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

1,3

Comments

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

Links

Table of n, a(n) for n=1..22.

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).

a(n) = A201949(n,n-1). - Alois P. Heinz, Dec 11 2025

Example

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.

Maple

T:= proc(n, k) option remember; `if`(k<0 or k>2*n, 0,
     `if`(n=0 and k=0, 1, T(n-1, k)+(n-1)*T(n-1, k-1)+T(n-1, k-2)))
    end:
a:= n-> T(n, n-1):
seq(a(n), n=1..23);  # Alois P. Heinz, Dec 11 2025

Prog

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

Crossrefs

Cf. A201949, A201950, A201951, A201953.

Sequence in context: A256325 A286743 A232318 * A221788 A111766 A368534

Adjacent sequences: A201949 A201950 A201951 * A201953 A201954 A201955

Keyword

nonn

Author

Paul D. Hanna, Dec 06 2011

Extensions

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

Status

approved