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A124373 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k*(k+1)/2*x).

%I A124373 #25 Feb 08 2022 02:16:23
%S A124373 1,1,2,6,25,135,909,7417,71698,806968,10427825,152915697,2519879761,
%T A124373 46276398129,940296067422,21007099850230,513172107841525,
%U A124373 13640345170943527,392780078386164389,12204609567437300313,407757149671568266678,14600807659376773500696
%N A124373 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k*(k+1)/2*x).
%C A124373 Starting (1, 2, 6, 25, ...) = row sums of triangle A080248. - _Gary W. Adamson_, Jul 11 2011
%H A124373 Alois P. Heinz, <a href="/A124373/b124373.txt">Table of n, a(n) for n = 0..358</a>
%F A124373 O.g.f.: A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-3x)) + x^3/((1-x)*(1-3x)*(1-6x)) + x^3/((1-x)*(1-3x)*(1-6x)*(1-10x)) + ...
%F A124373 G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-(k+1)*(k+2)*x/2)/(1-x/(x-1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 16 2013
%e A124373 Also generated by iterated binomial transforms in the following way:
%e A124373   [1,2,6,25,135,909,7417,71698,...]  = binomial([1,1,3,12,64,433,3567,...]);
%e A124373   [1,3,12,64,433,3567,34905,...]     = binomial^2([1,1,4,20,129,1045,...]);
%e A124373   [1,4,20,129,1045,10209,117069,...] = binomial^3([1,1,5,30,226,2121,...]);
%e A124373   [1,5,30,226,2121,23919,314605,...] = binomial^4([1,1,6,42,361,3835,...]);
%e A124373   [1,6,42,361,3835,48885,724569,...] = binomial^5([1,1,7,56,540,6385,...]);
%e A124373   [1,7,56,540,6385,90519,1490457,..] = binomial^6([1,1,8,72,769,9993,...]);
%e A124373 etc.
%p A124373 b:= proc(n, m) option remember; `if`(n=0, 1,
%p A124373       b(n-1, m)*m*(m+1)/2 +b(n-1, m+1))
%p A124373     end:
%p A124373 a:= n-> b(n, 0):
%p A124373 seq(a(n), n=0..23);  # _Alois P. Heinz_, Sep 10 2019
%t A124373 b[n_, m_] := b[n, m] = If[n == 0, 1, b[n-1, m] m(m+1)/2 + b[n-1, m+1]];
%t A124373 a[n_] := b[n, 0];
%t A124373 a /@ Range[0, 23] (* _Jean-François Alcover_, Nov 02 2020, after _Alois P. Heinz_ *)
%o A124373 (PARI) a(n)=polcoeff(sum(k=0,n,x^k/prod(j=0,k,1-j*(j+1)/2*x+x*O(x^n))),n)
%Y A124373 Cf. A000110, A000217, A080248.
%K A124373 nonn
%O A124373 0,3
%A A124373 _Paul D. Hanna_, Oct 28 2006