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A055356 Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.

%I A055356 #44 Apr 17 2025 19:11:44
%S A055356 1,1,0,1,1,0,1,4,2,0,1,11,18,6,0,1,26,98,96,24,0,1,57,424,874,600,120,
%T A055356 0,1,120,1614,6040,8244,4320,720,0,1,247,5682,35458,83500,83628,35280,
%U A055356 5040,0,1,502,19022,187288,701164,1169768,915984,322560,40320,0,1
%N A055356 Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.
%C A055356 In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.
%C A055356 Also related to the solution of the equation df/dt=f e^f (see the Maple code). - _F. Chapoton_, Jul 16 2004
%H A055356 Andrew Howroyd, <a href="/A055356/b055356.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%H A055356 Shi-Mei Ma, <a href="http://arxiv.org/abs/1208.3104">Some combinatorial sequences associated with context-free grammars</a>, arXiv:1208.3104v2 [math.CO], 2012. - _N. J. A. Sloane_, Aug 21 2012
%H A055356 <a href="/index/Mo#mobiles">Index entries for sequences related to mobiles</a>
%F A055356 Let p(n,x) be the polynomial with coefficients equal to the n-th row of the triangle in ascending powers of x, e.g., p(4,x) = 1+4*x+2*x^2; then p(n+1,x) = (1+(n-1)*x)*p(n,x) + x*p'(n,x). - _Ben Whitmore_, May 12 2021
%F A055356 Recurrence: T(n,k) = (n-2) * T(n-1,k-1) + k * T(n-1,k) for n >= 1, 1 <= k <= n with T(1,1) = 1 and T(n,k) = 0 for n < 1, k < 1 or k > n. - _Georg Fischer_, Oct 27 2021
%F A055356 Conjecture: row polynomials are R(n-2,0) for n > 1 where R(n,k) = R(n-1,k+1) + x*Sum_{i=0..n-1} Sum_{j=0..k} binomial(n-1, i)*R(n-i-1,j)*R(i,k-j) for n > 0, k >= 0 with R(0,k) = 1 for k >= 0. - _Mikhail Kurkov_, Apr 11 2025
%e A055356 Triangle begins
%e A055356   1;
%e A055356   1,  0;
%e A055356   1,  1,   0;
%e A055356   1,  4,   2,   0;
%e A055356   1, 11,  18,   6,   0;
%e A055356   1, 26,  98,  96,  24,   0;
%e A055356   1, 57, 424, 874, 600, 120, 0;
%e A055356   ...
%p A055356 P[1]:=1;for n from 1 to 8 do P[n+1]:=simplify((1+n*x)*P[n]+x*diff(P[n],x)) end; # _F. Chapoton_, Jul 16 2004
%t A055356 P[1][_] = 1;
%t A055356 P[n_][x_] := P[n][x] = (1 + (n-1) x) P[n-1][x] + x P[n-1]'[x] // Expand;
%t A055356 row[1] = {1};
%t A055356 row[n_] := Append[CoefficientList[P[n-1][x], x], 0];
%t A055356 Array[row, 10] // Flatten (* _Jean-François Alcover_, Nov 17 2018, after _F. Chapoton_ *)
%o A055356 (PARI)
%o A055356 A(n)={my(v=vector(n)); v[1]=y; for(n=2, #v, v[n]=v[n-1] + sum(k=1, n-2, binomial(n-2, k)*v[k]*v[n-k])); vector(#v, i, Vecrev(v[i]/y, i))}
%o A055356 { my(T=A(10)); for(i=1, #T, print(T[i])) } \\ _Andrew Howroyd_, Sep 23 2018
%Y A055356 Row sums give A029768 (p(n,1)).
%Y A055356 Alternating row sums give A089963 (p(n+1,-1)).
%Y A055356 Columns 2..8 are A000295, A055357, A055358, A055359, A055360, A055361, A055362.
%Y A055356 Cf. A055340, A055349, A055363.
%K A055356 nonn,tabl
%O A055356 1,8
%A A055356 _Christian G. Bower_, May 15 2000