A387036 a(n) = Sum_{k=0..n} binomial(4*n-2,k).
1, 3, 22, 176, 1471, 12616, 110056, 971712, 8656937, 77663192, 700614760, 6349125440, 57754842117, 527046644056, 4822774262296, 44235726874816, 406582639811581, 3743845040832376, 34529632747211560, 318931047174438720, 2949641596923575548, 27312107861301870368
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Crossrefs
Programs
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Magma
[&+[Binomial(4*n-2, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
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Mathematica
Table[Sum[Binomial[4*n-2,k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(4*n-2, k));
Formula
a(n) = [x^n] (1+x)^(4*n-2)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n-2) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-2,k) * binomial(4*n-k-3,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-3,n-k).
G.f.: 1/(g * (2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-7)*(2*n-3)*(4*n-9)*(22*n^2-17*n-15)*a(n-2) -8*(1892*n^5-11274*n^4+23326*n^3-18132*n^2+1323*n+2835)*a(n-1) +3*n*(3*n-4)*(3*n-5)*(22*n^2-61*n+24)*a(n) = 0. - Georg Fischer, Aug 17 2025