The On-Line Encyclopedia of Integer Sequences (OEIS)

A359927 revision #12

A359927

E.g.f.: lim_{N->oo} [ Sum_{n>=0} (N^2 + 3*N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).

1, 1, 7, 112, 2965, 111856, 5528419, 339433984, 24965493865, 2142654088960, 210377086601311, 23269631260880896, 2864038963868253373, 388330717110688399360, 57521524729462484086075, 9242821569458332441378816, 1601434996324769244061560529

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

OFFSET

0,3

COMMENTS

Related limits:

(C1) exp(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).

(C2) W(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n)^n * (x/N)^n/n! ]^(1/N), where W(x) = LambertW(-x)/(-x).

LINKS

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

FORMULA

E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following.

(1) A(x) = lim_{N->oo} [ Sum_{n>=0} (N + n)^n*(N + 2*n)^n * (x/N)^n/n! ]^(1/N).

(2) A(x) = exp( Sum_{n>=0} A359928(n)*x^n/n! ), where A359928(n) = (1/2) * [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n + y)^n*(n + 2*y)^n *x^n/n! ).

EXAMPLE

E.g.f.: A(x) = 1 + x + 7*x^2/2! + 112*x^3/3! + 2965*x^4/4! + 111856*x^5/5! + 5528419*x^6/6! + 339433984*x^7/7! + 24965493865*x^8/8! + 2142654088960*x^9/9! + 210377086601311*x^10/10! + 23269631260880896*x^11/11! + 2864038963868253373*x^12/12! + ...

where A(x) equals the limit, as N -> oo, of the series

[1 + (N^2+3*N+2)*(x/N) + (N^2+3*2*N+2*2^2)^2*(x/N)^2/2! + (N^2+3*3*N+2*3^2)^3*(x/N)^3/3! + (N^2+3*4*N+2*4^2)^4*(x/N)^4/4! + (N^2+3*5*N+2*5^2)^5*(x/N)^5/5! + (N^2+3*6*N+2*6^2)^6*(x/N)^6/6! + ...]^(1/N).

RELATED SERIES.

The logarithm of the g.f. A(x) begins:

(a) log(A(x)) = x + 6*x^2/2! + 93*x^3/3! + 2448*x^4/4! + 92505*x^5/5! + 4589568*x^6/6! + 283008621*x^7/7! + 20903023872*x^8/8! + ... + A359928(n)*x^n/n! + ...

where A359928(n) = [x^n*y^(n+1)/n!] (1/2) * log( Sum_{n>=0} (n^2 + 3*n*y + 2*y^2)^n * x^n/n! );

that is, the coefficient of x^n/n! in the logarithm of e.g.f A(x) equals the coefficient of y^(n+1)*x^n/n!, n >= 1, in the series given by

(b) (1/2) * log( Sum_{n>=0} (n^2 + 3*n*y + 2*y^2)^n * x^n/n! ) = x*(y^2 + 3/2*y + 1/2) + x^2/2!*(6*y^3 + 39/2*y^2 + 21*y + 15/2) + x^3/3!*(93*y^4 + 999/2*y^3 + 2055/2*y^2 + 1917/2*y + 683/2) + x^4/4!*(2448*y^5 + 19119*y^4 + 61704*y^3 + 102742*y^2 + 88080*y + 31019) + x^5/5!*(92505*y^6 + 1948347/2*y^5 + 8887325/2*y^4 + 11224575*y^3 + 16525750*y^2 + 26820135/2*y + 9342629/2) + x^6/6!*(4589568*y^7 + 61994772*y^6 + 374546664*y^5 + 1310466240*y^4 + 2862046080*y^3 + 3891543876*y^2 + 3041064504*y + 1050241608) + ...

PROG

(PARI) /* Using formula for the logarithm of g.f. A(x) */

{L(n) = (1/2) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^m*(m + 2*y)^m *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}

{a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}

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

(PARI) /* Using limit formula */

\p100

P(n) = sum(k=0, 31, ((n + k)*(n + 2*k))^k * x^k/k! +O(x^31))

Vec(round( serlaplace( subst(P(10^100)^(1/10^100), x, x/10^100) )*1.) )

CROSSREFS

Cf. A359928, A319147, A266481, A318633.

Sequence in context: A128576 A147631 A371330 * A010795 A293456 A099153

Adjacent sequences: A359924 A359925 A359926 * A359928 A359929 A359930

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 20 2023

STATUS

editing