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A000800 Sum of upward diagonals of Eulerian triangle.

%I A000800 #46 Sep 17 2023 21:40:36
%S A000800 1,1,1,2,5,13,38,125,449,1742,7269,32433,153850,772397,4088773,
%T A000800 22746858,132601933,807880821,5132235182,33925263901,232905588441,
%U A000800 1657807491222,12215424018837,93042845392105,731622663432978,5931915237693517,49535826242154973
%N A000800 Sum of upward diagonals of Eulerian triangle.
%D A000800 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
%D A000800 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.
%D A000800 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
%H A000800 Alois P. Heinz, <a href="/A000800/b000800.txt">Table of n, a(n) for n = 0..600</a> (first 201 terms from Vincenzo Librandi)
%F A000800 G.f.: 1/(1-x/(1-x^2/(1-2x/(1-2x^2/(1-3x/(1-3x^2/(1-... (continued fraction). - _Paul Barry_, Mar 24 2010
%F A000800 a(n) = Sum_{k} A173018(n-k, k). - _Michael Somos_, Mar 17 2011
%F A000800 G.f.: 1/Q(0), where Q(k) = 1 - x*(k+1)/(1 - x^2*(k+1)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Apr 14 2013
%F A000800 G.f.: 1/Q(0), where Q(k) = 1 - x - x*(x+1)*k - x^3*(k+1)^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Apr 14 2013
%F A000800 a(n) = Sum_{m=0..n} (-1)^(n-m)*m!*Sum_{k=0..floor((n-m)/2)} C(n-m-k,k)*Stirling2(n-k,m). - _Vladimir Kruchinin_, Jan 23 2018
%e A000800 1 = 1, 1 = 1, 1 = 1 + 0, 2 = 1 + 1, 5 = 1 + 4 + 0, etc.
%e A000800 G.f. = 1 + x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 38*x^6 + 125*x^7 + 449*x^8 + 1742*x^9 + ...
%p A000800 b:= proc(n, k) option remember; `if`(k=0 and n>=0, 1,
%p A000800       `if`(k<0 or k>n, 0, (n-k)*b(n-1, k-1)+(k+1)*b(n-1, k)))
%p A000800     end:
%p A000800 a:= n-> add(b(n-k, k), k=0..n):
%p A000800 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jan 23 2018
%t A000800 t[n_ /; n >= 0, 0] = 1; t[n_, k_] /; k < 0 || k > n = 0; t[n_, k_] := t[n, k] = (n-k)*t[n-1, k-1] + (k+1)*t[n-1, k]; a[n_] := Sum[t[n-k, k], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Dec 14 2011, after _Michael Somos_ *)
%t A000800 Table[Sum[Sum[(-1)^j*(k-j+1)^(n-k)*Binomial[n-k+1, j], {j, 0, k}], {k, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Aug 15 2015 *)
%o A000800 (Maxima)
%o A000800 a(n):=sum(m!*sum((binomial(n-m-k,k)*stirling2(n-k,m)*(-1)^(-n+m)),k,0,(n-m)/2),m,0,n); /* _Vladimir Kruchinin_, Jan 23 2018 */
%Y A000800 Cf. A173018.
%K A000800 nonn,easy,nice
%O A000800 0,4
%A A000800 Tony Harkin [ harkin(AT)mit.edu, tharkin(AT)vortex.weather.brockport.edu ]
%E A000800 More terms from _David W. Wilson_