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A089667
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a(n) = S2(n,4), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.
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5
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0, 4, 265, 5984, 85722, 944904, 8771462, 72095520, 541127988, 3785356752, 25032083230, 158102986624, 961123994220, 5656943319664, 32386277835772, 181019819948864, 990793669704552, 5323620638111136, 28137973407708174, 146552649537716992
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = (1/480)*( n*(n+1)*(93*n^3 + 132*n^2 + 53*n - 38)*4^n - 4*n*(n-1)*(15*n^5 - 99*n^3 + 116*n^2 - 34*n + 6)*binomial(2*n, n)/((2*n-1)*(2*n-3)). (See Wang and Zhang, p. 338.)
a(n) = (1/30)*( n*(n+1)*93*n^3 + 132*n^2 + 53*n -38)*4^(n-2) - (n-1)*(15*n^5 - 99*n^3 + 116*n^2 - 34*n + 6)*Catalan(n-2) ).
G.f.: x*( 4*(1 + 43*x + 160*x^2 + 96*x^3) - x*(3 + 62*x - 72*x^2 + 96*x^3 - 224*x^4 + 144*x^5)*sqrt(1-4*x) )/(1-4*x)^6. [Typo corrected by Georg Fischer, Nov 09 2022] (End)
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MATHEMATICA
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Table[(1/30)*(n*(n+1)*(93*n^3+132*n^2+53*n-38)*4^(n-2) -(n-1)*(15*n^5-99*n^3 + 116*n^2-34*n+6)*CatalanNumber[n-2]), {n, 0, 40}] (* G. C. Greubel, May 25 2022 *)
CoefficientList[Series[x*( 4*(1 + 43*x + 160*x^2 + 96*x^3) - x*(3 + 62*x - 72*x^2 + 96*x^3 - 224*x^4 + 144*x^5)*Sqrt[1-4*x] )/(1-4*x)^6, {x, 0, 35}], x] (* Georg Fischer, Nov 09 2022 *)
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PROG
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(SageMath) [(1/30)*(n*(n+1)*(93*n^3+132*n^2+53*n-38)*4^(n-2) - (n-1)*(15*n^5 - 99*n^3+116*n^2-34*n+6)*catalan_number(n-2) ) for n in (0..40)] # G. C. Greubel, May 25 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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