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
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
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
A365880
Expansion of (1/x) * Series_Reversion( x*(1+x)^4*(1-x)^5 ).
Original entry on oeis.org
1, 1, 6, 21, 116, 566, 3176, 17501, 101391, 590756, 3519782, 21163038, 128845344, 790810400, 4894134376, 30486741869, 191068074202, 1203710067455, 7619193325238, 48430121151156, 309011352878208, 1978450305442086, 12706836843595840
Offset: 0
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a(n) = sum(k=0, n, (-1)^k*binomial(4*n+k+3, k)*binomial(6*n-k+4, n-k))/(n+1);
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def A365880(n):
h = binomial(6*n + 4, n) * hypergeometric([-n, 4*(n + 1)], [-6 * n - 4], -1) / (n + 1)
return simplify(h)
print([A365880(n) for n in range(23)]) # Peter Luschny, Sep 21 2023
A370104
a(n) = Sum_{k=0..n} (-1)^k * binomial(2*n+k-1,k) * binomial(6*n-k-1,n-k).
Original entry on oeis.org
1, 3, 25, 219, 2025, 19253, 186469, 1829565, 18124521, 180886260, 1815946275, 18318160358, 185518492965, 1885157971596, 19211066004995, 196258973605094, 2009302383218409, 20610411795602760, 211768072490024440, 2179156980022097775, 22454554231950998275
Offset: 0
-
a(n) = sum(k=0, n, (-1)^k*binomial(2*n+k-1, k)*binomial(6*n-k-1, n-k));
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a(n, s=2, t=2, u=3) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((u+1)*n-s*k-1, n-s*k));
Showing 1-7 of 7 results.