A184553 a(n) = Sum_{k=0..n} C(3n+k,n-k)*C(4n-k,k).

1, 6, 79, 1158, 17851, 283246, 4579306, 75013112, 1240774907, 20677408080, 346638007264, 5839169781594, 98755770443674, 1675855850883818, 28520685212980872, 486589040917153648, 8319672542504635643

Offset

0,2

Links

Table of n, a(n) for n=0..16.

Formula

Self-convolution of A184554.

From Vaclav Kotesovec, Oct 05 2020: (Start)

Recurrence: 4608*n*(2*n - 1)*(3*n - 2)*(3*n - 1)*(6*n - 5)*(6*n - 1)*(824272*n^5 - 5898332*n^4 + 16800434*n^3 - 23808019*n^2 + 16784457*n - 4709052)*a(n) = 8*(5425780331264*n^11 - 55105585740928*n^10 + 246537716167440*n^9 - 639474746248560*n^8 + 1064922708464172*n^7 - 1190925449132724*n^6 + 908552008954195*n^5 - 470324138422910*n^4 + 160844796771909*n^3 - 34319567939418*n^2 + 4065509174760*n - 199264665600)*a(n-1) + 7*(7*n - 12)*(7*n - 11)*(7*n - 10)*(7*n - 9)*(7*n - 8)*(7*n - 6)*(824272*n^5 - 1776972*n^4 + 1449826*n^3 - 553989*n^2 + 97753*n - 6240)*a(n-2).

a(n) ~ 7^(7*n + 3/2) / (sqrt(Pi*n) * 2^(6*n + 4) * 3^(6*n + 1/2)). (End)

Example

G.f.: A(x) = 1 + 6*x + 79*x^2 + 1158*x^3 + 17851*x^4 + 283246*x^5 +...
A(x)^(1/2) = 1 + 3*x + 35*x^2 + 474*x^3 + 6891*x^4 + 104360*x^5 +...+ A184554(n)*x^n +...
Given triangle T(n,k) = C(4n-k,k), which begins:
1;
3, 1;
15, 7, 1;
84, 45, 11, 1;
495, 286, 91, 15, 1;
3003, 1820, 680, 153, 19, 1; ...
ILLUSTRATE formula a(n) = Sum_{k=0..n} T(n,k)*T(n,n-k):
a(2) = 79 = 15*1 + 7*7 + 1*15;
a(3) = 1158 = 84*1 + 45*11 + 11*45 + 1*84;
a(4) = 17851 = 495*1 + 286*15 + 91*91 + 15*286 + 1*495;
a(5) = 283246 = 3003*1 + 1820*19 + 680*153 + 153*680 + 19*1820 + 1*3003; ...

Mathematica

Table[Sum[Binomial[3*n + k, n - k]*Binomial[4*n - k, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 05 2020 *)

Prog

(PARI) {a(n)=sum(k=0, n, binomial(3*n+k, n-k)*binomial(4*n-k, k))}

Crossrefs

Cf. A184554, A183160.

Sequence in context: A093033 A156929 A123796 * A129524 A131237 A024082

Adjacent sequences: A184550 A184551 A184552 * A184554 A184555 A184556

Keyword

nonn

Author

Paul D. Hanna, Jan 16 2011

Status

approved