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A249018
Decimal expansion of the Flajolet-Prodinger constant 'K', a constant related to asymptotically enumerating level number sequences for binary trees.
0
2, 5, 4, 5, 0, 5, 5, 2, 3, 5, 6, 5, 3, 1, 9, 5, 1, 3, 3, 7, 0, 8, 8, 1, 7, 7, 0, 0, 3, 1, 5, 4, 6, 1, 5, 6, 0, 4, 6, 4, 9, 3, 7, 4, 1, 7, 2, 5, 0, 6, 1, 9, 4, 4, 4, 9, 8, 4, 5, 5, 0, 0, 0, 6, 3, 8, 6, 3, 8, 9, 2, 3, 9, 0, 0, 8, 8, 3, 1, 6, 8, 6, 0, 0, 2, 5, 8, 1, 2, 2, 6, 3, 5, 5, 8, 6, 1, 8, 7, 7
OFFSET
0,1
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5 Kalmar's composition constant, p. 294.
LINKS
Philippe Flajolet and Helmut Prodinger, Level number sequences for trees.
FORMULA
H(n) ~ K*nu^n, where H(n) is number of level number sequences associated to binary trees (Cf. A002572) and 'nu' is the constant A102375.
EXAMPLE
0.254505523565319513370881770031546156046493741725...
MATHEMATICA
digits = 105; m0 = 5; dm = 2; Clear[f, g, v, K]; v[c_, d_] := v[c, d] = If[d<0 || c<0, 0, If[d == c, 1, Sum[v[i, d-c], {i, 1, 2*c}]]]; H[n_] := v[1, n]; H[1] = 1; f[x_, m_] := Sum[((-1)^(j+1)*x^(2^(j+1)-2-j))/Product[1-x^(2^k-1), {k, 1, j}], {j, 1, m}] // N[#, digits]&; g[m_] := g[m] = (1/x /. FindRoot[f[x, m] == 1, {x, 5/9, 4/9, 6/9}, WorkingPrecision -> digits]); g[m0]; g[m = m0+dm]; While[RealDigits[g[m], 10, digits+5] != RealDigits[g[m-dm], 10, digits+5], m = m+dm]; nu = g[m]; K[m_] := K[m] = H[m]/nu^m; dm=100; K[m = 100]; K[m = m+dm]; While[Print[m]; RealDigits[K[m], 10, digits+5] != RealDigits[K[m-dm], 10, digits+5], m = m+dm]; RealDigits[K[m], 10, digits-5] // First
CROSSREFS
Sequence in context: A231730 A095758 A299212 * A235052 A102066 A279404
KEYWORD
nonn,cons
AUTHOR
STATUS
approved