OFFSET
1,7
COMMENTS
Appears to obey an 8-term hypergeometric recurrence with 4th-order polynomial coefficients.
LINKS
R. J. Mathar and Robert Israel, Table of n, a(n) for n = 1..2066 (1..105 from R. J. Mathar)
R. J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 (2012)
FORMULA
D-finite with recurrence: 5 n (5 n - 1) (5 n + 1) (5 n + 2) (5 n + 3) a(n) - (n + 1) (27906 n^4 + 198109 n^3 + 447051 n^2 + 405674 n + 128400) a(n + 1) + 25 (n + 2) (1875 n^4 + 28312 n^3 + 141513 n^2 + 287228 n + 204072) a(n + 2) + 250 (n + 2) (n + 3) (250 n^3 + 3031 n^2 + 11433 n + 13668) a(n + 3) + 625 (n + 2) (n + 3) (n + 4) (75 n^2 + 662 n + 1403) a(n + 4) + 3125 (n + 2) (n + 3) (n + 4) (n + 5) (6 n + 29) a(n + 5) + 3125(n + 2) (n + 3) (n + 4) (n + 5) (n + 6) a(n + 6) = 0. - Vladimir Reshetnikov, Jul 09 2015
From Robert Israel, Jul 10 2015: (Start)
G.f. G(x) satisfies G + G^2 + G^3 + G^4 + G^5 = x
and the differential equation
-2184*x^3+32760*x^2-163800*x+273000+(-10920*x^3+163800*x^2-819000*x+1365000)*G(x)+(2457000*x^4-24555336*x^3+33314040*x^2-22930200*x-10608000)*G'(x)+(11602500*x^5-88671024*x^4+64015500*x^3-18674400*x^2-27414000*x-9780000)*G''(x)+(7962500*x^6-48147528*x^5+12768480*x^4-2457200*x^3-7171500*x^2-6885000*x-1975000)*G'''(x)+(1137500*x^7-5485909*x^6-750720*x^5-1121525*x^4-654500*x^3-744375*x^2-475000*x-109375)*G''''(x) = 0
from which we can obtain the 8-term recurrence mentioned in the Comments:
1820*(5*n+1)*(5*n+2)*(5*n+3)*(5*n-1)*a(n)-13*(421993*n^4+2859670*n^3+6398855*n^2+5850050*n+1876272)*a(n+1)-60*(12512*n^4-187784*n^3-1717861*n^2-4206649*n-3230668)*a(n+2)-25*(44861*n^4+367454*n^3+1830175*n^2+6002422*n+7768608)*a(n+3)-500*(n+4)*(1309*n^3+22197*n^2+140942*n+279612)*a(n+4)-1875*(n+5)*(n+4)*(397*n^2+5657*n+18614)*a(n+5)-25000*(19*n+136)*(n+6)*(n+5)*(n+4)*a(n+6)-109375*(n+5)*(n+4)*(n+7)*(n+6)*a(n+7) = 0.
From the Lagrange inversion theorem,
a(n) = 1/n! * (d/dx)^(n-1) (p^n)(0) where p(x) = 1/(1+x+x^2+x^3+x^4).
(End)
Recurrence: 125*(n-3)*(n-2)*(n-1)*n*(3*n - 11)*(6*n - 23)*(6*n - 17)*a(n) = -100*(n-3)*(n-2)*(n-1)*(6*n - 23)*(144*n^3 - 1152*n^2 + 2921*n - 2310)*a(n-1) - 30*(n-3)*(n-2)*(7884*n^5 - 113004*n^4 + 636639*n^3 - 1760222*n^2 + 2386123*n - 1267420)*a(n-2) - 4*(n-3)*(6*n - 11)*(15048*n^5 - 240768*n^4 + 1533029*n^3 - 4855116*n^2 + 7647427*n - 4792620)*a(n-3) - 5*(3*n - 8)*(5*n - 21)*(5*n - 19)*(5*n - 18)*(5*n - 17)*(6*n - 17)*(6*n - 11)*a(n-4). - Vaclav Kotesovec, Aug 18 2015
EXAMPLE
If y=x+x^2+x^3+x^4+x^5, then x=y -y^2 +y^3 -y^4 +y^5 -6*y^7 +27*y^8 -83*y^9 +...
MAPLE
rec := 5*n*(5*n-1)*(5*n+1)*(5*n+2)*(5*n+3)*a(n)-(n+1)*(27906*n^4+198109*n^3+447051*n^2+405674*n+128400)*a(n+1)+(25*(n+2))*(1875*n^4+28312*n^3+141513*n^2+287228*n+204072)*a(n+2)+(250*(n+2))*(n+3)*(250*n^3+3031*n^2+11433*n+13668)*a(n+3)+(625*(n+2))*(n+3)*(n+4)*(75*n^2+662*n+1403)*a(n+4)+(3125*(n+2))*(n+3)*(n+4)*(n+5)*(6*n+29)*a(n+5)+(3125*(n+2))*(n+3)*(n+4)*(n+5)*(n+6)*a(n+6):
f:= gfun:-rectoproc({rec, a(1) = 1, a(2) = -1, a(3) = 1, a(4) = -1, a(5) = 1, a(6) = 0}, a(n), remember):
map(f, [$1..50]); # Robert Israel, Jul 10 2015
MATHEMATICA
InverseSeries[x + x^2 + x^3 + x^4 + x^5 + O[x]^50][[3]] (* Vladimir Reshetnikov, Jul 09 2015 *)
RecurrenceTable[{5 (-8+3 n) (-21+5 n) (-19+5 n) (-18+5 n) (-17+5 n) (-17+6 n) (-11+6 n) a[-4+n]+4 (-3+n) (-11+6 n) (-4792620+7647427 n-4855116 n^2+1533029 n^3-240768 n^4+15048 n^5) a[-3+n]+30 (-3+n) (-2+n) (-1267420+2386123 n-1760222 n^2+636639 n^3-113004 n^4+7884 n^5) a[-2+n]+100 (-3+n) (-2+n) (-1+n) (-23+6 n) (-2310+2921 n-1152 n^2+144 n^3) a[-1+n]+125 (-3+n) (-2+n) (-1+n) n (-11+3 n) (-23+6 n) (-17+6 n) a[n]==0, a[1]==1, a[2]==-1, a[3]==1, a[4]==-1}, a, {n, 1, 40}] (* Vaclav Kotesovec, Aug 18 2015 *)
PROG
(PARI) Vec(serreverse(x + x^2 + x^3 + x^4 + x^5 + O(x^50))) \\ Michel Marcus, Aug 03 2015
CROSSREFS
KEYWORD
sign
AUTHOR
R. J. Mathar, Oct 01 2012
STATUS
approved