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);
A368011
Expansion of (1/x) * Series_Reversion( x * ((1-x)^5-x^5) ).
Original entry on oeis.org
1, 5, 40, 385, 4095, 46377, 548380, 6691620, 83637450, 1065311665, 13777916774, 180451354720, 2388503030675, 31900445734050, 429369814375480, 5818270533841408, 79309912829992350, 1086768622818959100, 14961519902879613700, 206839961042385226110
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^5-x^5))/x)
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a(n) = sum(k=0, n\5, binomial(n+k, k)*binomial(6*n+4, n-5*k))/(n+1);
A369514
Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^3)^2 ).
Original entry on oeis.org
1, 6, 57, 652, 8250, 111228, 1566384, 22770990, 339136149, 5147965790, 79355002155, 1238845925070, 19546811164017, 311215082863152, 4993737492276384, 80673666233512572, 1311052196736963738, 21418709030787603984, 351563022864652061086, 5794815410347964694408
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3-x^3)^2)/x)
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a(n) = sum(k=0, n\3, binomial(2*n+k+1, k)*binomial(7*n+5, n-3*k))/(n+1);
A369694
Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^2) ).
Original entry on oeis.org
1, 3, 16, 106, 786, 6244, 51964, 447201, 3947306, 35538668, 325098696, 3013060258, 28232408848, 267003169668, 2545341982728, 24433290332007, 235967943943224, 2291147902820524, 22352525061549604, 219006814853751540, 2154083325737401740
Offset: 0
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CoefficientList[InverseSeries[Series[x*((1-x)^3 - x^2), {x, 0, 30}], x]/x, x](* Vaclav Kotesovec, Jan 29 2024 *)
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3-x^2))/x)
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a(n) = sum(k=0, n\2, binomial(n+k, k)*binomial(4*n+k+2, n-2*k))/(n+1);
A370284
Coefficient of x^n in the expansion of 1/( (1-x)^3 - x^3 )^n.
Original entry on oeis.org
1, 3, 21, 168, 1425, 12483, 111594, 1011636, 9264753, 85510590, 794087151, 7410887718, 69446624910, 653019755430, 6158495001960, 58226492157048, 551725482707505, 5238008159399163, 49814314319342424, 474467729545936650, 4525387365179378775
Offset: 0
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a(n) = sum(k=0, n\3, binomial(n+k-1, k)*binomial(4*n-1, n-3*k));
Showing 1-6 of 6 results.