A273925 G.f. satisfies: A( A(x)^2 - A(x)^3 ) = x^2, where A(x) = Sum_{n>=1} a(n)*x^n / 2^A273926(n).
1, 1, 3, 3, 175, 41, 4947, 321, 687611, 11403, 25132181, 107305, 1941554203, 2111325, 77643067507, 21427329, 25549683166419, 1782548851, 1073363084982753, 18891311061, 91744420207896017, 406630578535, 3975787925128277349, 4432136534071, 697211573846047770799, 195301983407647, 30867311449650538783337, 2171049926840877, 2756162894749311377078579, 48645967088000101
Offset: 1
Keywords
Examples
G.f.: A(x) = x + 1/2*x^2 + 3/8*x^3 + 3/4*x^4 + 175/128*x^5 + 41/16*x^6 + 4947/1024*x^7 + 321/32*x^8 + 687611/32768*x^9 + 11403/256*x^10 + 25132181/262144*x^11 + 107305/512*x^12 + 1941554203/4194304*x^13 + 2111325/2048*x^14 + 77643067507/33554432*x^15 + 21427329/4096*x^16 + 25549683166419/2147483648*x^17 + 1782548851/65536*x^18 + 1073363084982753/17179869184*x^19 + 18891311061/131072*x^20 + 91744420207896017/274877906944*x^21 + 406630578535/524288*x^22 + 3975787925128277349/2199023255552*x^23 + 4432136534071/1048576*x^24 +...+ a(n)*x^n/2^A273926(n) +... such that A( A(x)^2 - A(x)^3 ) = x^2 and A( +sqrt( A(x^2 - x^3) ) ) = x. RELATED SERIES. The g.f. is related to the Motzkin numbers by the relation: A( -sqrt( A(x^2 - x^3) ) ) = -x + x^2 - x^3 + 2*x^4 - 4*x^5 + 9*x^6 - 21*x^7 + 51*x^8 - 127*x^9 + 323*x^10 - 835*x^11 +...+ (-1)^n*A001006(n-2)*x^n +... which equals (1 - x - sqrt(1 + 2*x - 3*x^2))/2. Also, we have A( A(x)^3 - A(x)^2 ) = (1 - x^2 - sqrt(1 + 2*x^2 - 3*x^4))/2. A relevant series begins: A(x^2 - x^3) = x^2 - x^3 + 1/2*x^4 - x^5 + 7/8*x^6 - 9/8*x^7 + 15/8*x^8 - 27/8*x^9 + 751/128*x^10 - 1259/128*x^11 + 1087/64*x^12 - 1859/64*x^13 + 51307/1024*x^14 - 88509/1024*x^15 + 153519/1024*x^16 - 271065/1024*x^17 + 15515931/32768*x^18 - 27920307/32768*x^19 + 12582747/8192*x^20 - 22730223/8192*x^21 + 1317324005/262144*x^22 - 2391803575/262144*x^23 + 4354015459/262144*x^24 - 7946645097/262144*x^25 +... Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then B(x) = sqrt( A(x^2 - x^3) ) and begins sqrt( A(x^2 - x^3) ) = x - 1/2*x^2 + 1/8*x^3 - 7/16*x^4 + 27/128*x^5 - 103/256*x^6 + 629/1024*x^7 - 2535/2048*x^8 + 66835/32768*x^9 - 222155/65536*x^10 + 1517887/262144*x^11 - 5140113/524288*x^12 + 70575503/4194304*x^13 - 241166467/8388608*x^14 + 1663932701/33554432*x^15 - 5878842599/67108864*x^16 + 336833847555/2147483648*x^17 - 1211274078451/4294967296*x^18 + 8710075650043/17179869184*x^19 - 31385188980941/34359738368*x^20 + 453666114969205/274877906944*x^21 - 1644082529689977/549755813888*x^22 + 11949781587586819/2199023255552*x^23 +... Also, note that A(x)^2 - A(x)^3 = B(x^2) is an even function, where A(x)^2 = x^2 + x^3 + x^4 + 15/8*x^5 + 29/8*x^6 + 903/128*x^7 + 221/16*x^8 + 29559/1024*x^9 + 7851/128*x^10 + 4320363/32768*x^11 + 73433/256*x^12 + 165702201/262144*x^13 + 1439981/1024*x^14 + 13220447555/4194304*x^15 + 14569809/2048*x^16 + 542234209095/33554432*x^17 +... A(x)^3 = x^3 + 3/2*x^4 + 15/8*x^5 + 7/2*x^6 + 903/128*x^7 + 57/4*x^8 + 29559/1024*x^9 + 489/8*x^10 + 4320363/32768*x^11 + 1149/4*x^12 + 165702201/262144*x^13 + 179919/128*x^14 + 13220447555/4194304*x^15 + 1821543/256*x^16 + 542234209095/33554432*x^17 +... The bisections of this sequence begin: odd bisection: [1, 3, 175, 4947, 687611, 25132181, 1941554203, 77643067507, 25549683166419, 1073363084982753, 91744420207896017, 3975787925128277349, ...]; even bisection: [1, 3, 41, 321, 11403, 107305, 2111325, 21427329, 1782548851, 18891311061, 406630578535, 4432136534071, 195301983407647, 2171049926840877, ...].
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..261
Programs
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PARI
{a(n) = my(A=x); for(i=0,n, A = serreverse( sqrt(subst(A,x,x^2 - x^3 +x^2*O(x^n) )) )); numerator(polcoeff(A,n))} for(n=1,30,print1(a(n),", "))
Formula
G.f. A(x) = Sum_{n>=1} a(n)*x^n / 2^A273926(n) satisfies:
(1) A( +sqrt( A(x^2 - x^3) ) ) = x.
(2) A( -sqrt( A(x^2 - x^3) ) ) = (1 - x - sqrt(1 + 2*x - 3*x^2))/2.
(3) A( A(x)^3 - A(x)^2 ) = (1 - x^2 - sqrt(1 + 2*x^2 - 3*x^4))/2.
Comments