A387034 a(n) = Sum_{k=0..n} binomial(4*n-4,k).
1, 1, 11, 93, 794, 6885, 60460, 536155, 4791323, 43081973, 389329652, 3533047572, 32174057272, 293874981603, 2691171713924, 24700051833634, 227150464141969, 2092620625940629, 19308393192688804, 178406554524801820, 1650535921328322392, 15287533448476027572
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Crossrefs
Programs
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Magma
[&+[Binomial(4*n-4, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
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Mathematica
Table[Sum[Binomial[4*n-4,k], {k,0,n}], {n,0,25}] (* Vaclav Kotesovec, Aug 20 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(4*n-4, k));
Formula
a(n) = [x^n] (1+x)^(4*n-4)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n-4) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-4,k) * binomial(4*n-k-5,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-5,n-k).
G.f.: 1/(g^3 * (2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-11)*(2*n-5)*(4*n-9)*(44*n^3-122*n^2+18*n+105)*a(n-2)-8*(3784*n^6-37684*n^5+141548*n^4-238406*n^3+145758*n^2+37290*n-51975)*a(n-1)+3*n*(3*n-5)*(3*n-7)*(44*n^3-254*n^2+394*n-79)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 2^(8*n - 17/2) / (sqrt(Pi*n) * 3^(3*n - 9/2)). - Vaclav Kotesovec, Aug 20 2025