A186185 Expansion of 1/(1 - x*A001764(x/(1-x))/(1-x)).

1, 1, 3, 11, 48, 239, 1306, 7612, 46436, 292875, 1894365, 12496864, 83753165, 568628232, 3902600850, 27031069848, 188709211952, 1326456525471, 9379857716098, 66680723764051, 476269444919163, 3416178576731504

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

a(n) = Sum_{m=1..n} Sum_{k=m..n} binomial(n-1,k-1)*m/(2*k-m)*binomial(3*k-2*m-1,k-m), n>0, a(0)=1.

Conjecture: D-finite with recurrence 2*(2*n-1)*(n-1)*a(n) +(9*n^2-127*n+170)*a(n-1) +10*(-57*n^2+368*n-559)*a(n-2) +(2579*n^2-18451*n+32892)*a(n-3) +3*(-1633*n^2+13195*n-26878)*a(n-4) +6*(712*n-3111)*(n-5)*a(n-5) -1395*(n-5)*(n-6)*a(n-6)=0, n>6. - R. J. Mathar, May 07 2024

a(n) ~ 31^(n + 1/2) / (49 * sqrt(Pi) * n^(3/2) * 2^(2*n)). - Vaclav Kotesovec, May 07 2024

Example

G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 48*x^4 + 239*x^5 + ...
The g.f. of A001764, where A001764(x) = 1 + x*A001764(x)^3, begins:
A001764(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ...
The g.f. of the binomial transform of A001764 begins:
A001764(x/(1-x))/(1-x) = 1 + 2*x + 6*x^2 + 25*x^3 + 126*x^4 + 704*x^5 + ...
where A(x) = 1/(1 - x*A001764(x/(1-x))/(1-x)).

Prog

(PARI) {a(n)=if(n<0, 0, if(n==0, 1, sum(m=1, n, sum(k=m, n, binomial(n-1, k-1)*binomial(3*k-2*m-1, k-m)*m/(2*k-m)))))}
(PARI) {a(n)=local(A001764=sum(m=0, n, binomial(3*m, m)*x^m/(2*m+1))+O(x^n)); polcoeff(1/(1-x*subst(A001764, x, x/(1-x))/(1-x)), n)}

Crossrefs

Cf. A001764.

Sequence in context: A192399 A233162 A345341 * A367874 A317170 A127087

Adjacent sequences: A186182 A186183 A186184 * A186186 A186187 A186188

Keyword

nonn

Author

Vladimir Kruchinin, Feb 14 2011

Status

approved