A365879
Expansion of (1/x) * Series_Reversion( x*(1+x)^3*(1-x)^5 ).
Original entry on oeis.org
1, 2, 10, 54, 332, 2162, 14734, 103630, 746857, 5486206, 40926152, 309202686, 2361065920, 18192978966, 141280871840, 1104603758526, 8687878404289, 68692224882620, 545681467048132, 4353096328518810, 34858239962177764, 280095777427596008
Offset: 0
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a(n) = sum(k=0, n, (-1)^k*binomial(3*n+k+2, k)*binomial(6*n-k+4, n-k))/(n+1);
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def A365879(n):
h = binomial(6*n + 4, n) * hypergeometric([-n, 3*(n + 1)], [-6 * n - 4], -1) / (n + 1)
return simplify(h)
print([A365879(n) for n in range(22)]) # Peter Luschny, Sep 21 2023
A368079
Expansion of (1/x) * Series_Reversion( x * (1-x)^3 * (1-x^2)^3 ).
Original entry on oeis.org
1, 3, 18, 127, 996, 8322, 72644, 654615, 6043455, 56866028, 543368586, 5258196762, 51426990112, 507537393600, 5048033356128, 50549237164615, 509197913456922, 5156339940802941, 52460340305220466, 535976129228082972, 5496745175387480976
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3*(1-x^2)^3)/x)
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a(n, s=2, t=3, u=3) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((u+1)*(n+1)-s*k-2, n-s*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
A370269
Coefficient of x^n in the expansion of 1/( (1-x) * (1-x^2)^3 )^n.
Original entry on oeis.org
1, 1, 9, 37, 233, 1251, 7461, 43219, 257769, 1534096, 9224259, 55607850, 336885029, 2046705428, 12472585155, 76185639162, 466380345065, 2860318763352, 17571932737128, 108111252582449, 666049600308483, 4108363051479346, 25369393216077370
Offset: 0
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a(n, s=2, t=3, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((u+1)*n-s*k-1, n-s*k));
Showing 1-4 of 4 results.