A201950 - OEIS

%I #33 May 04 2024 07:06:07

%S 1,0,2,6,28,160,1078,8358,73260,716112,7721844,91039740,1164932470,

%T 16077368580,238037983558,3763371442530,63276351409092,

%U 1127406030014112,21218146474666864,420611921077524912,8759617763834095796,191208185756772875880

%N Central coefficients in Product_{k=0..n-1} (1 + k*x + x^2).

%H Vaclav Kotesovec, <a href="/A201950/b201950.txt">Table of n, a(n) for n = 0..300</a>

%F Central terms of rows in irregular triangle A201949.

%F a(n) = (n-1)*a(n-1) + 2*A201952(n-1) for n>0. [corrected by _Vaclav Kotesovec_, May 04 2024]

%F E.g.f.: BesselI(0, 2*log(1 - x)). - _Ilya Gutkovskiy_, Feb 22 2019

%F E.g.f.: Sum_{n>=0} log(1 - x)^(2*n) / n!^2. [After _Ilya Gutkovskiy_ - _Paul D. Hanna_, Feb 24 2019]

%e The coefficients in Product_{k=0..n-1} (1+k*x+x^2) form triangle A201949:

%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 1, 15, 91, 300, 629, 945, (1078), 945, 629, 300, 91, 15, 1;

%e 1, 21, 182, 861, 2520, 5019, 7377, (8358), 7377, 5019, 2520, 861, 182, 21, 1;

%e 1, 28, 330, 2156, 8729, 23520, 45030, 65016, (73260), 65016, 45030, 23520, 8729, 2156, 330, 28, 1; ...

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

%t Flatten[{1,Table[Coefficient[Expand[Product[1 + k*x + x^2,{k,0,n-1}]],x^n],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 10 2015 *)

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

%o for(n=0,30, print1(a(n),", "))

%o (PARI) /* From series BesselI(0, 2*log(1 - x)), after _Ilya Gutkovskiy_ */

%o {a(n) = n!*polcoeff( sum(m=0,n, log(1 - x +x*O(x^n))^(2*m)/m!^2), n)}

%o for(n=0,30, print1(a(n),", ")) \\ _Paul D. Hanna_, Feb 24 2019

%Y Cf. A201949, A201951, A201952, A201953.

%Y Cf. A086672, A324304 (variant).

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 06 2011