cp's OEIS Frontend

A201952 A diagonal of irregular triangle A201949.

%I A201952 #12 Feb 25 2019 00:15:10
%S A201952 1,1,5,24,139,945,7377,65016,638418,6910650,81747665,1049089470,
%T A201952 14516096009,215419836359,3412889885571,57492203734320,
%U A201952 1026121982213480,19342642266760680,383995631680561234,8007915240045479980,175020604366224762038,4000551483475536398178
%N A201952 A diagonal of irregular triangle A201949.
%C A201952 G.f. of row n in triangle A201949 equals Product_{k=0..n-1} (1 + k*x + x^2).
%F A201952 E.g.f.: Sum_{n>=0} -log(1 - x)^(2*n+1) / (n!*(n+1)!). - _Paul D. Hanna_, Feb 25 2019
%F A201952 a(n) = [x^(n-1)] Product_{k=0..n-1} (1 + k*x + x^2).
%F A201952 a(n) = (n-1)*a(n-1) + A201950(n-1) + A201953(n-1).
%e A201952 E.g.f.: A(x) = x + x^2/2! + 5*x^3/3! + 24*x^4/4! + 139*x^5/5! + 945*x^6/6! + 7377*x^7/7! + 65016*x^8/8! + 638418*x^9/9! + 6910650*x^10/10! + ...
%e A201952 Triangle A201949 begins:
%e A201952 [1],
%e A201952 [(1), 0, 1],
%e A201952 [1,(1), 2, 1, 1],
%e A201952 [1, 3, (5), 6, 5, 3, 1],
%e A201952 [1, 6, 15, (24), 28, 24, 15, 6, 1],
%e A201952 [1, 10, 40, 90,(139), 160, 139, 90, 40, 10, 1], ...
%e A201952 where coefficients in parenthesis form the initial terms of this sequence.
%o A201952 (PARI) {a(n) = polcoeff( prod(j=0, n-1, 1 + j*x + x^2), n-1)}
%o A201952 for(n=1,30,print1(a(n),", "))
%Y A201952 Cf. A201949, A201950, A201951, A201953.
%K A201952 nonn
%O A201952 1,3
%A A201952 _Paul D. Hanna_, Dec 06 2011
%E A201952 Offset changed to 1 to agree with the e.g.f. - _Paul D. Hanna_, Feb 25 2019