A000800 - OEIS

%I #46 Sep 17 2023 21:40:36

%S 1,1,1,2,5,13,38,125,449,1742,7269,32433,153850,772397,4088773,

%T 22746858,132601933,807880821,5132235182,33925263901,232905588441,

%U 1657807491222,12215424018837,93042845392105,731622663432978,5931915237693517,49535826242154973

%N Sum of upward diagonals of Eulerian triangle.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.

%H 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 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 a(n) = Sum_{k} A173018(n-k, k). - _Michael Somos_, Mar 17 2011

%F 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 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 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 1 = 1, 1 = 1, 1 = 1 + 0, 2 = 1 + 1, 5 = 1 + 4 + 0, etc.

%e 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 b:= proc(n, k) option remember; `if`(k=0 and n>=0, 1,

%p       `if`(k<0 or k>n, 0, (n-k)*b(n-1, k-1)+(k+1)*b(n-1, k)))

%p     end:

%p a:= n-> add(b(n-k, k), k=0..n):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Jan 23 2018

%t 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 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 (Maxima)

%o 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 Cf. A173018.

%K nonn,easy,nice

%O 0,4

%A Tony Harkin [ harkin(AT)mit.edu, tharkin(AT)vortex.weather.brockport.edu ]

%E More terms from _David W. Wilson_