A391466 Expansion of g/(2 - g^2)^2, where g = 1+x*g^4 is the g.f. of A002293.

1, 5, 38, 312, 2651, 22972, 201640, 1786192, 15931889, 142871524, 1286804904, 11631656288, 105459841793, 958650722196, 8733984073432, 79730279613824, 729113571351117, 6678008216229236, 61250624197094888, 562507296727181792, 5171903809038602028

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n) = (1/(2*n)) * Sum_{k=1..n} k * (k+1) * Pell(k+2) * binomial(4*n,n-k) for n > 0.

D-finite with recurrence: (32505856*n^3 + 48758784*n^2 + 22347776*n + 3047424)*a(n) + (1321185280*n^3 + 7193825280*n^2 + 12668395520*n + 7165255680)*a(n + 1) + (-587447296*n^3 - 8473500672*n^2 - 30246010880*n - 31676700672)*a(n + 2) + (-364244992*n^3 - 1125076992*n^2 + 4153927168*n + 12971079168)*a(n + 3) + (208428928*n^3 + 1924702080*n^2 + 5416832000*n + 4193957760)*a(n + 4) + (-37905168*n^3 - 468623472*n^2 - 1894425312*n - 2499670944)*a(n + 5) + (3118980*n^3 + 46853700*n^2 + 232728528*n + 383618808)*a(n + 6) + (-112446*n^3 - 1988982*n^2 - 11649294*n - 22654998)*a(n + 7) + (1215*n^3 + 25515*n^2 + 177660*n + 410400)*a(n + 8) = 0. - Robert Israel, Dec 10 2025

Maple

f:= gfun:-rectoproc({(32505856*n^3 + 48758784*n^2 + 22347776*n + 3047424)*a(n) + (1321185280*n^3 + 7193825280*n^2 + 12668395520*n + 7165255680)*a(n + 1) + (-587447296*n^3 - 8473500672*n^2 - 30246010880*n - 31676700672)*a(n + 2) + (-364244992*n^3 - 1125076992*n^2 + 4153927168*n + 12971079168)*a(n + 3) + (208428928*n^3 + 1924702080*n^2 + 5416832000*n + 4193957760)*a(n + 4) + (-37905168*n^3 - 468623472*n^2 - 1894425312*n - 2499670944)*a(n + 5) + (3118980*n^3 + 46853700*n^2 + 232728528*n + 383618808)*a(n + 6) + (-112446*n^3 - 1988982*n^2 - 11649294*n - 22654998)*a(n + 7) + (1215*n^3 + 25515*n^2 + 177660*n + 410400)*a(n + 8), a(0) = 1, a(1) = 5, a(2) = 38, a(3) = 312, a(4) = 2651, a(5) = 22972, a(6) = 201640, a(7) = 1786192}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Dec 10 2025

Mathematica

Table[If[n==0, 1, Sum[k (k+1) PellP[k+2] Binomial[4 n, n-k], {k, 1, n}]/(2 n)], {n, 0, 20}] (* Vincenzo Librandi, Dec 10 2025 *)

Prog

(PARI) pell(n) = ([2, 1; 1, 0]^n)[2, 1];
a(n) = if(n==0, 1, sum(k=1, n, k*(k+1)*pell(k+2)*binomial(4*n, n-k))/(2*n));
(Magma) function Pell(n)
    a := 0; b := 1;
    for i in [2..n] do
        tmp := b;
        b := 2*b + a;
        a := tmp;
    end for;
    return n eq 0 select 0 else b;
end function;
vals := [ n eq 0 select 1 else
    (&+[ k*(k+1)*Pell(k+2)*Binomial(4*n, n-k) : k in [1..n] ]) div (2*n)
    : n in [0..25] ];
print vals; // Vincenzo Librandi, Dec 10 2025

Crossrefs

Cf. A391460, A391463.

Cf. A000129.

Sequence in context: A110082 A073508 A282964 * A389361 A357163 A357224

Adjacent sequences: A391463 A391464 A391465 * A391467 A391468 A391469

Keyword

nonn

Author

Seiichi Manyama, Dec 10 2025

Status

approved