A326550 E.g.f. A(x) satisfies: Sum_{n>=0} (exp(n*x) + A(x))^n * x^n / n! = Sum_{n>=0} (exp((n+1)*x) + 1)^n * x^n / n!.

1, 1, 3, 19, 183, 2451, 44523, 1003955, 27132591, 862477603, 31585560483, 1312509666051, 61199375982759, 3171627767105747, 181223609188848411, 11340823889035215187, 772846532507982245727, 57069311173494600701763, 4546598329397176113578067, 389300199395007408056468579, 35704214147724534934522349655, 3496767016630336049148287129971, 364696725110047554147731853993291

Offset

0,3

Comments

RELATED IDENTITY: 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!.

Links

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

Formula

E.g.f. A(x) allows the following sums to be equal:

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

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

(3) B(x) = Sum_{n>=0} exp(n^2*x) * exp(exp(n*x)*x * A(x)) * x^n / n!,

(4) B(x) = Sum_{n>=0} exp(n^2*x) * exp(exp(n*x)*x + n*x ) * x^n / n!,

where B(x) is the e.g.f. of A326009.

a(n) = 1 (mod 2) for n >= 0.

a(6*n+1) = 2 (mod 3) for n >= 1;

a(6*n+3) = 1 (mod 3) for n >= 0;

a(6*n+k) = 0 (mod 3) when k = {0,2,4,5} for n >= 1.

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 183*x^4/4! + 2451*x^5/5! + 44523*x^6/6! + 1003955*x^7/7! + 27132591*x^8/8! + 862477603*x^9/9! + 31585560483*x^10/10! + 1312509666051*x^11/11! + ...
such that the following sums are equal
(1) B(x) = 1 + (exp(x) + A(x)) + (exp(2*x) + A(x))^2*x^2/2! + (exp(3*x) + A(x))^3*x^3/3! + (exp(4*x) + A(x))^4*x^4/4! + (exp(5*x) + A(x))^5*x^5/5! + ...
and
(2) B(x) = 1 + (exp(2*x) + 1)*x + (exp(3*x) + 1)^2*x^2/2! + (exp(4*x) + 1)^3*x^3/3! + (exp(5*x) + 1)^4*x^4/4! + (exp(6*x) + 1)^5*x^5/5! + ...
also
(3) B(x) = exp(x*A(x)) + exp(x)*exp(exp(x)*x*A(x))*x + exp(4*x)*exp(exp(2*x)*x*A(x))*x^2/2! + exp(9*x)*exp(exp(3*x)*x*A(x))*x^3/3! + exp(16*x)*exp(exp(4*x)*x*A(x))*x^4/4! + ...
and
(4) B(x) = exp(x) + exp(x)*exp(exp(x)*x + x)*x + exp(4*x)*exp(exp(2*x)*x + 2*x)*x^2/2! + exp(9*x)*exp(exp(3*x)*x + 3*x)*x^3/3! + exp(16*x)*exp(exp(4*x)*x + 4*x)*x^4/4! + ...
where
B(x) = 1 + 2*x + 8*x^2/2! + 56*x^3/3! + 564*x^4/4! + 7452*x^5/5! + 124126*x^6/6! + 2527646*x^7/7! + 61337576*x^8/8! + 1740438008*x^9/9! + 56893173354*x^10/10! + ... + A326009(n)*x^n/n! + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, (exp((m+1)*x +x*O(x^#A)) + 1)^m*x^m/m! - (exp(m*x +x*O(x^#A)) + Ser(A))^m*x^m/m! ), #A)); n!*A[n+1]}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, exp(m*(m+1)*x + exp(m*x +x*O(x^#A))*x )*x^m/m! - exp(m^2*x + exp(m*x +x*O(x^#A))*x * Ser(A))*x^m/m! ), #A)); n!*A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A326009.

Sequence in context: A213533 A203133 A006531 * A352236 A362205 A242369

Adjacent sequences: A326547 A326548 A326549 * A326551 A326552 A326553

Keyword

nonn

Author

Paul D. Hanna, Jul 13 2019

Status

approved