A367639 -id:A367639 - cp's OEIS frontend

A367640 G.f. A(x) satisfies A(x) = (1 + x)^2 + x*A(x)^3 / (1 + x)^2.

1, 3, 8, 36, 200, 1220, 7896, 53220, 369528, 2624772, 18981864, 139287588, 1034475624, 7761249476, 58735359032, 447827171556, 3436759851672, 26526255859716, 205782644595912, 1603655203428900, 12548225647402248, 98548826076070596, 776552629964300952

Offset: 0

Seiichi Manyama, Nov 25 2023

nonn

Cf. A367639, A367641.

Cf. A366221, A366266.

a(n) = sum(k=0, n, binomial(2*k+2, n-k)*binomial(3*k, k)/(2*k+1));

a(n) = Sum_{k=0..n} binomial(2*k+2,n-k) * binomial(3*k,k)/(2*k+1).

D-finite with recurrence 2*n*(14*n+71)*(2*n+1)*a(n) +3*(-150*n^3-209*n^2-379*n+228)*a(n-1) +9*(-30*n^3-981*n^2+4297*n-3624)*a(n-2) +27*(n-4)*(22*n^2-491*n+1151)*a(n-3) +81*(n-4)*(n-5)*(6*n-49)*a(n-4)=0. - R. J. Mathar, Dec 04 2023

A367641 G.f. A(x) satisfies A(x) = (1 + x)^2 + x*A(x)^4 / (1 + x)^3.

Original entry on oeis.org

1, 3, 10, 64, 504, 4368, 40208, 385728, 3813888, 38590208, 397648384, 4158436864, 44020882944, 470804670464, 5079479547904, 55217003536384, 604200374845440, 6649658071007232, 73560096496779264, 817467602640830464, 9121818467786162176

Offset: 0

Seiichi Manyama, Nov 25 2023

nonn

Cf. A367639, A367640.

Cf. A366267, A366272.

a(n) = sum(k=0, n, binomial(3*k+2, n-k)*binomial(4*k, k)/(3*k+1));

a(n) = Sum_{k=0..n} binomial(3*k+2,n-k) * binomial(4*k,k)/(3*k+1).

D-finite with recurrence 3*n*(5589*n-14914)*(3*n-1)*(3*n+1)*a(n) +(150903*n^4 -5939762*n^3 +21653157*n^2 -22049842*n +6856944)*a(n-1) +6*(-2312427*n^4 +15333754*n^3 -28367401*n^2 +6040114*n +14892656)*a(n-2) +24*(-3942141*n^4 +46541449*n^3 -199851671*n^2 +367766019*n -243569600)*a(n-3) -32*(n-5)*(8043984*n^3 -85808428*n^2 +305023231*n -361082892)*a(n-4) -384*(n-5)*(n-6)*(885234*n^2 -6808468*n +12951185)*a(n-5) -1536*(n-6)*(n-7)*(144699*n^2 -1203919*n +2337211)*a(n-6) -2048*(n-6)*(n-7)*(n-8)*(27819*n-74186)*a(n-7)=0. - R. J. Mathar, Dec 04 2023

cp's OEIS Frontend

A367640 G.f. A(x) satisfies A(x) = (1 + x)^2 + x*A(x)^3 / (1 + x)^2.

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