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A331987
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a(n) = ((n + 1) - 9*(n + 1)^2 + 8*(n + 1)^3)/6.
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4
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0, 5, 23, 62, 130, 235, 385, 588, 852, 1185, 1595, 2090, 2678, 3367, 4165, 5080, 6120, 7293, 8607, 10070, 11690, 13475, 15433, 17572, 19900, 22425, 25155, 28098, 31262, 34655, 38285, 42160, 46288, 50677, 55335, 60270, 65490, 71003, 76817, 82940, 89380, 96145
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OFFSET
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0,2
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COMMENTS
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The start values of the partial rows on the main diagonal of A332662 in the representation in the example section.
Apparently the sum of the hook lengths over the partitions of 2*n + 1 with exactly 2 parts (cf. A180681).
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LINKS
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FORMULA
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a(n) = [x^n] (x*(5 + 3*x)/(1 - x)^4).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = binomial(n+2, 3) + binomial(n+1, 3) + 2*(n+1)*binomial(n+1, 2).
a(n) = 3*binomial(n+1,1) - 11*binomial(n+2,2) + 8*binomial(n+3,3).
a(n) = n*binomial(8*n+8,2)/24.
a(n) = n*(n+1)*(8*n+7)/6.
E.g.f.: (1/6)*x*(30 + 39*x + 8*x^2)*exp(x). (End)
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MAPLE
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a := n -> ((n+1) - 9*(n+1)^2 + 8*(n+1)^3)/6: seq(a(n), n=0..41);
gf := (x*(3*x + 5))/(x - 1)^4: ser := series(gf, x, 44):
seq(coeff(ser, x, n), n=0..41);
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MATHEMATICA
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LinearRecurrence[{4, -6, 4, -1}, {0, 5, 23, 62}, 42]
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PROG
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(Magma) [n*(n+1)*(8*n+7)/6: n in [0..50]]; // G. C. Greubel, Apr 19 2023
(SageMath)
def A331987(n): return n*(n+1)*(8*n+7)/6
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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