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%I #44 Jan 31 2024 08:05:44
%S 1,1,3,9,33,141,651,3333,18369,108153,678771,4495041,31324833,
%T 228803589,1744475643,13852095741,114235118721,976176336753,
%U 8627940414819,78726234866553,740440277799201,7168107030092541,71331617341611243,728811735008913909,7637128289949856833,81995144342947130601
%N Row sums of triangle A049404.
%H Seiichi Manyama, <a href="/A049425/b049425.txt">Table of n, a(n) for n = 0..616</a> (terms 0..200 from Vincenzo Librandi)
%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Munarini/muna4.html">Shifting Property for Riordan, Sheffer and Connection Constants Matrices</a>, J. Integer Seqs., Vol. 20 (2017), #17.8.2.
%F E.g.f.: exp(x+x^2+(x^3)/3).
%F a(n) = n! * sum(k=0..n, sum(j=0..k, binomial(3*j,n) * (-1)^(k-j)/(3^k * (k-j)!*j!))). [_Vladimir Kruchinin_, Feb 07 2011]
%F Conjecture: -a(n) +a(n-1) +(2*n-2)*a(n-2) + (2-3*n+n^2)*a(n-3)=0. - _R. J. Mathar_, Nov 14 2011
%F a(n) ~ exp(n^(2/3)+n^(1/3)/3-2*n/3-2/9)*n^(2*n/3)/sqrt(3)*(1+59/(162*n^(1/3))). - _Vaclav Kotesovec_, Oct 08 2012
%F From _Emanuele Munarini_, Oct 20 2014: (Start)
%F Recurrence: a(n+3) = a(n+2)+2*(n+2)*a(n+1)+(n+2)*(n+1)*a(n).
%F It derives from the differential equation for the e.g.f.: A'(x) = (1+2*x+x^2)*A(x).
%F So, the above conjecture is true.
%F b(n) = a(n+1) = sum((n!/k!)*sum(bin(k,i)*bin(k-i+2,n-2*i-k)/3^i,i=0..k),k=0..n).
%F E.g.f. for b(n) = a(n+1): (1+t)^2*exp(t+t^2+t^3/3).
%F (End)
%F a(n) = Sum_{k=0..n} Stirling1(n,k) * A004212(k). - _Seiichi Manyama_, Jan 31 2024
%t Table[n!*SeriesCoefficient[E^(x+x^2+(x^3)/3),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 08 2012 *)
%o (PARI) x='x+O('x^66); Vec(serlaplace(exp(x+x^2+(x^3)/3))) \\ _Joerg Arndt_, May 04 2013
%o (Maxima) /* for b(n) = a(n+1) */
%o b(n) := sum((n!/k!)*sum(binomial(k,i)*binomial(k-i+2,n-2*i-k)/3^i,i,0,k),k,0,n);
%o makelist(b(n),n,0,24); /* _Emanuele Munarini_, Oct 20 2014 */
%Y Column k=2 of A293991.
%Y Cf. A004212.
%K easy,nonn
%O 0,3
%A _Wolfdieter Lang_
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