A326096 -id:A326096

Search: a326096 -id:a326096

Displaying 1-4 of 4 results found.

page 1

A326092

E.g.f.: Sum_{n>=0} ((1+x)^n + 2)^n * x^n/n!.

+10
5

1, 3, 11, 63, 525, 5883, 84519, 1494783, 31854489, 800205075, 23315862339, 777867156927, 29384670476709, 1245177345486987, 58718905551858015, 3060140159517853887, 175176443950054714161, 10955959246057628397987, 745058168844977314910331, 54857350105041217492956735, 4356213264604432880789346621

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

More generally, the following sums are equal:

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

(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

here, q = (1+x) and p = 2, r = x.

In general, let F(x) be a formal power series in x such that F(0)=1, then

Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! =

Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);

here, F(x) = exp(x), q = 1+x, p = 2, r = x, m = 1.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

E.g.f.: Sum_{n>=0} ((1+x)^n + 2)^n * x^n/n!,

E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(2*x*(1+x)^n) * x^n/n!.

a(n) = 0 (mod 3) for n > 2.

EXAMPLE

E.g.f.: A(x) = 1 + 3*x + 11*x^2/2! + 63*x^3/3! + 525*x^4/4! + 5883*x^5/5! + 84519*x^6/6! + 1494783*x^7/7! + 31854489*x^8/8! + 800205075*x^9/9! + 23315862339*x^10/10! + ...

such that

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

also

A(x) = 1 + (1+x)*exp(2*x*(1+x))*x + (1+x)^4*exp(2*x*(1+x)^2)*x^2/2! + (1+x)^9*exp(2*x*(1+x)^3)*x^3/3! + (1+x)^16*exp(2*x*(1+x)^4)*x^4/4! + (1+x)^25*exp(2*x*(1+x)^5)*x^5/5! + (1+x)^36*exp(2*x*(1+x)^6)*x^6/6! + ...

PROG

(PARI) /* E.g.f.: Sum_{n>=0} ((1+x)^n + 2)^n * x^n/n! */

{a(n) = my(A = sum(m=0, n, ((1+x)^m + 2 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

(PARI) /* E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(2*x*(1+x)^n) * x^n/n! */

{a(n) = my(A = sum(m=0, n, (1+x +x*O(x^n))^(m^2) * exp(2*x*(1+x)^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A326096, A326093, A326094.

Cf. A326272.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 21 2019

STATUS

approved

A326093

E.g.f.: Sum_{n>=0} ((1+x)^n + 3)^n * x^n/n!.

+10
5

1, 4, 18, 112, 976, 11424, 169936, 3101032, 67876608, 1746757504, 52034505376, 1771434644544, 68180144988928, 2939951026982272, 140920461751138176, 7457658363325181824, 433145750643704774656, 27464893679743640343552, 1892311278990953945563648, 141074242336048184406390784, 11336870115013701213795557376

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

More generally, the following sums are equal:

(1) Sum_{n>=0} (q^n + p)^n * x^n/n!,

(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*x) * x^n/n!;

here, q = (1+x) and p = 3.

In general, let F(x) be a formal power series in x such that F(0)=1, then

Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! =

Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);

here, F(x) = exp(x), q = 1+x, p = 3, r = x, m = 1.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

E.g.f.: Sum_{n>=0} ((1+x)^n + 3)^n * x^n/n!,

E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(3*x*(1+x)^n) * x^n/n!.

a(n) = 0 (mod 4) for n > 2.

EXAMPLE

E.g.f.: A(x) = 1 + 4*x + 18*x^2/2! + 112*x^3/3! + 976*x^4/4! + 11424*x^5/5! + 169936*x^6/6! + 3101032*x^7/7! + 67876608*x^8/8! + 1746757504*x^9/9! + 52034505376*x^10/10! + ...

such that

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

also

A(x) = 1 + (1+x)*exp(3*x*(1+x))*x + (1+x)^4*exp(3*x*(1+x)^2)*x^2/2! + (1+x)^9*exp(3*x*(1+x)^3)*x^3/3! + (1+x)^16*exp(3*x*(1+x)^4)*x^4/4! + (1+x)^25*exp(3*x*(1+x)^5)*x^5/5! + (1+x)^36*exp(3*x*(1+x)^6)*x^6/6! + ...

PROG

(PARI) /* E.g.f.: Sum_{n>=0} ((1+x)^n + 3)^n * x^n/n! */

{a(n) = my(A = sum(m=0, n, ((1+x)^m + 3 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

(PARI) /* E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(3*x*(1+x)^n) * x^n/n! */

{a(n) = my(A = sum(m=0, n, (1+x +x*O(x^n))^(m^2) * exp(3*x*(1+x)^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A326096, A326092, A326094.

Cf. A326273.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 21 2019

STATUS

approved

A326094

E.g.f.: Sum_{n>=0} ((1+x)^n + 4)^n * x^n/n!.

+10
5

1, 5, 27, 185, 1693, 20565, 316375, 5948465, 133579065, 3517749125, 107024710675, 3714813650025, 145570443534805, 6383184292589525, 310815510350462415, 16694390352153656225, 983323269272332915825, 63186890982241624232325, 4409134435821084657726475, 332714992062735780407411225

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

More generally, the following sums are equal:

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

(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*x) * r^n/n!;

here, q = (1+x) and p = 4, r = x.

In general, let F(x) be a formal power series in x such that F(0)=1, then

Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! =

Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);

here, F(x) = exp(x), q = 1+x, p = 4, r = x, m = 1.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

E.g.f.: Sum_{n>=0} ((1+x)^n + 4)^n * x^n/n!,

E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(4*x*(1+x)^n) * x^n/n!.

a(n) = 0 (mod 5) for n > 4.

EXAMPLE

E.g.f.: A(x) = 1 + 5*x + 27*x^2/2! + 185*x^3/3! + 1693*x^4/4! + 20565*x^5/5! + 316375*x^6/6! + 5948465*x^7/7! + 133579065*x^8/8! + 3517749125*x^9/9! + 107024710675*x^10/10! + ...

such that

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

also

A(x) = 1 + (1+x)*exp(4*x*(1+x))*x + (1+x)^4*exp(4*x*(1+x)^2)*x^2/2! + (1+x)^9*exp(4*x*(1+x)^3)*x^3/3! + (1+x)^16*exp(4*x*(1+x)^4)*x^4/4! + (1+x)^25*exp(4*x*(1+x)^5)*x^5/5! + (1+x)^36*exp(4*x*(1+x)^6)*x^6/6! + ...

PROG

(PARI) /* E.g.f.: Sum_{n>=0} ((1+x)^n + 4)^n * x^n/n! */

{a(n) = my(A = sum(m=0, n, ((1+x)^m + 4 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

(PARI) /* E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(4*x*(1+x)^n) * x^n/n! */

{a(n) = my(A = sum(m=0, n, (1+x +x*O(x^n))^(m^2) * exp(4*x*(1+x)^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A326096, A326092, A326093.

Cf. A326274.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 21 2019

STATUS

approved

A259208

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

+10
4

1, 1, 3, 1, 25, 1, 721, 1, 6721, 181441, 151201, 1, 203575681, 1, 121080961, 108972864001, 3491282995201, 1, 133541574566401, 1, 304119455447808001, 212878925715456001, 309744468633601, 1, 17665560315112008499201, 646300418472124416000001, 841941782922240001

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

More generally, the following sums are equal:

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

(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

here, q = x, p = 1, r = x. - Paul D. Hanna, Jul 03 2019

In general, let F(x) be a formal power series in x such that F(0)=1, then

Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! =

Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);

here, F(x) = exp(x), q = , p = 1, r = x, m = 1. - Paul D. Hanna, Jul 06 2019

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..1025

FORMULA

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

a(n) = Sum_{d|n} binomial(d, n/d - 1) * n!/d!.

EXAMPLE

E.g.f.: A(x) = 1 + x + 3*x^2/2! + x^3/3! + 25*x^4/4! + x^5/5! + 721*x^6/6! + x^7/7! + 6721*x^8/8! + 181441*x^9/9! +...

where

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

also

A(x) = exp(x) + x^2*exp(x^2) + x^6*exp(x^3)/2! + x^12*exp(x^4)/3! + x^20*exp(x^5)/4! + x^30*exp(x^6)/5! + x^42*exp(x^7)/6! +...

RELATED SERIES.

Below we illustrate the following identity at specific values of x:

Sum_{n>=0} x^n * (1 + x^n)^n / n! = Sum_{n>=0} x^(n*(n+1)) * exp(x^(n+1)) / n!.

(1) At x = 1/2, the following sums are equal

S1 = Sum_{n>=0} 2^(-n*(n+1)) * (2^n + 1)^n / n!,

S1 = Sum_{n>=0} 2^(-n*(n+1)) * exp( 1/2^(n+1) ) / n!,

where S1 = 1.97862370255774939923047215233920359461142155926360261512472...

(2) At x = 1/3, the following sums are equal

S2 = Sum_{n>=0} 3^(-n*(n+1)) * (3^n + 1)^n / n!,

S2 = Sum_{n>=0} 3^(-n*(n+1)) * exp( 1/3^(n+1) ) / n!,

where S2 = 1.52049327799122758174016893855657751957768595647287750170026...

PROG

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

for(n=0, 30, print1(a(n), ", "))

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

for(n=0, 30, print1(a(n), ", "))

(PARI) {a(n) = if(n==0, 1, sumdiv(n, d, binomial(d, n/d-1) * n!/d! ) )}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A326096.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 21 2015

STATUS

approved

page 1

Search completed in 0.005 seconds