|
|
A112029
|
|
a(n) = Sum_{k=0..n} binomial(n+k, k)^2.
|
|
12
|
|
|
1, 5, 46, 517, 6376, 82994, 1119210, 15475205, 217994860, 3115374880, 45035696036, 657153097330, 9663914317396, 143050882063262, 2129448324373546, 31853280798384645, 478503774600509620, 7215090439396842572, 109154411037070011504, 1656268648035559711392
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
LINKS
|
|
|
FORMULA
|
Recurrence: 2*(2*n+1)*(21*n-13)*n^2*a(n) = (1365*n^4 - 1517*n^3 + 240*n^2 + 216*n - 64)*a(n-1) - 4*(n-1)*(2*n-1)^2*(21*n+8)*a(n-2). - Vaclav Kotesovec, Nov 23 2012
a(p-1) == 1 (mod p^3) for all primes p >= 5. See the comments in A173774. - Peter Bala, Jul 12 2024
a(n-1) = 1/(4*n) * binomial(2*n, n)^2 * ( 1 + 3*((n - 1)/(n + 1))^3 + 5*((n - 1)*(n - 2)/((n + 1)*(n + 2)))^3 + 7*((n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)))^3 + ... ) for n >= 1. - Peter Bala, Jul 22 2024
|
|
MAPLE
|
f := 64*x^2/(16*x-1); S := sqrt(x)*sqrt(4-x);
H := ((10*x-5/8)*hypergeom([1/4, 1/4], [1], f)-(21*x-21/8)*hypergeom([1/4, 5/4], [1], f))/(S*(1-16*x)^(5/4));
ord := 30;
ogf := series(int(series(H, x=0, ord), x)/S, x=0, ord);
|
|
MATHEMATICA
|
Table[Sum[Binomial[n+k, k]^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 23 2012 *)
|
|
PROG
|
(Magma) [(&+[Binomial(n+j, j)^2: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Jul 06 2021
(Sage) [sum(binomial(n+j, j)^2 for j in (0..n)) for n in (0..20)] # G. C. Greubel, Jul 06 2021
(PARI) a(n) = sum(k=0, n, binomial(n+k, k)^2); \\ Michel Marcus, Jul 07 2021
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|