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A101476
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Sum[k=0..n, Sum[i>=0, C(2n,n-i(k+2))-C(2n,n+i(k+2)+k+1) ]].
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1
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1, 3, 14, 64, 287, 1265, 5500, 23660, 100931, 427669, 1802170, 7559473, 31586889, 131548937, 546297862, 2263042926, 9354234285, 38590884287, 158932839678, 653540335951, 2683657672278, 11006131763414, 45086288225520
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