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);
A379172
G.f. A(x) satisfies A(x) = (1 + x*A(x)^3)/(1 - x*A(x))^3.
Original entry on oeis.org
1, 4, 33, 358, 4445, 59745, 846023, 12430941, 187753479, 2896929975, 45465112431, 723520554096, 11647721390271, 189352106241567, 3104046096391902, 51254005259550753, 851674902290491936, 14231191062537888864, 238978853442142491358, 4030889937027642017872
Offset: 0
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a(n) = sum(k=0, n, binomial(n+2*k+1, k)*binomial(4*n+5*k+2, n-k)/(n+2*k+1));
A365766
Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x) ).
Original entry on oeis.org
1, 6, 56, 626, 7721, 101322, 1387648, 19606874, 283711805, 4183074796, 62618441024, 949174260118, 14539621490403, 224721722650224, 3500129695446816, 54882906729334378, 865664769346769005, 13725517938819785298, 218639429113140366968
Offset: 0
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a(n) = sum(k=0, n, binomial(n+1, k)*binomial(6*n-k+4, n-k))/(n+1);
A365765
Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1+x) ).
Original entry on oeis.org
1, 5, 39, 365, 3772, 41491, 476410, 5644477, 68493324, 846937140, 10633195119, 135185288475, 1736883987836, 22516798984946, 294169295918996, 3869084306851933, 51189853304834940, 680816769653570044, 9097058255214149068, 122064057533865334100
Offset: 0
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CoefficientList[(1/x) *InverseSeries[Series[x*(1-x)^4/(1+x),{x,0,20}]],x] (* Stefano Spezia, May 04 2025 *)
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a(n) = sum(k=0, n, binomial(n+1, k)*binomial(5*n-k+3, n-k))/(n+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));
Showing 1-5 of 5 results.