A280621 E.g.f. C(x) satisfies: C(x)^2 - S(x)^2 = 1 and D(x)^3 - S(x)^3 = 1, where functions S(x) and D(x) are described by A280620 and A280622, respectively.

1, 0, 1, 0, 1, 20, 1, 420, 1841, 7140, 190001, 555940, 12774881, 141201060, 946212001, 25228809060, 202847031121, 3740829095780, 66881800434001, 733452394335780, 19147386646802561, 273971147946411300, 5322649952666824001, 124410236296546608100, 2010649742531247641201, 53865539929721514523620, 1113617365649653498950001

Offset

0,6

Links

Table of n, a(n) for n=0..26.

Formula

E.g.f. C(x), 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^3 - S^3 = 1.

Integrals.

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

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

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

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

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

Exponential.

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

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

(3.c) C = cosh( Integral D^2 dx ).

(3.d) S = sinh( Integral D^2 dx ).

Derivatives.

(4.a) S' = C*D^2.

(4.b) C' = S*D^2.

(4.c) D' = C*S^2.

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

(4.e) (D' - S')/(D - S) = -(D + S) * C.

Example

E.g.f.: C(x) = 1 + x^2/2! + x^4/4! + 20*x^5/5! + x^6/6! + 420*x^7/7! + 1841*x^8/8! + 7140*x^9/9! + 190001*x^10/10! + 555940*x^11/11! + 12774881*x^12/12! + 141201060*x^13/13! + 946212001*x^14/14! + 25228809060*x^15/15! + 202847031121*x^16/16! + 3740829095780*x^17/17! + 66881800434001*x^18/18! +...
such that
(1) C(x)^2 - S(x)^2 = 1,
(2) D(x)^3 - S(x)^3 = 1,
where functions S(x) and D(x) are illustrated below.
RELATED SERIES.
S(x) = x + x^3/3! + 4*x^4/4! + x^5/5! + 100*x^6/6! + 161*x^7/7! + 1764*x^8/8! + 22001*x^9/9! + 49700*x^10/10! + 1649921*x^11/11! + 10057124*x^12/12! + 105372001*x^13/13! + 2044251300*x^14/14! + 12879413281*x^15/15! + 315936586084*x^16/16! + 3892292034001*x^17/17! + 49987743460900*x^18/18! +...
D(x) = 1 + 2*x^3/3! + 20*x^5/5! + 40*x^6/6! + 182*x^7/7! + 3360*x^8/8! + 5320*x^9/9! + 165480*x^10/10! + 1193962*x^11/11! + 7681520*x^12/12! + 182657020*x^13/13! + 1028347320*x^14/14! + 21430373342*x^15/15! + 296385660480*x^16/16! + 2926954283120*x^17/17! + 74104327031560*x^18/18! +...
S(x)^2 = 2*x^2/2! + 8*x^4/4! + 40*x^5/5! + 32*x^6/6! + 1680*x^7/7! + 3808*x^8/8! + 49560*x^9/9! + 646912*x^10/10! + 2192960*x^11/11! + 65759008*x^12/12! + 475555080*x^13/13! + 5786067392*x^14/14! + 114473289840*x^15/15! + 891694992608*x^16/16! + 21934824868600*x^17/17! + 298444830841472*x^18/18! +...
such that C(x)^2 = 1 + S(x)^2.
D(x)^2 = 1 + 4*x^3/3! + 40*x^5/5! + 160*x^6/6! + 364*x^7/7! + 11200*x^8/8! + 24080*x^9/9! + 519120*x^10/10! + 5344724*x^11/11! + 27288800*x^12/12! + 752580920*x^13/13! + 5142016880*x^14/14! + 86718961084*x^15/15! + 1483995676800*x^16/16! + 13774998062560*x^17/17! + 356032443815440*x^18/18! +...
such that D(x)^2 = S'(x)/C(x) = C'(x)/S(x).
S(x)^3 = 6*x^3/3! + 60*x^5/5! + 360*x^6/6! + 546*x^7/7! + 23520*x^8/8! + 69720*x^9/9! + 1060920*x^10/10! + 14669886*x^11/11! + 67692240*x^12/12! + 1957699380*x^13/13! + 16377040680*x^14/14! + 228086752026*x^15/15! + 4642872212160*x^16/16! + 43205148425040*x^17/17! + 1084693228559640*x^18/18! +...
such that D(x)^3 = 1 + S(x)^3.
C(x) + S(x) = 1 + x + x^2/2! + x^3/3! + 5*x^4/4! + 21*x^5/5! + 101*x^6/6! + 581*x^7/7! + 3605*x^8/8! + 29141*x^9/9! + 239701*x^10/10! + 2205861*x^11/11! + 22832005*x^12/12! + 246573061*x^13/13! + 2990463301*x^14/14! + 38108222341*x^15/15! + 518783617205*x^16/16! + 7633121129781*x^17/17! + 116869543894901*x^18/18! + 1918479435194021*x^19/19! + 33025793008567205*x^20/20! + 595507639576003301*x^21/21! +...
such that C(x) + S(x) = exp( Integral D(x)^2 dx ).
1/(D(x) - S(x)) = 1 + x + 2*x^2/2! + 5*x^3/3! + 20*x^4/4! + 81*x^5/5! + 452*x^6/6! + 2765*x^7/7! + 19460*x^8/8! + 156121*x^9/9! + 1368052*x^10/10! + 13327125*x^11/11! + 141326500*x^12/12! + 1616350561*x^13/13! + 20040895252*x^14/14! + 264759181085*x^15/15! + 3740415315140*x^16/16! + 56164918735401*x^17/17! + 891038080096052*x^18/18! + 14957788277468645*x^19/19! + 263869908657105380*x^20/20! + 4889789934063374641*x^21/21! +...
such that 1/(D(x) - S(x)) = exp( Integral (D(x) + S(x)) * C(x) dx.

Prog

(PARI) {a(n) = my(S=x, C=1, D=1); for(i=1, n, S = intformal( C*D^2 + x*O(x^n)); C = 1 + intformal( S*D^2 ); D = 1 + intformal( C*S^2 )); n!*polcoeff(C, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A280620 (S), A280622 (D), A280623 (C+S), A280624 (1/(C-S)).

Sequence in context: A164812 A332258 A327023 * A223522 A040395 A040394

Adjacent sequences: A280618 A280619 A280620 * A280622 A280623 A280624

Keyword

nonn

Author

Paul D. Hanna, Jan 06 2017

Status

approved