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A055363 Triangle of asymmetric mobiles (circular rooted trees) with n nodes and k leaves. 12
1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 6, 10, 5, 1, 0, 0, 1, 9, 22, 19, 7, 1, 0, 0, 1, 12, 42, 53, 31, 8, 1, 0, 0, 1, 16, 73, 130, 109, 45, 10, 1, 0, 0, 1, 20, 119, 280, 321, 190, 63, 11, 1, 0, 0, 1, 25, 184, 556, 833, 672, 310, 83, 13, 1, 0, 0, 1, 30, 272 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,12
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
FORMULA
G.f. satisfies A(x, y) = x*(y - Sum_{i>0} moebius(i)/i * log(1 - A(x^i, y^i))). - Michael Somos, Aug 19 2015
Sum_k T(n, k) = A032171(n). - Michael Somos, Aug 24 2015
EXAMPLE
G.f. = x*(y + x*y + x^2*y + x^3*(y + y^2) + x^4*(y + 2*y^2 + y^3) + x^5*(y + 4*y^2 + 4*y^3 + y^4) + ...).
n\k 1 2 3 4 5 6 7 8
--:-- -- -- -- -- -- -- --
1: 1
2: 1 0
3: 1 0 0
4: 1 1 0 0
5: 1 2 1 0 0
6: 1 4 4 1 0 0
7: 1 6 10 5 1 0 0
8: 1 9 22 19 7 1 0 0
MATHEMATICA
T[n_, k_] := Module[{A}, A[_, _] = 0; If[k<1 || k>n, 0, For[j=1, j <= n, j++, A[x_, y_] = x*y-x*Sum[MoebiusMu[i]/i * Log[1 - A[x^i, y^i]] + O[x]^j // Normal, {i, 1, j}]]; Coefficient[Coefficient[A[x, y], x, n], y, k]]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 30 2017, after Michael Somos *)
PROG
(PARI) {T(n, k) = my(A = O(x)); if(k<1 || k>n, 0, for(j=1, n, A = x*y - x*sum(i=1, j, moebius(i)/i * log(1 - subst( subst( A + x * O(x^min(j, n\i)), x, x^i), y, y^i) ) )); polcoeff( polcoeff(A, n), k))}; /* Michael Somos, Aug 24 2015 */
CROSSREFS
Row sums give A032171.
Sequence in context: A188816 A168312 A076837 * A363845 A350681 A110855
KEYWORD
nonn,tabl
AUTHOR
Christian G. Bower, May 15 2000
STATUS
approved

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Last modified June 15 03:12 EDT 2024. Contains 373402 sequences. (Running on oeis4.)