OFFSET
0,3
COMMENTS
Conjecture: a(n) also counts permutations of [n+1] in which the number of left-to-right maxima equals the number of right-to-left maxima. Verified via direct enumeration for n=0..11 and Stirling-number-based calculation for n = 0..29. - Austen M. Fletcher, Dec 10 2025
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
FORMULA
a(n) = A201949(n,n).
a(n) = (n-1)*a(n-1) + 2*A201952(n-1) for n>0. [corrected by Vaclav Kotesovec, May 04 2024]
E.g.f.: BesselI(0, 2*log(1 - x)). - Ilya Gutkovskiy, Feb 22 2019
E.g.f.: Sum_{n>=0} log(1 - x)^(2*n) / n!^2. [After Ilya Gutkovskiy - Paul D. Hanna, Feb 24 2019]
EXAMPLE
The coefficients in Product_{k=0..n-1} (1+k*x+x^2) form triangle A201949:
(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;
1, 15, 91, 300, 629, 945, (1078), 945, 629, 300, 91, 15, 1;
1, 21, 182, 861, 2520, 5019, 7377, (8358), 7377, 5019, 2520, 861, 182, 21, 1;
1, 28, 330, 2156, 8729, 23520, 45030, 65016, (73260), 65016, 45030, 23520, 8729, 2156, 330, 28, 1;
...
where coefficients in parenthesis form the initial terms of this sequence.
MAPLE
b:= proc(n) b(n):= `if`(n=0, 1, expand((1+(n-1)*x+x^2)*b(n-1))) end:
a:= n-> coeff(b(n), x, n):
seq(a(n), n=0..23); # Alois P. Heinz, Dec 11 2025
# Alternative:
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$2):
seq(a(n), n=0..23); # Alois P. Heinz, Dec 11 2025
MATHEMATICA
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 *)
PROG
(PARI) {a(n) = polcoeff( prod(k=1, n, 1+(k-1)*x+x^2+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From series BesselI(0, 2*log(1 - x)), after Ilya Gutkovskiy */
{a(n) = n!*polcoeff( sum(m=0, n, log(1 - x +x*O(x^n))^(2*m)/m!^2), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 24 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 06 2011
STATUS
approved