A201952 A diagonal of irregular triangle A201949.

1, 1, 5, 24, 139, 945, 7377, 65016, 638418, 6910650, 81747665, 1049089470, 14516096009, 215419836359, 3412889885571, 57492203734320, 1026121982213480, 19342642266760680, 383995631680561234, 8007915240045479980, 175020604366224762038, 4000551483475536398178

Offset: 1

Author

Paul D. Hanna, Dec 06 2011

Keywords

• nonn

Comments

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

Examples

			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! + ...
Triangle A201949 begins:
[1],
[(1), 0, 1],
[1,(1), 2, 1, 1],
[1, 3, (5), 6, 5, 3, 1],
[1, 6, 15, (24), 28, 24, 15, 6, 1],
[1, 10, 40, 90,(139), 160, 139, 90, 40, 10, 1], ...
where coefficients in parenthesis form the initial terms of this sequence.
		

Crossrefs

Cf. A201949, A201950, A201951, A201953.

Programs

PARI

{a(n) = polcoeff( prod(j=0, n-1, 1 + j*x + x^2), n-1)}
for(n=1,30,print1(a(n),", "))

Formula

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

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

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

Extensions

Offset changed to 1 to agree with the e.g.f. - Paul D. Hanna, Feb 25 2019