A396209 a(n) = Sum_{i=1..n} Sum_{j=1..i} binomial(j+1,2) * binomial(n+2*i-1,3*i-1) * binomial(i+2*j-1,3*j-1).

0, 1, 9, 72, 543, 3933, 27648, 189888, 1280037, 8497635, 55695799, 361110834, 2319610209, 14780020446, 93508249422, 587886009414, 3675355218708, 22862145873975, 141565603698549, 872978145149842, 5363042950323864, 32833615637339547, 200376192427825201

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (39, -696, 7573, -56505, 308418, -1283292, 4189581, -10965693, 23401729, -41276241, 60838386, -75602015, 79761162, -71814357, 55376019, -36637668, 20800974, -10116315, 4197834, -1476390, 435631, -106215, 20919, -3213, 363, -27, 1).

Formula

G.f.: B(B(B(x))), the third iteration of B(x) = x/(1-x)^3.

G.f.: x * ((1-x) * ((1-x)^3 - x))^6 / (((1-x)^3 - x)^3 - x*(1-x)^6)^3.

a(n) = 39*a(n-1) - 696*a(n-2) + 7573*a(n-3) - 56505*a(n-4) + 308418*a(n-5) - 1283292*a(n-6) + 4189581*a(n-7) - 10965693*a(n-8) + 23401729*a(n-9) - 41276241*a(n-10) + 60838386*a(n-11) - 75602015*a(n-12) + 79761162*a(n-13) - 71814357*a(n-14) + 55376019*a(n-15) - 36637668*a(n-16) + 20800974*a(n-17) - 10116315*a(n-18) + 4197834*a(n-19) - 1476390*a(n-20) + 435631*a(n-21) - 106215*a(n-22) + 20919*a(n-23) - 3213*a(n-24) + 363*a(n-25) - 27*a(n-26) + a(n-27).

G.f.: x / ( (1-x) * (1-B(x)) * (1-B(B(x))) )^3, where B(x) = x/(1-x)^3.

Prog

(PARI) a(n) = sum(i=1, n, sum(j=1, i, binomial(j+1, 2)*binomial(n+2*i-1, 3*i-1)*binomial(i+2*j-1, 3*j-1)));

Crossrefs

Cf. A396093, A396210.

Cf. A000217, A393908.

Sequence in context: A319873 A110396 A162755 * A378483 A045993 A084327

Adjacent sequences: A396206 A396207 A396208 * A396210 A396211 A396212

Keyword

nonn

Author

Seiichi Manyama, May 19 2026

Status

approved