A368961
Expansion of (1/x) * Series_Reversion( x * (1-x-x^2)^2 ).
Original entry on oeis.org
1, 2, 9, 48, 286, 1820, 12116, 83334, 587537, 4223582, 30840355, 228111390, 1705509981, 12868775056, 97867753424, 749401318160, 5772939358590, 44708058004740, 347879528717526, 2718400037837988, 21323471768334120, 167844335760482220, 1325332432687278960
Offset: 0
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a[n_] := (1/(n+1)) * Sum[Binomial[2*n+k+1,k] * Binomial[3*n-k+1,n-2*k],{k,0,Floor[n/2]}]; Array[a,23,0] (* Stefano Spezia, Aug 11 2025 *)
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x-x^2)^2)/x)
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a(n, s=2, t=2, u=0) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);
A368968
Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x-x^3)^2 ).
Original entry on oeis.org
1, 4, 26, 206, 1813, 17030, 167229, 1695920, 17624932, 186722580, 2009077416, 21894695420, 241170873096, 2680761546396, 30032284769832, 338744791093796, 3843699928567438, 43844993166845920, 502497843180361288, 5783367971991398760, 66815895492710846218
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2*(1-x-x^3)^2)/x)
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a(n, s=3, t=2, u=2) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);
A368965
Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^2)^2 ).
Original entry on oeis.org
1, 3, 17, 117, 895, 7309, 62410, 550431, 4975297, 45846977, 429095387, 4067760593, 38977419018, 376901628882, 3673226867356, 36043590216621, 355800292078095, 3530878133357175, 35205183620396571, 352505713454687599, 3543078943592291301
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*(1-x-x^2)^2)/x)
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a(n, s=2, t=2, u=1) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);
A368975
Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x+x^2)^2 ).
Original entry on oeis.org
1, 4, 24, 170, 1320, 10868, 93199, 823548, 7446480, 68567202, 640757920, 6061477500, 57933260067, 558580920160, 5426644737984, 53069206438226, 522004849765080, 5161083186971000, 51262685633583970, 511272660117154692, 5118240198221249088
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2*(1-x+x^2)^2)/x)
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a(n, s=2, t=2, u=2) = sum(k=0, n\s, (-1)^k*binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);
A365855
Expansion of (1/x) * Series_Reversion( x*(1+x)^2*(1-x)^4 ).
Original entry on oeis.org
1, 2, 9, 46, 264, 1612, 10291, 67830, 458109, 3153744, 22049065, 156127140, 1117369884, 8069610992, 58735003740, 430416574918, 3172987081311, 23514565653058, 175083678670264, 1309132916709168, 9825882638364144, 74003924059921940, 559112987425763365
Offset: 0
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a(n) = sum(k=0, n, (-1)^k*binomial(2*n+k+1, k)*binomial(5*n-k+3, n-k))/(n+1);
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def A365855(n):
h = binomial(5*n + 3, n) * hypergeometric([-n, 2*n + 2], [-5 * n - 3], -1) / (n + 1)
return simplify(h)
print([A365855(n) for n in range(23)]) # Peter Luschny, Sep 20 2023
A369486
Expansion of (1/x) * Series_Reversion( x / (1-x) * (1-x-x^2)^2 ).
Original entry on oeis.org
1, 1, 4, 15, 67, 314, 1547, 7865, 41004, 217953, 1176832, 6436676, 35587416, 198569471, 1116741601, 6323669519, 36024382515, 206315985386, 1187205083042, 6860598312545, 39797882898452, 231666709974264, 1352813494962672, 7922553881534274, 46520280837291427
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1-x)*(1-x-x^2)^2)/x)
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a(n, s=2, t=2, u=1) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t-u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);
A372460
Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-x-x^2) )^(2*n).
Original entry on oeis.org
1, 4, 40, 442, 5136, 61424, 748462, 9240480, 115194720, 1446820588, 18279806600, 232071505120, 2958062657550, 37831613904036, 485233557808704, 6239148779539472, 80397210629541696, 1037970502613332320, 13423439565585274180, 173859642721737225552
Offset: 0
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a(n, s=2, t=2, u=2) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t+u+1)*n-(s-1)*k-1, n-s*k));
Showing 1-7 of 7 results.