cp's OEIS Frontend

0! - 1! + 2! - 3! + 4! - 5! + ... = (Borel) Sum_{n>=0} (-y)^n n! = KummerU(1,1,1/y)/y.

Decimal expansion of phi(1) where phi(x) = Integral_{t>=0} e^-t/(x+t) dt. - Benoit Cloitre, Apr 11 2003

The divergent series g(x=1,m) = 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ..., m => -1, is intimately related to Gompertz's constant. We discovered that g(x=1,m) = (-1)^m * (A040027(m) - A000110(m+1) * A073003) with A000110 the Bell numbers and A040027 a sequence that was published by Gould, see for more information A163940. - Johannes W. Meijer, Oct 16 2009

Named by Le Lionnais (1983) after the English self-educated mathematician and actuary Benjamin Gompertz (1779 - 1865). It was named the Euler-Gompertz constant by Finch (2003). Lagarias (2013) noted that he has not located this constant in Gompertz's writings. - Amiram Eldar, Aug 15 2020

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. The number of unbounded relaxed compacted binary trees of size n is A082161(n). The number of relaxed compacted binary trees of right height at most one of size n is A001147(n). See the Genitrini et al. and Wallner link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where node 3 is at depth 1 on the right of node 2 and where the node n+1 has a left sibling. See the Wallner link. - Michael Wallner, Apr 20 2017

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. A branch node is a node with a left and right edge (no pointer). See the Genitrini et al. link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where in the growth process induced by the labels a maximal young leaf has to be followed by a non-maximal young leaf. A young leaf is a leaf with no left sibling. A maximal young leaf is a young leaf with maximal label. See the Wallner link. - Michael Wallner, Apr 20 2017

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. A branch node is a node with a left and right edge (no pointer). See the Genitrini et al. link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where in the growth process induced by the labels maximal young leaves and non-maximal young leaves alternate except for a sequence of maximal young leaves at the beginning. A young leaf is a leaf with no left sibling. A maximal young leaf is a young leaf with maximal label. See the Wallner link. - Michael Wallner, Apr 20 2017

See A199577 for general comments on associated Laguerre-Sonin(e) polynomials of order m, and the Ismail reference.

The monic row polynomials are La_n(1;1,x) = sum(a(n,k)*x^k,k=0..n), with the three term recurrence

La_n(1;1,x) = (x-2*(n+1))*La_{n-1}{1;1,x) - n*(n+1)*La_{n-2}{1;1,x), La_{-1}{1;1,x)=0, La_0(1;1,x)=1.

In the Ismail reference the non-monic associated Laguerre polynomials of order 1 appear on p. 160 in Theorem 5.6.1, eq. 5.6.11. The connection is: La_n(1;1,x)= L_n^{(alpha=1)}(x;1)*(n+1)!*(-1)^n.

The e.g.f. gLa(1;1,z,x) for La_n(1;1,x) can be obtained from the o.g.f. G(1;1,z,x) for the non-monic version L_n^{alpha=1}(x;1) by gLa(1;1,z,x)= (d/dz) (z*G(1;1,-z,x)).

G(1;1,z,x) satisfies an ordinary first order inhomogeneous differential equation:

(d/dz) G(1;1,z,x) = (3/(1-z)+(1-x)/(1-z)^2-1/(z*(1-z)^2 z)^2))* G(1;1,z,x) + 1/(z*(1-z)^2), with G(1;1,z=0,x)=1. The standard solution is:

G(1;1,z,x) = (exp(-x*z/(1-z))-1+z-x*exp(-x/(1-z))* (Ei(1,-x/(1-z))-Ei(1,-x)))/(z*(1-z)^2), with the exponential integral Ei(1,y)=int(exp(-t)/t,t=y..infty).

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. A branch node is a node with a left and right edge (no pointer). See the Genitrini et al. link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where in the growth process induced by the labels a maximal young leaves and non-maximal young leaves alternate except for a sequence of maximal young leaves at the begininning and at the end. A young leaf is a leaf with no left sibling. A maximal young leaf is a young leaf with maximal label. See the Wallner link. - Michael Wallner, Apr 20 2017

a(n) satisfies the recurrence a(n) = (2n-1)*a(n-1) - (n-1)*(n-2)*a(n-2) for n > 2, with initial conditions a(1)=1, a(2)=2.

The same recurrence is satisfied by A000262(n), but with different initial conditions.

The limit of a(n)/A000262(n) as n tends to infinity is the Euler-Gompertz constant G = e*E1(1), where E1 is an exponential integral. The decimal representation of G is given by A073003.

The convergents of the c.f. G = 1-1/(3-1*2/(5-2*3/(7-3*4/(9-...)))) are (a(n)/A000262(n)) = (1, 2/3, 8/13, 44/73, ...). The c.f. is equivalent to Bala's c.f. for 1-G given in the entry for A073003.

a(n)/A000262(n) - G ~ 2*Pi*exp(1-4*sqrt(n)) as n tends to infinity.

a(n)/n! ~ G*exp(2*sqrt(n))/(2*n^(3/4)*sqrt(Pi*e)) as n tends to infinity.

a(n) = A000262(n) - |A201203(n-2)| for n >= 2.

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and at most n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. It is called simple if for nodes with two pointers both point to the same node. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. See the Wallner link.

a(n) is one of two "basis" sequences for sequences of the form a(0)=a, a(1)=b, a(n) = n*a(n-1) + (n-1)*a(n-2), the second basis sequence being A096654 (with 0 appended as a(0)). The sum of these sequences is listed as A000255. - Gary Detlefs, Dec 11 2018