A317133 G.f.: Sum_{n>=0} binomial(4*(n+1), n)/(n+1) * x^n / (1+x)^(n+1).

1, 3, 15, 85, 526, 3438, 23358, 163306, 1167235, 8490513, 62648451, 467769217, 3527692298, 26832220834, 205601792340, 1585604105312, 12297768490441, 95861469636203, 750611119223931, 5901214027721577, 46564408929573723, 368644188180241449, 2927350250765841801, 23310167641788680947, 186089697960587977233, 1489085453187335910243

Offset

0,2

Comments

Note that: binomial(4*(n+1), n)/(n+1) = A002293(n+1) for n >= 0, where F(x) = Sum_{n>=0} A002293(n)*x^n satisfies F(x) = 1 + x*F(x)^4.

Compare the g.f. to:

(C1) M(x) = Sum_{n>=0} binomial(2*(n+1), n)/(n+1) * x^n / (1+x)^(n+1) where M(x) = 1 + M(x) + M(x)^2 is the g.f. of Motzkin numbers (A001006).

(C2) 1 = Sum_{n>=0} binomial(m*(n+1), n)/(n+1) * x^n / (1+x)^(m*(n+1)) holds for fixed m.

(C3) If S(x,p,q) = Sum_{n>=0} binomial(p*(n+1),n)/(n+1) * x^n/(1+x)^(q*(n+1)), then Series_Reversion ( x*S(x,p,q) ) = x*S(x,q,p) holds for fixed p and q.

Links

Table of n, a(n) for n=0..25.

Formula

G.f. A(x) satisfies:

(1) A(x) = (1 + x*A(x))^4 / (1+x).

(2) A(x) = (1/x) * Series_Reversion( x/((1+x)^4 - x) ).

(3) A(x) = Sum_{n>=0} binomial(4*(n+1), n)/(n+1) * x^n / (1+x)^(n+1).

a(n) ~ 229^(n + 3/2) / (sqrt(Pi) * 2^(7/2) * n^(3/2) * 3^(3*n + 9/2)). - Vaclav Kotesovec, Jul 22 2018

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(n+1,k) * binomial(4*n-4*k+4,n-k). - Seiichi Manyama, Mar 23 2024

Example

G.f.: A(x) = 1 + 3*x + 15*x^2 + 85*x^3 + 526*x^4 + 3438*x^5 + 23358*x^6 + 163306*x^7 + 1167235*x^8 + 8490513*x^9 + 62648451*x^10 + ...
such that
A(x) = 1/(1+x) + 4*x/(1+x)^2 + 22*x^2/(1+x)^3 + 140*x^3/(1+x)^4 + 969*x^4/(1+x)^5 + 7084*x^5/(1+x)^6 + ... + A002293(n+1)*x^n/(1+x)^(n+1) + ...
RELATED SERIES.
Series_Reversion( x*A(x) )  =  x/((1+x)^4 - x)  =  x - 3*x^2 + 3*x^3 + 5*x^4 - 22*x^5 + 27*x^6 + 28*x^7 - 163*x^8 + 235*x^9 + 134*x^10 + ...
which equals the sum:
Sum_{n>=0} binomial(n+1, n)/(n+1) * x^(n+1)/(1+x)^(4*(n+1)).

Mathematica

Rest[CoefficientList[InverseSeries[Series[x/((1 + x)^4 - x), {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Jul 22 2018 *)

Prog

(PARI) {a(n) = my(A = sum(m=0, n, binomial(4*(m+1), m)/(m+1) * x^m / (1+x +x*O(x^n))^(1*(m+1)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A = (1/x) * serreverse( x/((1+x)^4 - x +x*O(x^n)) ) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A316371, A127897, A317134, A349361, A349362, A349363, A349364.

Sequence in context: A346374 A118342 A084209 * A182016 A127085 A326275

Adjacent sequences: A317130 A317131 A317132 * A317134 A317135 A317136

Keyword

nonn

Author

Paul D. Hanna, Jul 21 2018

Status

approved