A369215
Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x) ).
Original entry on oeis.org
1, 4, 29, 261, 2627, 28315, 319648, 3731037, 44663058, 545312504, 6764556591, 85015779095, 1080185111768, 13852183882612, 179058158369828, 2330621446075640, 30519758687849439, 401806204894374041, 5315243189757111099, 70613088335938995385, 941714812929017751855
Offset: 0
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CoefficientList[InverseSeries[Series[x((1-x)^3-x),{x,0,21}],x]/x,x] (* Stefano Spezia, Mar 31 2025 *)
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3-x))/x)
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a(n) = sum(k=0, n, binomial(n+k, k)*binomial(4*n+2*k+2, n-k))/(n+1);
A369214
Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^3) ).
Original entry on oeis.org
1, 2, 7, 31, 155, 833, 4696, 27393, 163944, 1001022, 6211049, 39048685, 248213672, 1592561156, 10300192220, 67083304750, 439571860881, 2895898913453, 19169805142929, 127442939722175, 850536450459795, 5696270624620125, 38271171118343550
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^2-x^3))/x)
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a(n) = sum(k=0, n\3, binomial(n+k, k)*binomial(3*n-k+1, n-3*k))/(n+1);
A370280
Coefficient of x^n in the expansion of 1/( (1-x)^2 - x )^n.
Original entry on oeis.org
1, 3, 25, 234, 2305, 23373, 241486, 2527920, 26720529, 284555700, 3048323135, 32812937820, 354619072990, 3845377105794, 41817926091120, 455893204069944, 4980851709418353, 54521955043418925, 597823622561048020, 6564929893462467450, 72189820135528858455
Offset: 0
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R:=PowerSeriesRing(Rationals(), 100);
A370280:= func< n | Coefficient(R!( 1/(1-3*x+x^2)^n ), n) >;
[A370280(n): n in [0..30]]; // G. C. Greubel, Feb 07 2025
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A370280[n_]:= Coefficient[Series[1/(1-3*x+x^2)^n, {x,0,100}], x, n];
Table[A370280[n], {n,0,40}] (* G. C. Greubel, Feb 07 2025 *)
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a(n) = sum(k=0, n, binomial(n+k-1, k)*binomial(3*n+k-1, n-k));
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def A370280(n): return sum(binomial(n+j-1,j)*binomial(3*n+j-1,n-j) for j in range(n+1))
print([A370280(n) for n in range(31)]) # G. C. Greubel, Feb 07 2025
A379174
G.f. A(x) satisfies A(x) = (1 + x*A(x)^3)/(1 - x*A(x))^2.
Original entry on oeis.org
1, 3, 20, 176, 1772, 19309, 221651, 2640016, 32322122, 404256442, 5142846467, 66341063274, 865723122919, 11408144684248, 151593390664710, 2029025599194394, 27330120599494110, 370183683091079836, 5038997387800717228, 68896081533831380702, 945747379824209853435
Offset: 0
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a(n) = sum(k=0, n, binomial(n+2*k+1, k)*binomial(3*n+3*k+1, n-k)/(n+2*k+1));
A369216
Expansion of (1/x) * Series_Reversion( x * ((1-x)^4-x) ).
Original entry on oeis.org
1, 5, 44, 479, 5827, 75887, 1034980, 14593794, 211031650, 3112385177, 46636714566, 707983562624, 10865572966703, 168306274609798, 2627854427929448, 41314461126179272, 653481096161664690, 10391753978329136808, 166040704868503173384
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serreverse(x*((1-x)^4-x))/x)
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a(n) = sum(k=0, n, binomial(n+k, k)*binomial(5*n+3*k+3, n-k))/(n+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));
Showing 1-6 of 6 results.