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);
A379188
G.f. A(x) satisfies A(x) = 1/((1 - x*A(x)^3) * (1 - x*A(x))^3).
Original entry on oeis.org
1, 4, 34, 392, 5271, 77530, 1208602, 19620262, 328167191, 5616065633, 97867738285, 1730732539345, 30981439344096, 560293394484145, 10221582080782452, 187884236846039893, 3476266045318846245, 64690833375603622619, 1210026171180264742927, 22736845507710710652858
Offset: 0
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a(n) = sum(k=0, n, binomial(n+3*k+1, k)*binomial(4*n+5*k+2, n-k)/(n+3*k+1));
A379191
G.f. A(x) satisfies A(x) = (1 + x*A(x))^3/(1 - x*A(x)^3).
Original entry on oeis.org
1, 4, 31, 338, 4356, 61603, 923958, 14433315, 232298914, 3825260332, 64140203645, 1091364139213, 18796605318655, 327056343952311, 5740466392321499, 101516213938082457, 1807045676161156515, 32352346658163940698, 582185299986049977601, 10524395285312191583304, 191034444423571726099486
Offset: 0
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a(n) = sum(k=0, n, binomial(n+3*k+1, k)*binomial(3*n+6*k+3, n-k)/(n+3*k+1));
A379171
G.f. A(x) satisfies A(x) = (1 + x)/(1 - x*A(x))^3.
Original entry on oeis.org
1, 4, 21, 139, 1021, 8010, 65708, 556751, 4834686, 42800265, 384832083, 3504693519, 32261240127, 299685628629, 2805773759322, 26448278629697, 250806022116194, 2390973659474304, 22901157688878983, 220279614235505630, 2126890041331033797, 20606993367985131716
Offset: 0
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a(n) = sum(k=0, n, binomial(n-k+1, k)*binomial(4*n-4*k+2, n-k)/(n-k+1));
A379190
G.f. A(x) satisfies A(x) = (1 + x*A(x)^3) * (1 + x*A(x))^3.
Original entry on oeis.org
1, 4, 30, 304, 3557, 45150, 604222, 8393282, 119872890, 1749183075, 25964512607, 390828464403, 5951561595889, 91523131078999, 1419293428538496, 22169968253466467, 348507676062911520, 5509187208564734328, 87522347516801353980, 1396619714730284551913, 22375420057050167868366
Offset: 0
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a(n) = sum(k=0, n, binomial(n+2*k+1, k)*binomial(3*n+6*k+3, n-k)/(n+2*k+1));
A379246
G.f. A(x) satisfies A(x) = ( (1 + x*A(x)^3)/(1 - x*A(x)) )^3.
Original entry on oeis.org
1, 6, 90, 1838, 43362, 1111878, 30101786, 846703950, 24501770370, 724733787206, 21813611057562, 665947742487342, 20571682188676450, 641823879285627654, 20195381326042866138, 640146274715559742670, 20421757641058980395010, 655181707585675667750790, 21125606434257067959841242
Offset: 0
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a(n) = sum(k=0, n, binomial(3*n+6*k+3, k)*binomial(4*n+5*k+2, n-k)/(n+2*k+1));
Showing 1-6 of 6 results.