3)) - a(m-2)^2 for all m > 3.
%F A358735 If x=1, then a(n) = A058797(n+2) = Sum_{k=0..n} T(n, k).
%F A358735 If x=2, then a(n) = A093986(n+2).
%F A358735 T(n, n) = n!, T(n, 0) = 1, T(n, 1) = A000292(n). T(n, 2) = 2*A040977(n-2).
%e A358735 a(3) = 1, a(4) = 1 + x, a(5) = 1 + 4*x + 2*x^2.
%e A358735 Triangular array T(n, k) starts:
%e A358735 n\k | 0   1   2   3   4   5
%e A358735 --- + - --- --- --- --- ---
%e A358735  0  | 1
%e A358735  1  | 1   1
%e A358735  2  | 1   4   2
%e A358735  3  | 1  10  16   6
%e A358735  4  | 1  20  70  76  24
%e A358735  5  | 1  35 224 496 428 120
%t A358735 T[ n_, k_] := If[ n<0, 0, Module[{a = Table[1, n+3], x}, Do[ a[[m]] = a[[m-1]] *(a[[m-1]] + x*a[[m-2]] + a[[m-3]])/a[[m-2]] - a[[m-2]] //Factor//Expand, {m, 4, n+3}]; Coefficient[ a[[n+3]], x, k]]];
%o A358735 (PARI) {T(n, k) = if( n<0, 0, my(a = vector(n+3, i, 1)); for(m = 4, n+3, a[m] = a[m-1]*(a[m-1] + 'x*a[m-2] + a[m-3])/a[m-2] - a[m-2]); polcoeff( a[n+3], k))};
%Y A358735 Cf. A000292, A040977, A058797, A093986, A204024.
%K A358735 nonn,tabl
%O A358735 0,5
%A A358735 _Michael Somos_, Mar 15 2023