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A256462
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Double sum of the product of two binomials with even arguments.
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0
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1, 14, 238, 4092, 70070, 1192516, 20177164, 339653880, 5692584870, 95050101300, 1581953021220, 26255495764680, 434697697648188, 7181635051211432, 118422830335911640, 1949458102569167344, 32043155434056810246, 525974795270875804308
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OFFSET
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0,2
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COMMENTS
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Summing the product binomial(k + j, j) * binomial(2n - k - j, n - j), without even arguments restriction, one gets the sequence A002457.
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LINKS
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FORMULA
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a(n) = Sum_{k = 0..n} (Sum_{j = 0..n} binomial(2*k + 2*j, 2*j) * binomial(4*n - 2*k - 2*j, 2*n - 2*j)).
a(n) = (2*n^2 + 4*n + 1)*C(2*n), where C(n) is the n-th Catalan number. In other words, a(n) = A056220(n+1)*A000108(2*n).
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EXAMPLE
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For n = 2, the sum ((70+15+1) + (15+36+15) + (1+15+70)) evaluates to a(2) = 238.
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MATHEMATICA
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Table[(2*n^2 + 4*n + 1)*CatalanNumber[2*n], {n, 0, 25}]
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PROG
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(Magma) [(2*n^2 + 4*n + 1)*Catalan(2*n): n in [0..20]]; // Vincenzo Librandi, Apr 01 2015
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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