A259209
E.g.f.: Sum_{n>=1} x^(n^2) * exp(x^n) / n!.
Original entry on oeis.org
1, 2, 3, 16, 5, 366, 7, 10088, 60489, 302410, 11, 89812812, 13, 363242894, 108972864015, 886312627216, 17, 178478870169618, 19, 101401086923136020, 354798209525760021, 1548722343168022, 23, 13787827750211997081624, 129260083694424883200025, 5051650697533440026
Offset: 1
E.g.f.: A(x) = x + 2*x^2/2! + 3*x^3/3! + 16*x^4/4! + 5*x^5/5! + 366*x^6/6! +...
where
A(x) = x*exp(x) + x^4*exp(x^2)/2! + x^9*exp(x^3)/3! + x^16*exp(x^4)/4! + x^25*exp(x^5)/5! + x^36*exp(x^6)/6! +...
also
A(x) = -exp(1) + 1 + (1+x) + (1+x^2)^2/2! + (1+x^3)^3/3! + (1+x^4)^4/4! + (1+x^5)^5/5! + (1+x^6)^6/6! +...
Particular values.
A(1) = exp(2) - exp(1).
A(-1) = cosh(2) - exp(1).
A(1/2) = 0.8648559700938957468696599588156983897723576531...
A(1/2) = exp(1/2)/2 + exp(1/2^2)/(2!*2^4) + exp(1/2^3)/(3!*2^9) + exp(1/2^4)/(4!*2^16) + exp(1/2^5)/(5!*2^25) +...
A(1/2) = -exp(1) + 1 + (1+1/2) + (1+1/2^2)^2/2! + (1+1/2^3)^3/3! + (1+1/2^4)^4/4! + (1+1/2^5)^5/5! + (1+1/2^6)^6/6! +...
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Table[Sum[Binomial[d, n/d]*n!/d!, {d, Divisors[n]}], {n, 1, 30}] (* Vaclav Kotesovec, Oct 20 2020 *)
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{a(n) = my(A=1); A = sum(m=1, n, x^(m^2) * exp(x^m +x*O(x^n)) / m!); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
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{a(n) = my(A=1); A = -exp(1) + sum(m=0, n, (1 + x^m +x*O(x^n))^m/m!); if(n==0,0, n!*polcoeff(A, n))}
for(n=1, 30, print1(a(n), ", "))
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{a(n) = if(n<1,0, sumdiv(n,d, binomial(d, n/d) * n!/d! ) )}
for(n=1, 30, print1(a(n), ", "))
A326096
E.g.f.: Sum_{n>=0} ((1+x)^n + 1)^n * x^n/n!.
Original entry on oeis.org
1, 2, 6, 32, 256, 2712, 37744, 645752, 13371264, 327748832, 9332342944, 304875611328, 11298403070464, 470279355784448, 21809054992366464, 1118931830122060928, 63115145120561606656, 3892675200470654980608, 261242029823318546162176, 18994387868664467440590848, 1490356266852194536099393536, 125747158151444491631754033152
Offset: 0
E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 32*x^3/3! + 256*x^4/4! + 2712*x^5/5! + 37744*x^6/6! + 645752*x^7/7! + 13371264*x^8/8! + 327748832*x^9/9! + 9332342944*x^10/10! + 304875611328*x^11/11! + 11298403070464*x^12/12! + ...
such that
A(x) = 1 + ((1+x) + 1)*x + ((1+x)^2 + 1)^2*x^2/2! + ((1+x)^3 + 1)^3*x^3/3! + ((1+x)^4 + 1)^4*x^4/4! + ((1+x)^5 + 1)^5*x^5/5! + ((1+x)^6 + 1)^6*x^6/6! + ((1+x)^7 + 1)^7*x^7/7! + ...
also
A(x) = 1 + (1+x)*exp(x*(1+x))*x + (1+x)^4*exp(x*(1+x)^2)*x^2/2! + (1+x)^9*exp(x*(1+x)^3)*x^3/3! + (1+x)^16*exp(x*(1+x)^4)*x^4/4! + (1+x)^25*exp(x*(1+x)^5)*x^5/5! + (1+x)^36*exp(x*(1+x)^6)*x^6/6! + ...
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} ((1+x)^n + 1)^n * x^n/n! = Sum_{n>=0} (1+x)^(n^2) * exp(x*(1+x)^n) * x^n/n!.
(1) At x = -1/2, the following sums are equal
S1 = Sum_{n>=0} (-1)^n * 2^(-n*(n+1)) * (2^n + 1)^n / n!,
S1 = Sum_{n>=0} (-1)^n * 2^(-n*(n+1)) * exp( -1/2^(n+1) ) / n!,
where S1 = 0.41868678468707099609788224908427981408329845879700862624389...
(2) At x = -2/3, the following sums are equal
S2 = Sum_{n>=0} (-1)^n * 2^n * 3^(-n*(n+1)) * (3^n + 1)^n / n!,
S2 = Sum_{n>=0} (-1)^n * 2^n * 3^(-n*(n+1)) * exp( -2/3^(n+1) ) / n!,
where S2 = 0.33802063384093377391547056494398131361711992142768124149541...
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/* E.g.f.: Sum_{n>=0} ((1+x)^n + 1)^n * x^n/n! */
{a(n) = my(A = sum(m=0,n, ((1+x)^m + 1 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A,n)}
for(n=0,25, print1(a(n),", "))
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/* E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(x*(1+x)^n) * x^n/n! */
{a(n) = my(A = sum(m=0,n, (1+x +x*O(x^n))^(m^2) * exp(x*(1+x)^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A,n)}
for(n=0,25, print1(a(n),", "))
A326098
E.g.f.: Sum_{n>=0} (n+1) * (1 + x^n)^n * x^n/n!
Original entry on oeis.org
1, 2, 7, 4, 77, 6, 2527, 8, 33609, 725770, 907211, 12, 946028173, 14, 968647695, 653837184016, 17473020364817, 18, 935267389056019, 20, 1723379337808128021, 1703031405723648022, 3716933623603223, 24, 124520749358323872153625, 3877802510832746496000026, 11787184960911360027, 10802449851605508096000028, 16938242101749730412851200029, 30, 51981534567681070815925862400031
Offset: 0
E.g.f.: A(x) = 1 + 2*x + 7*x^2 + 4*x^3 + 77*x^4 + 6*x^5 + 2527*x^6 + 8*x^7 + 33609*x^8 + 725770*x^9 + 907211*x^10 + 12*x^11 + 946028173*x^12 + 14*x^13 + ...
such that
A(x) = 1 + 2*(1+x)*x + 3*(1+x^2)^2*x^2/2! + 4*(1+x^3)^3*x^3/3! + 5*(1+x^4)^4*x^4/4! + 6*(1+x^5)^5*x^5/5! + 7*(1+x^6)^6*x^6/6! + 8*(1+x^7)^7*x^7/7! + ...
also
A(x) = (1 + x)*exp(x) + (2 + x^2)*exp(x^2)*x^2 + (3 + x^3)*exp(x^3)*x^6/2! + (4 + x^4)*exp(x^4)*x^12/3! + (5 + x^5)*exp(x^5)*x^20/4! + (6 + x^6)*exp(x^6)*x^30/5! + (7 + x^7)*exp(x^7)*x^42/6! + ...
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{a(n) = if(n==0,1, sumdiv(n,d, (d + 1) * binomial(d, n/d - 1) * n!/d! ) )}
for(n=0,30, print1(a(n),", "))
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/* E.g.f.: Sum_{n>=0} (n+1) * (1 + x^n)^n * x^n/n! */
{a(n) = my(A = sum(m=0,n, (m+1) * (x^m + 1 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))
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/* E.g.f.: Sum_{n>=1} (n + x^n) * exp(x^n) * x^(n*(n-1))/(n-1)! */
{a(n) = my(A = sum(m=1,sqrtint(2*n+1), (m + x^m) * exp(x^m +x*O(x^n)) * x^(m*(m-1))/(m-1)! )); n!*polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))
Showing 1-3 of 3 results.
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