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A193196 G.f.: A(x) = Sum_{n>=0} x^n / Product_{k=1..n} (1 - k*x^k).

%I A193196 #25 Jun 18 2019 03:58:33
%S A193196 1,1,2,3,6,9,19,29,57,94,172,280,519,833,1472,2433,4185,6800,11666,
%T A193196 18816,31686,51340,84929,136561,225476,359746,586133,936243,1511650,
%U A193196 2397400,3856698,6084186,9711492,15299490,24247456,38016261,60079125,93752706,147284928
%N A193196 G.f.: A(x) = Sum_{n>=0} x^n / Product_{k=1..n} (1 - k*x^k).
%C A193196 Number of rooted ordered trees with n non-root nodes such that both successive branch heights and the lengths of the branches are weakly increasing; see example. - _Joerg Arndt_, Aug 27 2014
%H A193196 Robert Israel, <a href="/A193196/b193196.txt">Table of n, a(n) for n = 0..5627</a>
%H A193196 Joerg Arndt, <a href="/A193196/a193196.txt">trees as in comment for 1<=n<=7</a>
%F A193196 G.f.: G(0) - 1 where G(k) =  1 + (1-x)/(1-x^k*k)/(1-x/(x+(1-x)/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 22 2013
%F A193196 a(n) = sum( prod(j=2..m, min(C[j-1], C[j]))) where the sum is over all partitions C[1..m] (m parts) of n, see example. - _Joerg Arndt_, Sep 03 2014
%F A193196 From _Vaclav Kotesovec_, Jun 18 2019: (Start)
%F A193196 a(n) ~ c * 3^(n/3), where
%F A193196 c = 9390.8440644933535486959046639452060731482141... if mod(n,3)=0
%F A193196 c = 9390.7389359914729419715573277079935321683397... if mod(n,3)=1
%F A193196 c = 9390.7321933046037554603013237581369727858708... if mod(n,3)=2
%F A193196 (End)
%e A193196 G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 19*x^6 + 29*x^7 +...
%e A193196 where:
%e A193196 A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-2*x^2)) + x^3/((1-x)*(1-2*x^2)*(1-3*x^3)) + x^4/((1-x)*(1-2*x^2)*(1-3*x^3)*(1-4*x^4)) +...
%e A193196 From _Joerg Arndt_, Aug 27 2014: (Start)
%e A193196 The a(4) = 5 trees described in the comment are:
%e A193196 :
%e A193196 :     1:
%e A193196 :    [ 1 1 1 1 ]  <--= branch lengths
%e A193196 :    [ 0 0 0 0 ]  <--= branch heights
%e A193196 :
%e A193196 :  O--o
%e A193196 :  .--o
%e A193196 :  .--o
%e A193196 :  .--o
%e A193196 :
%e A193196 :
%e A193196 :     2:
%e A193196 :    [ 1 1 2 ]
%e A193196 :    [ 0 0 0 ]
%e A193196 :
%e A193196 :  O--o
%e A193196 :  .--o
%e A193196 :  .--o--o
%e A193196 :
%e A193196 :
%e A193196 :     3:
%e A193196 :    [ 1 3 ]
%e A193196 :    [ 0 0 ]
%e A193196 :
%e A193196 :  O--o
%e A193196 :  .--o--o--o
%e A193196 :
%e A193196 :
%e A193196 :     4:
%e A193196 :    [ 2 2 ]
%e A193196 :    [ 0 0 ]
%e A193196 :
%e A193196 :  O--o--o
%e A193196 :  .--o--o
%e A193196 :
%e A193196 :
%e A193196 :     5:
%e A193196 :    [ 2 2 ]
%e A193196 :    [ 0 1 ]
%e A193196 :
%e A193196 :  O--o--o
%e A193196 :     .--o--o
%e A193196 :
%e A193196 :
%e A193196 :     6:
%e A193196 :    [ 4 ]
%e A193196 :    [ 0 ]
%e A193196 :
%e A193196 :  O--o--o--o--o
%e A193196 :
%e A193196 See the Arndt link for all examples for 1 <= n <= 7.
%e A193196 (End)
%e A193196 a(6) = 19 because the 11 partitions of 6 with the products as in the comment are
%e A193196 01:  [ 1 1 1 1 1 1 ]   1*1*1*1*1 = 1
%e A193196 02:  [ 1 1 1 1 2 ]     1*1*1*1   = 1
%e A193196 03:  [ 1 1 1 3 ]       1*1*1     = 1
%e A193196 04:  [ 1 1 2 2 ]       1*1*2     = 2
%e A193196 05:  [ 1 1 4 ]         1*1       = 1
%e A193196 06:  [ 1 2 3 ]         1*2       = 1
%e A193196 07:  [ 1 5 ]           1         = 1
%e A193196 08:  [ 2 2 2 ]         2*2       = 4
%e A193196 09:  [ 2 4 ]           2         = 2
%e A193196 10:  [ 3 3 ]           3         = 3
%e A193196 11:  [ 6 ]         (empty prod.) = 1
%e A193196 and the sum of the products is 19. - _Joerg Arndt_, Sep 03 2014
%p A193196 N:= 100: # to get all terms up to a(N)
%p A193196 gN:= add(x^n/mul(1-k*x^k,k=1..n),n=0..N):
%p A193196 S:= series(gN,x,N+1):
%p A193196 seq(coeff(S,x,n), n=0..N); # _Robert Israel_, Aug 28 2014
%o A193196 (PARI) {a(n)=my(A=1);polcoeff(sum(m=0,n,x^m/prod(k=1,m,1-k*x^k +x*O(x^n))),n)}
%K A193196 nonn
%O A193196 0,3
%A A193196 _Paul D. Hanna_, Jul 17 2011