login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A323680 G.f.: Sum_{n>=0} x^n * ((1+x)^n + 1)^n / (1 + x*(1+x)^n)^(n+1). 8
1, 1, 2, 3, 10, 27, 109, 427, 1958, 9467, 49459, 274712, 1614199, 9996580, 64940226, 441179351, 3125044744, 23021059143, 175976694409, 1393077001768, 11400165893604, 96286628620151, 838123560744653, 7508677200329118, 69152466448641019, 653972815019717914, 6344196087718370108, 63073829812214409363, 642093553544993640915, 6687618467901426663337, 71209887695115322487153, 774636418450000537370791 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

More generally, the following sums are equal:

(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (q^n + p)^n / (1 + p*q^n*r)^(n+k),

(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * (q^n - p)^n / (1 - p*q^n*r)^(n+k),

for any fixed integer k; here, k = 1 and r = x, p = 1, q = (1+x). See other examples for k = 2 (A326006), k = 3 (A326007), k = 4 (A326008).

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..500

FORMULA

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

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

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

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

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

FORMULAS INVOLVING TERMS.

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

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

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 10*x^4 + 27*x^5 + 109*x^6 + 427*x^7 + 1958*x^8 + 9467*x^9 + 49459*x^10 + 274712*x^11 + 1614199*x^12 + ...

such that

A(x) = 1/(1+x) + x*((1+x) + 1)/(1 + x*(1+x))^2 + x^2*((1+x)^2 + 1)^2/(1 + x*(1+x)^2)^3 + x^3*((1+x)^3 + 1)^3/(1 + x*(1+x)^3)^4 + x^4*((1+x)^4 + 1)^4/(1 + x*(1+x)^4)^5 + x^5*((1+x)^5 + 1)^5/(1 + x*(1+x)^5)^6 + x^6*((1+x)^6 + 1)^6/(1 + x*(1+x)^6)^7 + x^7*((1+x)^7 + 1)^7/(1 + x*(1+x)^7)^8 + ...

also,

A(x) = 1/(1-x) + x*((1+x) - 1)/(1 - x*(1+x))^2 + x^2*((1+x)^2 - 1)^2/(1 - x*(1+x)^2)^3 + x^3*((1+x)^3 - 1)^3/(1 - x*(1+x)^3)^4 + x^4*((1+x)^4 - 1)^4/(1 - x*(1+x)^4)^5 + x^5*((1+x)^5 - 1)^5/(1 - x*(1+x)^5)^6 + x^6*((1+x)^6 - 1)^6/(1 - x*(1+x)^6)^7 + x^7*((1+x)^7 - 1)^7/(1 - x*(1+x)^7)^8 + ...

RELATED INFINITE SERIES.

At x = -1/2, the g.f. as a power series in x diverges, but the related series converges:

S = Sum_{n>=0} (-1/2)^n*(1/2^n + 1)^n / (1 - 1/2^(n+1))^(n+1), and

S = Sum_{n>=0} (-1/2)^n*(1/2^n - 1)^n / (1 + 1/2^(n+1))^(n+1).

Equivalently,

S = Sum_{n>=0} (-2)^n * (2^n + 1)^n / (2^(n+1) - 1)^(n+1), and

S = Sum_{n>=0} 2^n * (2^n - 1)^n / (2^(n+1) + 1)^(n+1) ;

written explicitly,

S = 1/(2-1) - 2*(2+1)/(2^2-1)^2 + 2^2*(2^2+1)^2/(2^3-1)^3 - 2^3*(2^3+1)^3/(2^4-1)^4 + 2^4*(2^4+1)^4/(2^5-1)^5 - 2^5*(2^5+1)^5/(2^6-1)^6 + 2^6*(2^6+1)^6/(2^7-1)^7 + ...

also,

S = 1/(2+1) + 2*(2-1)/(2^2+1)^2 + 2^2*(2^2-1)^2/(2^3+1)^3 + 2^3*(2^3-1)^3/(2^4+1)^4 + 2^4*(2^4-1)^4/(2^5+1)^5 + 2^5*(2^5-1)^5/(2^6+1)^6 + 2^6*(2^6-1)^6/(2^7+1)^7 + ...

where

S = 0.54250659711853510199583159448775795614278675261848614946772936514239222...

PROG

(PARI) {a(n) = my(A = sum(m=0, n+1, x^m*((1+x +x*O(x^n) )^m - 1)^m/(1 - x*(1+x +x*O(x^n) )^m )^(m+1) )); polcoeff(A, n)}

for(n=0, 35, print1(a(n), ", "))

(PARI) {a(n) = sum(i=0, n, sum(j=0, n-i, sum(k=0, n-i-j, (-1)^k * binomial(n-i, j) * binomial(n-i-j, k) * binomial((n-i-j)*(n-i-k), i) )))}

for(n=0, 35, print1(a(n), ", "))

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

for(n=0, 35, print1(a(n), ", "))

CROSSREFS

Cf. A323681, A323695, A325059, A326006, A326007, A326008.

Sequence in context: A000060 A089752 A264759 * A171190 A216332 A007029

Adjacent sequences:  A323677 A323678 A323679 * A323681 A323682 A323683

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 11 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 16:24 EST 2019. Contains 329808 sequences. (Running on oeis4.)