This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A319204 #17 Mar 18 2025 07:29:13
%S A319204 0,-2,-3,6,20,-5,-105,-98,420,1008,-990,-6501,-2574,31603,52052,
%T A319204 -107250,-411944,81328,2343042,2413456,-9883800,-25327722,23371634,
%U A319204 168185131,77113020,-835281800,-1452148815,2847865635,11561517870,-1613666430,-66318892875,-72637680690,280330495200,750725215020
%N A319204 Sequence used for the Boas-Buck type recurrence for Riordan triangle A319203.
%C A319204 See A319203 for the Boas-Buck type recurrence.
%F A319204 O.g.f.: (log(f(x)))' = (1/(1/f(x) + x^2*f(x) + 2*x^3*f(x)^2) - 1)/x, with the expansion of f given in A319201. f(x) = F^{[-1]}(x)/x, where F(t) = t/(1 - t^2 - t^3).
%F A319204 a(n) = (1/(n+1)!)*[d^(n+1)/dx^(n+1) (1 - x^2 - x^3)^(n+1)] evaluated at x = 0, for n >= 0. (Cf. _Joerg Arndt_'s conjecture for A176806, which is proved there.)
%F A319204 a(n-1) = Sum_{2*e + 3*e3 = n} (-1)^(e2+e3)*n!/((n - (e2+e3))!*e2!*e3!), n >= 2, with a(0) = 0. The pairs (e2, e3) are given in A321201; see also the multinomial coefficient table A321203 and add the sign factors.
%e A319204 a(5) = (1/6!)*[d^6/dx^6 (1 - x^2 - x^3)^6] for x = 0, which is -5.
%e A319204 a(5) = +15 - 20 = -5; from the sum of the signed row n=6 in A321203, with parity of e2 + e3 from A321201 even and odd.
%Y A319204 Cf. A319203, A321201, A321203.
%K A319204 sign
%O A319204 0,2
%A A319204 _Wolfdieter Lang_, Oct 29 2018