A316703 O.g.f. A(x) satisfies: [x^n] exp(-n*x*A(x)) * (n+1 - n/A(x)) = 0 for n>=1.

1, 1, 3, 11, 48, 231, 1198, 6571, 37708, 224612, 1381047, 8728357, 56520580, 374049962, 2524760084, 17352434291, 121271358844, 860832917836, 6200469605740, 45281350853036, 335040385844140, 2510109678504943, 19031562100423046, 145961670086604701, 1131893950714288692, 8871861944975204172

Offset

0,3

Comments

It is remarkable that this sequence should consist entirely of integers.

Compare to: [x^n] exp(-n^2 * x*G(x)) * (n + 1 - n/G(x)) = 0, which holds for n>=1 when G(x) = 1 + x*G(x)^2 + x^2*G(x)*G'(x), the o.g.f. of A088716.

Links

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

Formula

O.g.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 + x^2*A(x)^3 + x^3*A(x)^2*A'(x).

Example

O.g.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 48*x^4 + 231*x^5 + 1198*x^6 + 6571*x^7 + 37708*x^8 + 224612*x^9 + 1381047*x^10 + ...
such that A(x) = 1 + x*A(x)^2 + x^2*A(x)^3 + x^3*A(x)^2*A'(x).
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*x*A(x)) * (n+1 - n/A(x)) begins:
n=1: [1, 0, 1, 8, 189, 5024, 173725, 7248744, 358001497, ...];
n=2: [1, 0, 0, 4, 192, 5568, 210880, 9271680, 476620032, ...];
n=3: [1, 0, -3, 0, 117, 4032, 180225, 8532864, 462998025, ...];
n=4: [1, 0, -8, 8, 0, 1856, 122560, 6538752, 384283648, ...];
n=5: [1, 0, -15, 40, -195, 0, 59725, 4165800, 280894425, ...];
n=6: [1, 0, -24, 108, -576, -576, 0, 1897344, 175779072, ...];
n=7: [1, 0, -35, 224, -1323, 1568, -60095, 0, 80665417, ...];
n=8: [1, 0, -48, 400, -2688, 8832, -137600, -1330176, 0, ...];
n=9: [1, 0, -63, 648, -4995, 25056, -268515, -1821528, -66997287, 0, ...]; ...
in which the zero coefficient of x^n is shown as a diagonal of zeros.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 361*x^4/4! + 7701*x^5/5! + 218851*x^6/6! + 7835773*x^7/7! + 339872625*x^8/8! + ...
1/A(x) = 1 - x - 2*x^2 - 6*x^3 - 25*x^4 - 118*x^5 - 612*x^6 - 3382*x^7 - 19639*x^8 - 118618*x^9 + ...
A'(x)/A(x) = 1 + 5*x + 25*x^2 + 141*x^3 + 836*x^4 + 5183*x^5 + 33202*x^6 + 218613*x^7 + 1473064*x^8 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A-1; A[#A] = -Vec( exp(-m * x*Ser(A) ) * (m+1 - m/Ser(A)))[#A]/m ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Differential equation: */
{a(n) = my(A=1); for(i=1, n, A = 1 + x*A^2 + x^2*A^3 + x^3*A^2*A' + x*O(x^n) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A088716.

Sequence in context: A181278 A126180 A121139 * A362741 A192399 A233162

Adjacent sequences: A316700 A316701 A316702 * A316704 A316705 A316706

Keyword

nonn

Author

Paul D. Hanna, Jul 20 2018

Status

approved