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A243116 a(n) = Sum_{k=0..n} C(n + 2*k, 3*k) * C(3*k, 2*k). 1
1, 4, 28, 220, 1816, 15424, 133456, 1169872, 10354528, 92331904, 828204928, 7464652672, 67547774464, 613295870464, 5584367987968, 50974595472640, 466307503244800, 4273832891668480, 39237007284226048, 360768875975526400, 3321625537178669056, 30619908430235828224, 282578914501599305728 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Compare to: Sum_{k=0..n} (-1)^k * C(n+2*k,3*k) * C(3*k,2*k) = (-2)^n for n>=0.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..1027

Hacène Belbachir, Abdelghani Mehdaoui, Diagonal sums in Pascal pyramid (1, 2, r), Les Annales RECITS (2019) Vol. 6, 45-52.

FORMULA

G.f.: Sum_{n>=0} C(3*n, n) * x^n / (1-x)^(3*n+1).  - Paul D. Hanna, Aug 30 2014

G.f.: 1/(1-x) / ( 3 / G(x/(1-x)^3) - 2 ), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.  - Paul D. Hanna, Aug 30 2014

G.f. satisfies: A(x) = 1 + (4-3*x)*A(x) - (4 - 39*x + 12*x^2 - 4*x^3)*A(x)^3. - Paul D. Hanna, Sep 05 2014

a(n) = Sum_{k=0..n} A109955(n,k) * A005809(k).

a(n) = -(-2)^n + 2*Sum_{k=0..[n/2]} C(n+4*k, 6*k) * C(6*k, 4*k).

Recurrence: 2*n*(2*n-1)*(3*n-4)*a(n) = (3*n-2)*(39*n^2 - 65*n + 18)*a(n-1) - 2*(n-1)*(18*n^2 - 33*n + 10)*a(n-2) + 4*(n-2)*(n-1)*(3*n-1)*a(n-3). - Vaclav Kotesovec, Aug 21 2014

EXAMPLE

G.f.: A(x) = 1 + 4*x + 28*x^2 + 220*x^3 + 1816*x^4 + 15424*x^5 +...

where

A(x) = 1/(1-x) + 3*x/(1-x)^4 + 15*x^2/(1-x)^7 + 84*x^3/(1-x)^10 + 495*x^4/(1-x)^13 + 3003*x^5/(1-x)^16 + 18564*x^6/(1-x)^19 + 116280*x^7/(1-x)^22 + 735471*x^8/(1-x)^25 +...+ C(3*n, n)*x^n/(1-x)^(3*n+1) +...

ILLUSTRATION OF TERMS.

The sequence A005809(k) = C(3*k, 2*k) begins:

[1, 3, 15, 84, 495, 3003, 18564, 116280, 735471, 4686825, ...];

the triangle A109955(n,k) = C(n + 2*k, 3*k) begins:

1;

1, 1;

1, 4, 1;

1, 10, 7, 1;

1, 20, 28, 10, 1;

1, 35, 84, 55, 13, 1;

1, 56, 210, 220, 91, 16, 1;

1, 84, 462, 715, 455, 136, 19, 1; ...

where a(n) = Sum_{k=0..n} A109955(n,k) * A005809(k):

a(1) = 1*1 + 1*3 = 4;

a(2) = 1*1 + 4*3 + 1*15 = 28;

a(3) = 1*1 + 10*3 + 7*15 + 1*84 = 220;

a(4) = 1*1 + 20*3 + 28*15 + 10*84 + 1*495 = 1816; ...

compare to: Sum_{k=0..n} (-1)^k * A109955(n,k) * A005809(k) = (-2)^n.

MATHEMATICA

Table[Sum[Binomial[n + 2*k, 3*k] * Binomial[3*k, 2*k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 21 2014 *)

PROG

(PARI) {a(n)=sum(k=0, n, binomial(n+2*k, 3*k) * binomial(3*k, 2*k))}

for(n=0, 30, print1(a(n), ", "))

(PARI) {a(n)=-(-2)^n + 2*sum(k=0, n\2, binomial(n+4*k, 6*k) * binomial(6*k, 4*k))}

for(n=0, 30, print1(a(n), ", "))

(PARI) {a(n)=local(A=1); A=sum(m=0, n, binomial(3*m, m) * x^m/(1-x +x*O(x^n))^(3*m+1)); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A109955, A005809.

Sequence in context: A130185 A182432 A026020 * A026033 A005810 A121203

Adjacent sequences:  A243113 A243114 A243115 * A243117 A243118 A243119

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 20 2014

STATUS

approved

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Last modified September 20 20:02 EDT 2020. Contains 337265 sequences. (Running on oeis4.)