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A351145
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Triangle T(n,m) read by rows: Sum_{k=1..m} binomial(2*n, n+k)*d(k), m<=n, with d(k)=A000005(k).
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1
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1, 4, 6, 15, 27, 29, 56, 112, 128, 131, 210, 450, 540, 570, 572, 792, 1782, 2222, 2420, 2444, 2448, 3003, 7007, 9009, 10101, 10283, 10339, 10341, 11440, 27456, 36192, 41652, 42772, 43252, 43284, 43288, 43758, 107406, 144534, 170238, 176358, 179622, 179928, 180000, 180003
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OFFSET
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1,2
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COMMENTS
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Exercise 52 in chapter 5.2.2 of Knuth's TAOCP 3 asks: "What is the asymptotic behavior of the sum S_n = Sum_{t>=1} binomial(2n,n+t)*d(t)?" and mentions "This question arises in connection with the analysis of a tree traversal algorithm, exercise 2.3.1-11."
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REFERENCES
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D. E. Knuth, The Art of Computer Programming Second Edition. Vol. 3, Sorting and Searching. Chapter 5.2.2 Sorting by Exchanging, pages 138, 637 (answer to exercise 52). Addison-Wesley, Reading, MA, 1998.
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LINKS
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EXAMPLE
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The triangle begins:
1;
4, 6;
15, 27, 29;
56, 112, 128, 131;
210, 450, 540, 570, 572;
792, 1782, 2222, 2420, 2444, 2448;
3003, 7007, 9009, 10101, 10283, 10339, 10341;
11440, 27456, 36192, 41652, 42772, 43252, 43284, 43288;
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MATHEMATICA
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T[n_, m_] := Sum[Binomial[2*n, n + k] * DivisorSigma[0, k], {k, 1, m}]; Table[T[n, m], {n, 1, 9}, {m, 1, n}] // Flatten (* Amiram Eldar, Feb 02 2022 *)
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PROG
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(PARI) for(n=1, 10, for(m=1, n, my(s=sum(t=1, m, binomial(2*n, n+t)*numdiv(t))); print1(s, ", ")))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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