A379186
G.f. A(x) satisfies A(x) = 1/((1 - x*A(x)^3) * (1 - x*A(x))^2).
Original entry on oeis.org
1, 3, 21, 202, 2270, 27903, 363412, 4927840, 68834941, 983680783, 14312988289, 211329419670, 3158263216267, 47682769300288, 726188701482730, 11142842570134264, 172101193009427174, 2673445730846829604, 41742159037922167264, 654721526817143247304, 10311337739352708700427
Offset: 0
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terms = 21; A[] = 0; Do[A[x] = 1/((1-x*A[x]^3)*(1 -x*A[x])^2) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jun 14 2025 *)
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a(n) = sum(k=0, n, binomial(n+3*k+1, k)*binomial(3*n+3*k+1, n-k)/(n+3*k+1));
A379173
G.f. A(x) satisfies A(x) = (1 + x)/(1 - x*A(x))^2.
Original entry on oeis.org
1, 3, 11, 53, 284, 1630, 9794, 60830, 387390, 2515892, 16599051, 110943779, 749603067, 5111606801, 35133394554, 243146923574, 1692918638012, 11850006727400, 83341778073920, 588646472669454, 4173607638548291, 29694593381322531, 211941668053441490, 1517087043428034420
Offset: 0
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a(n) = sum(k=0, n, binomial(n-k+1, k)*binomial(3*n-3*k+1, n-k)/(n-k+1));
A379244
G.f. A(x) satisfies A(x) = ( (1 + x*A(x)^3)/(1 - x*A(x)) )^2.
Original entry on oeis.org
1, 4, 40, 540, 8400, 141876, 2528760, 46815116, 891483808, 17350187364, 343578992328, 6900588813564, 140230648164720, 2878066866407316, 59571280942854808, 1242093725341221996, 26064579113472078144, 550041399791036747460, 11665771061882347813224, 248527169321049466503132
Offset: 0
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a(n) = sum(k=0, n, binomial(2*n+4*k+2, k)*binomial(3*n+3*k+1, n-k)/(n+2*k+1));
Showing 1-3 of 3 results.