A304863 O.g.f. A(x) satisfies: [x^n] exp( n*(n+2) * x ) / A(x) = 0 for n>0.

1, 3, 17, 180, 3079, 74271, 2308940, 87438684, 3888682559, 198073751505, 11348409001233, 721483807171188, 50361931297722244, 3827114191186713588, 314413091556481490640, 27761835704580647457012, 2621495363274661266785679, 263593068966612639018287637, 28117066903131481643928647363, 3171150259810035292799245555884, 377044852592342586608552585592079

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Example

O.g.f.: A(x) = 1 + 3*x + 17*x^2 + 180*x^3 + 3079*x^4 + 74271*x^5 + 2308940*x^6 + 87438684*x^7 + 3888682559*x^8 + 198073751505*x^9 + 11348409001233*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+2)*x) / A(x) begins:
n=0: [1, -3, -16, -630, -50112, -6665400, -1325134080, -366674636160, ...];
n=1: [1, 0, -25, -828, -58779, -7479072, -1452239685, -395811449100, ...];
n=2: [1, 5, 0, -1078, -78464, -9183672, -1700942720, -450843184000, ...];
n=3: [1, 12, 119, 0, -99387, -12381300, -2151101205, -544666984560, ...];
n=4: [1, 21, 416, 6858, 0, -15533496, -2923952256, -703585823616, ...];
n=5: [1, 32, 999, 29540, 730213, 0, -3653179205, -962999633260, ...];
n=6: [1, 45, 2000, 86922, 3589056, 124275528, 0, -1200826684800, ...];
n=7: [1, 60, 3575, 210672, 12162501, 668679228, 30900268395, 0, ...];
n=8: [1, 77, 5904, 449930, 33949888, 2513449800, 177544721920, 10559736679040, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n+2)*x ) / A(x) = 0 for n>=0.
RELATED SERIES.
The logarithmic derivative of A(x) yields:
A'(x)/A(x) = 3 + 25*x + 414*x^2 + 10109*x^3 + 320253*x^4 + 12346720*x^5 + 557708406*x^6 + 28786882117*x^7 + 1668054884229*x^8 + 107077380781005*x^9 + ...
1 - 1/A(x) = 3*x + 8*x^2 + 105*x^3 + 2088*x^4 + 55545*x^5 + 1840464*x^6 + 72752904*x^7 + 3334122880*x^8 + 173569203225*x^9 + 10108800765000*x^10 + ...

Prog

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

Crossrefs

Cf. A304322, A304318, A304319, A304864, A304865.

Sequence in context: A159592 A126443 A054976 * A163886 A340881 A163879

Adjacent sequences: A304860 A304861 A304862 * A304864 A304865 A304866

Keyword

nonn

Author

Paul D. Hanna, May 19 2018

Status

approved