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A000800 Sum of upward diagonals of Eulerian triangle. 6
1, 1, 1, 2, 5, 13, 38, 125, 449, 1742, 7269, 32433, 153850, 772397, 4088773, 22746858, 132601933, 807880821, 5132235182, 33925263901, 232905588441, 1657807491222, 12215424018837, 93042845392105, 731622663432978, 5931915237693517, 49535826242154973 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

REFERENCES

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

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

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..600 (first 201 terms from Vincenzo Librandi)

FORMULA

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

a(n) = Sum_{k} A173018(n-k, k). - Michael Somos, Mar 17 2011

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

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

a(n) = Sum_{m=0..n} (-1)^(n-m)*m!*Sum_{k=0..(n-m)/2} C(n-m-k,k)*stirling2(n-k,m). - Vladimir Kruchinin, Jan 23 2018

EXAMPLE

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

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 + ...

MAPLE

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

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

    end:

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

seq(a(n), n=0..30);  # Alois P. Heinz, Jan 23 2018

MATHEMATICA

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-Fran├žois Alcover, Dec 14 2011, after Michael Somos *)

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 *)

PROG

(Maxima)

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 */

CROSSREFS

Cf. A173018.

Sequence in context: A148303 A148304 A149859 * A149860 A006823 A319378

Adjacent sequences:  A000797 A000798 A000799 * A000801 A000802 A000803

KEYWORD

nonn,easy,nice

AUTHOR

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

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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Last modified December 15 20:00 EST 2019. Contains 330000 sequences. (Running on oeis4.)