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Search: id:A108643
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| A108643 |
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Number of binary rooted trees with n nodes and internal path length n. |
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+0
3
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| 1, 1, 0, 2, 0, 1, 4, 0, 4, 2, 8, 6, 8, 8, 8, 40, 4, 29, 40, 52, 56, 64, 116, 112, 200, 86, 296, 366, 360, 432, 652, 800, 840, 1470, 1116, 2048, 2356, 3052, 3524, 4220, 5648, 6964, 9660, 8688, 14128, 17024, 19432, 23972, 32784, 37873, 44912, 59672, 67560, 93684
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Self-convolution equals A095830 (number of binary trees of path length n). - Paul D. Hanna (pauldhanna(AT)juno.com), Aug 20 2007
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REFERENCES
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Knuth Vol. 1 Sec. 2.3.4.5, Problem 5.
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FORMULA
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G.f. = B(w, w) where B(w, z) is defined in A095830.
G.f.: A(x) = 1 + x*(A_2)^2; A_2 = 1 + x^2*(A_3)^2; A_3 = 1 + x^3*(A_4)^2; ... A_n = 1 + x^n*(A_{n+1})^2 for n>=1 with A_1 = A(x). - Paul D. Hanna (pauldhanna(AT)juno.com), Aug 20 2007
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MAPLE
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A := proc(n, k) option remember; if n=0 then 1 else convert (series (1+ x^k*A(n-1, k+1)^2, x, n+1), polynom) fi end: a:= n-> coeff (A(n, 1), x, n): seq (a(n), n=0..53); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 22 2008]
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PROGRAM
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(PARI) {a(n)=local(A=1+x*O(x^n)); for(j=0, n-1, A=1+x^(n-j)*A^2); polcoeff(A, n)} - Paul D. Hanna (pauldhanna(AT)juno.com), Aug 20 2007
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CROSSREFS
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Adjacent sequences: A108640 A108641 A108642 this_sequence A108644 A108645 A108646
Sequence in context: A131487 A039991 A081265 this_sequence A133838 A121583 A124915
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KEYWORD
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nonn
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AUTHOR
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njas and Nadia Heninger (nadiah(AT)cs.princeton.edu), Jul 08 2005
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 08 2005
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