A324954 E.g.f.: Sum_{n>=0} x^n * (exp(n*x) + 1)^n / (1 + x*exp(n*x))^(n+1).

1, 1, 4, 21, 268, 4185, 100146, 2944921, 110303880, 4976190225, 267276082150, 16769865552561, 1212712561009500, 99940859707310089, 9294528076388524074, 967395346945752464265, 111872727929427271336336, 14283181659779238236208801, 2002001211803983519973104974, 306524554411560349543442894017, 51035879163206907059508782146980

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

E.g.f.: Sum_{n>=0} x^n * (exp(n*x) + 1)^n / (1 + x*exp(n*x))^(n+1).

E.g.f.: Sum_{n>=0} x^n * (exp(n*x) - 1)^n / (1 - x*exp(n*x))^(n+1).

E.g.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (exp(n*x) - exp(k*x))^(n-k).

E.g.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (-1)^k * (exp(n*x) + exp(k*x))^(n-k).

E.g.f.: Sum_{n>=0} x^n*Sum_{k=0..n} binomial(n,k) * Sum_{j=0..n-k} binomial(n-k,j) * (-1)^j * exp((n-j)*(n-k)*x).

FORMULAS INVOLVING TERMS.

a(n) = Sum_{i=0..n} n!/i! * Sum_{j=0..n-i} binomial(n-i,j) * Sum_{k=0..n-i-j} (-1)^k * binomial(n-i-j,k) * (n-i-j)^i * (n-i-k)^i.

a(n) = Sum_{i=0..n} Sum_{j=0..n-i} Sum_{k=0..n-i-j} (-1)^k * n!*(n-i)! / ((n-i-j-k)! * i!*j!*k!) * (n-i-j)^i * (n-i-k)^i.

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 21*x^3/3! + 268*x^4/4! + 4185*x^5/5! + 100146*x^6/6! + 2944921*x^7/7! + 110303880*x^8/8! + 4976190225*x^9/9! + 267276082150*x^10/10! + ...
such that
A(x) = 1/(1+x) + x*(exp(x) + 1)/(1 + x*exp(x))^2 + x^2*(exp(2*x) + 1)^2/(1 + x*exp(2*x))^3 + x^3*(exp(3*x) + 1)^3/(1 + x*exp(3*x))^4 + x^4*(exp(4*x) + 1)^4/(1 + x*exp(4*x))^5 + x^5*(exp(5*x) + 1)^5/(1 + x*exp(5*x))^6 + x^6*(exp(6*x) + 1)^6/(1 + x*exp(6*x))^7 + x^7*(exp(x)^7 + 1)^7/(1 + x*exp(x)^7)^8 + ...
also,
A(x) = 1/(1-x) + x*(exp(x) - 1)/(1 - x*exp(x))^2 + x^2*(exp(2*x) - 1)^2/(1 - x*exp(2*x))^3 + x^3*(exp(3*x) - 1)^3/(1 - x*exp(3*x))^4 + x^4*(exp(4*x) - 1)^4/(1 - x*exp(4*x))^5 + x^5*(exp(5*x) - 1)^5/(1 - x*exp(5*x))^6 + x^6*(exp(6*x) - 1)^6/(1 - x*exp(6*x))^7 + x^7*(exp(x)^7 - 1)^7/(1 - x*exp(x)^7)^8 + ...

Prog

(PARI) {a(n) = my(A = sum(m=0, n+1, x^m*(exp(m*x +x*O(x^n) ) + 1)^m/(1 + x*exp(m*x +x*O(x^n) ) )^(m+1) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n) = my(A = sum(m=0, n+1, x^m*(exp(m*x +x*O(x^n) ) - 1)^m/(1 - x*exp(m*x +x*O(x^n) ) )^(m+1) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n) = sum(i=0, n, n!/i! * sum(j=0, n-i, binomial(n-i, j) * sum(k=0, n-i-j, (-1)^k * binomial(n-i-j, k) * (n-i-j)^i * (n-i-k)^i )))}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n) = sum(i=0, n, sum(j=0, n-i, sum(k=0, n-i-j, (-1)^j*n!*(n-i)!/((n-i-j-k)!*i!*j!*k!) * (n-i-j)^i * (n-i-k)^i )))}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A324955, A325995, A325996.

Sequence in context: A184961 A006822 A165627 * A198050 A270482 A126458

Adjacent sequences: A324951 A324952 A324953 * A324955 A324956 A324957

Keyword

nonn

Author

Paul D. Hanna, Mar 24 2019

Status

approved