A280625 E.g.f. S(x) satisfies: C(x)^2 - S(x)^2 = 1 and D(x)^4 - S(x)^4 = 1, where functions C(x) and D(x) are described by A280626 and A280627, respectively.

1, 1, 19, 739, 35641, 3753721, 500577499, 91718242219, 22737318482161, 6983681901945841, 2676021948941279779, 1243547540389481251699, 686920343453752746986281, 446624144083900575607651561, 336962029373243047762784366059, 292073105918297230422637244241979, 288329217381806896076167481522680801, 321507277164646598826580516711872346081, 402176512693739754388880316622683052031539

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Formula

E.g.f. S(x) = Series_Reversion( Integral 1/( (1+x^2)^(1/2) * (1+x^4)^(3/4) ) dx ).

E.g.f. S(x) = Sum_{n>=1} a(n)*x^(2*n-1)/(2*n-1)!, where related functions S = S(x), C = C(x), and D = D(x) possess the following properties.

(1.a) C^2 - S^2 = 1.

(1.b) D^4 - S^4 = 1.

Integrals.

(2.a) S = Integral C*D^3 dx.

(2.b) C = 1 + Integral S*D^3 dx.

(2.c) D = 1 + Integral C*S^3 dx.

(2.d) C + S = 1 + Integral D^3 * (C + S) dx.

(2.e) D^2 + S^2 = 1 + Integral 2*S*C*D * (D^2 + S^2) dx.

(2.f) D - S = 1 - Integral C * (D^3 - S^3) dx.

Exponential.

(3.a) C + S = exp( Integral D^3 dx ).

(3.b) D^2 + S^2 = exp( Integral 2*S*C*D dx ).

(3.d) C = cosh( Integral D^3 dx ).

(3.e) S = sinh( Integral D^3 dx ).

(3.f) D^2 = cosh( Integral 2*S*C*D dx ).

(3.g) S^2 = sinh( Integral 2*S*C*D dx ).

(3.h) sinh( Integral D^3 dx )^2 = sinh( Integral 2*S*C*D dx ).

Derivatives.

(4.a) S' = C*D^3.

(4.b) C' = S*D^3.

(4.c) D' = C*S^3.

(4.d) S'*D - D'*S = C.

(4.e) S'*C - C'*S = D^3.

(4.f) (C' + S')/(C + S) = D^3.

(4.g) (D^2 + S^2)'/(D^2 + S^2) = 2*S*C*D.

(4.h) (D' - S')/(D - S) = -C * (D^2 + D*S + S^2).

Example

E.g.f.: S(x) = x + x^3/3! + 19*x^5/5! + 739*x^7/7! + 35641*x^9/9! + 3753721*x^11/11! + 500577499*x^13/13! + 91718242219*x^15/15! + 22737318482161*x^17/17! + 6983681901945841*x^19/19! + 2676021948941279779*x^21/21! + 1243547540389481251699*x^23/23! + 686920343453752746986281*x^25/25! + 446624144083900575607651561*x^27/27! +...
such that
(1) C(x)^2 - S(x)^2 = 1,
(2) D(x)^4 - S(x)^4 = 1,
where functions C(x) and D(x) are illustrated below.
RELATED SERIES.
C(x) = 1 + x^2/2! + x^4/4! + 109*x^6/6! + 3889*x^8/8! + 292681*x^10/10! + 37275121*x^12/12! + 5709311029*x^14/14! + 1254902705569*x^16/16! + 350061261777361*x^18/18! + 120872805166945441*x^20/20! + 51564789352080559549*x^22/22! + 26284030671328082426449*x^24/24! + 15848108292907342195314841*x^26/26! + 11161807217694742818283238161*x^28/28! +...
D(x) = 1 + 6*x^4/4! + 120*x^6/6! + 4284*x^8/8! + 382560*x^10/10! + 40975176*x^12/12! + 6524350560*x^14/14! + 1420005102864*x^16/16! + 386400824613120*x^18/18! + 133774424157792096*x^20/20! + 56530740636066364800*x^22/22! + 28642309445854790698944*x^24/24! + 17209537237868777504801280*x^26/26! + 12062425479867549597010598016*x^28/28! +...
C(x) + S(x) = 1 + x + x^2/2! + x^3/3! + x^4/4! + 19*x^5/5! + 109*x^6/6! + 739*x^7/7! + 3889*x^8/8! + 35641*x^9/9! + 292681*x^10/10! + 3753721*x^11/11! + 37275121*x^12/12! + 500577499*x^13/13! + 5709311029*x^14/14! + 91718242219*x^15/15! + 1254902705569*x^16/16! + 22737318482161*x^17/17! + 350061261777361*x^18/18! + 6983681901945841*x^19/19! + 120872805166945441*x^20/20! +...
such that C(x) + S(x) = exp( Integral D(x)^3 dx ).
C(x)^2 = 1 + 2*x^2/2! + 8*x^4/4! + 248*x^6/6! + 13952*x^8/8! + 981152*x^10/10! + 128012288*x^12/12! + 21334590848*x^14/14! + 4721317609472*x^16/16! + 1369528258007552*x^18/18! + 487519312215277568*x^20/20! + 212815485425900238848*x^22/22! + 111362541450468672929792*x^24/24! + 68655437948261593572810752*x^26/26! +...
such that C(x)^2 = 1 + S(x)^2.
D(x)^2 = 1 + 12*x^4/4! + 240*x^6/6! + 11088*x^8/8! + 1067520*x^10/10! + 120702912*x^12/12! + 20731576320*x^14/14! + 4706356447488*x^16/16! + 1338363800125440*x^18/18! + 482064458680691712*x^20/20! + 210556245001175040000*x^22/22! + 110103167770187282239488*x^24/24! + 68059391373987458643394560*x^26/26! +...
D(x)^3 = 1 + 18*x^4/4! + 360*x^6/6! + 20412*x^8/8! + 2054880*x^10/10! + 246667608*x^12/12! + 45345998880*x^14/14! + 10711766694672*x^16/16! + 3182147454332160*x^18/18! + 1190153458696009248*x^20/20! + 536990828063228035200*x^22/22! + 289633988053086885277632*x^24/24! + 184083367623416380788963840*x^26/26! +...
D(x)^4 = 1 + 24*x^4/4! + 480*x^6/6! + 32256*x^8/8! + 3344640*x^10/10! + 426353664*x^12/12! + 83091939840*x^14/14! + 20370678153216*x^16/16! + 6310701707796480*x^18/18! + 2444823498480943104*x^20/20! + 1138286636773997568000*x^22/22! + 632578480424353976549376*x^24/24! + 413014933705057627523973120*x^26/26! +...
such that D(x)^4 = 1 + S(x)^4.
D(x)^2 + S(x)^2 = 1 + 2*x^2/2! + 20*x^4/4! + 488*x^6/6! + 25040*x^8/8! + 2048672*x^10/10! + 248715200*x^12/12! + 42066167168*x^14/14! + 9427674056960*x^16/16! + 2707892058132992*x^18/18! + 969583770895969280*x^20/20! + 423371730427075278848*x^22/22! + 221465709220655955169280*x^24/24! + 136714829322249052216205312*x^26/26! +...
sqrt(D(x)^2 + S(x)^2) = 1 + x^2/2! + 7*x^4/4! + 139*x^6/6! + 6913*x^8/8! + 508921*x^10/10! + 57888967*x^12/12! + 9313574419*x^14/14! + 1984690709953*x^16/16! + 547467006437041*x^18/18! + 188946742298214727*x^20/20! + 79783392959511537499*x^22/22! + 40498043815904027702593*x^24/24! + 24314800861291379306213161*x^26/26! +...
such that sqrt(D(x)^2 + S(x)^2) = exp( Integral S(x)*C(x)*D(x) dx ).

Prog

(PARI) {a(n) = my(S=x, C=1, D=1); for(i=0, 2*n, S = intformal( C*D^3 + x*O(x^(2*n))); C = 1 + intformal( S*D^3 ); D = 1 + intformal( C*S^3 )); (2*n-1)!*polcoeff(S, 2*n-1)}
for(n=1, 20, print1(a(n), ", "))
(PARI) /* Explicit formula for the e.g.f. S(x) */
{a(n) = (2*n-1)!*polcoeff( serreverse( intformal( 1/((1+x^2 +x*O(x^(2*n)))^(1/2)*(1+x^4 +x*O(x^(2*n)))^(3/4)) )), 2*n-1)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A280626 (C), A280627 (D), A280628 (C+S), A280629 (sqrt(D^2+S^2)).

Sequence in context: A041684 A157165 A201708 * A183441 A067621 A206308

Adjacent sequences: A280622 A280623 A280624 * A280626 A280627 A280628

Keyword

nonn

Author

Paul D. Hanna, Jan 06 2017

Status

approved