A201952 - OEIS

%I #12 Feb 25 2019 00:15:10

%S 1,1,5,24,139,945,7377,65016,638418,6910650,81747665,1049089470,

%T 14516096009,215419836359,3412889885571,57492203734320,

%U 1026121982213480,19342642266760680,383995631680561234,8007915240045479980,175020604366224762038,4000551483475536398178

%N A diagonal of irregular triangle A201949.

%C G.f. of row n in triangle A201949 equals Product_{k=0..n-1} (1 + k*x + x^2).

%F E.g.f.: Sum_{n>=0} -log(1 - x)^(2*n+1) / (n!*(n+1)!). - _Paul D. Hanna_, Feb 25 2019

%F a(n) = [x^(n-1)] Product_{k=0..n-1} (1 + k*x + x^2).

%F a(n) = (n-1)*a(n-1) + A201950(n-1) + A201953(n-1).

%e 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 Triangle A201949 begins:

%e [1],

%e [(1), 0, 1],

%e [1,(1), 2, 1, 1],

%e [1, 3, (5), 6, 5, 3, 1],

%e [1, 6, 15, (24), 28, 24, 15, 6, 1],

%e [1, 10, 40, 90,(139), 160, 139, 90, 40, 10, 1], ...

%e where coefficients in parenthesis form the initial terms of this sequence.

%o (PARI) {a(n) = polcoeff( prod(j=0, n-1, 1 + j*x + x^2), n-1)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A201949, A201950, A201951, A201953.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Dec 06 2011

%E Offset changed to 1 to agree with the e.g.f. - _Paul D. Hanna_, Feb 25 2019