|
| |
|
|
A302113
|
|
a(n) = (4/(2*n-3))*(2*(n-1)*(2*n-1)*a(n-1) + (-1)^n*Product_{k=0..n-1} (2*k+1)) with a(0) = 0.
|
|
1
|
|
|
|
0, 4, 108, 2860, 96180, 3956580, 193437420, 10973128140, 709033518900, 51428555381700, 4138486257710700, 365929308313512300, 35268615299594546100, 3680203334234934622500, 413360438535421144267500, 49725729790306916413567500, 6378610855886528420493832500, 869137169523850497054287002500
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = (-1)^(n-1)*f1(n-1)*5*Product_{k=0..n-1} (2*k+1) where f1(n) corresponds to the x values such that Sum_{k>=0} (-1)^k/(binomial(2*k,k)*(2*k+(2*n+1))) = x*sqrt(5)*log((1+sqrt(5))/2) + y. (See examples for connection with a(n) in terms of material at Links section).
|
|
|
EXAMPLE
|
Examples ((3.48) - (3.52)) at page 11 in Links section as follows, respectively.
For n=0, f1(0) = 4/5, so a(1) = 4.
For n=1, f1(1) = -36/5, so a(2) = 108.
For n=2, f1(2) = 572/15, so a(3) = 2860.
For n=3, f1(3) = -916/5, so a(4) = 96180.
For n=4, f1(4) = 29308/35, so a(5) = 3956580.
|
|
|
MATHEMATICA
|
RecurrenceTable[{a[m+1] == (4/(2*m - 1))*(2*m*(2*m + 1)*a[m] + (-1)^(m + 1) * Product[2*k + 1, {k, 0, m}]), a[0] == 0}, a, {m, 0, 15}] (* Vaclav Kotesovec, Apr 11 2018 *)
nmax = 15; Flatten[{0, Table[CoefficientList[1/Sqrt[5] * TrigToExp[Expand[ FunctionExpand[Table[FullSimplify[Sum[(-1)^j/(Binomial[2*j, j]*(2*j + (2*m + 1))), {j, 0, Infinity}]]*(-1)^m * 5 * Product[2*k + 1, {k, 0, m}], {m, 0, nmax}]]]], Log[1/2 + Sqrt[5]/2]][[n, 2]], {n, 1, nmax}]}] (* Vaclav Kotesovec, Apr 11 2018 *)
|
|
|
PROG
|
(PARI) a=vector(20); a[1]=4; for(n=2, #a, a[n]=(4/(2*n-3))*(2*(n-1)*(2*n-1)*a[n-1]+((-1)^n)*prod(k=0, n-1, (2*k+1)))); concat(0, a) \\ Altug Alkan, Apr 01 2018
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
