A365752
Expansion of (1/x) * Series_Reversion( x*(1+x)*(1-x)^4 ).
Original entry on oeis.org
1, 3, 16, 103, 735, 5592, 44452, 364815, 3067558, 26290517, 228819168, 2016953848, 17968790029, 161536295244, 1463535347928, 13349907110367, 122499957767130, 1130001670577730, 10472708110616136, 97468774074103041, 910582642690819351
Offset: 0
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a(n) = sum(k=0, n, (-1)^k*binomial(n+k, k)*binomial(5*n-k+3, n-k))/(n+1);
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def A365752(n):
h = binomial(5*n + 3, n) * hypergeometric([-n, n + 1], [-5 * n - 3], -1) / (n + 1)
return simplify(h)
print([A365752(n) for n in range(21)]) # Peter Luschny, Sep 20 2023
A365753
Expansion of (1/x) * Series_Reversion( x*(1+x)*(1-x)^5 ).
Original entry on oeis.org
1, 4, 27, 220, 1984, 19064, 191325, 1981932, 21031965, 227463808, 2498039219, 27782561352, 312281382836, 3541879743840, 40484779373060, 465888833819532, 5393215780225983, 62761359573224612, 733784067570047400, 8615217370731224160, 101533102164551821896
Offset: 0
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a(n) = sum(k=0, n, (-1)^k*binomial(n+k, k)*binomial(6*n-k+4, n-k))/(n+1);
A365764
Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1+x) ).
Original entry on oeis.org
1, 4, 25, 188, 1563, 13840, 127972, 1221260, 11938471, 118936100, 1203155633, 12325599632, 127611357300, 1333153669632, 14035828918560, 148773617605036, 1586305110768863, 17002975960876300, 183102052226442475, 1980078493171083292, 21493846031259095539
Offset: 0
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CoefficientList[(1/x) *InverseSeries[Series[x*(1-x)^3/(1+x),{x,0,21}]],x] (* Stefano Spezia, May 04 2025 *)
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a(n) = sum(k=0, n, binomial(n+1, k)*binomial(4*n-k+2, n-k))/(n+1);
A365854
Expansion of (1/x) * Series_Reversion( x*(1+x)^2*(1-x)^3 ).
Original entry on oeis.org
1, 1, 4, 13, 55, 232, 1052, 4869, 23206, 112519, 554560, 2767336, 13959941, 71060356, 364569352, 1883143669, 9785481498, 51118097686, 268294595396, 1414106565611, 7481787454031, 39721596068000, 211549545257760, 1129912319370600, 6050931114958080
Offset: 0
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a(n) = sum(k=0, n, (-1)^k*binomial(2*n+k+1, k)*binomial(4*n-k+2, n-k))/(n+1);
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def A365854(n):
h = binomial(2*(2*n + 1), n) * hypergeometric([-n, 2*(n + 1)], [-2*(2*n + 1)], -1) / (n + 1)
return simplify(h)
print([A365854(n) for n in range(25)]) # Peter Luschny, Sep 20 2023
Showing 1-4 of 4 results.