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

1, 0, 0, 2, 0, 20, 40, 182, 3360, 5320, 165480, 1193962, 7681520, 182657020, 1028347320, 21430373342, 296385660480, 2926954283120, 74104327031560, 836520209938322, 16747875178596240, 336576896391971620, 4854838471934266200, 128099320528299728902, 2250302805832188984800, 49372680864950984970520, 1245420698064315926211240

Offset

0,4

Links

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

Formula

E.g.f. D(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.: 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! +...
such that
(1) C(x)^2 - S(x)^2 = 1,
(2) D(x)^3 - S(x)^3 = 1,
where functions S(x) and C(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! +...
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! +...
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(D, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A280620 (S), A280621 (C), A280623 (C+S), A280624 (1/(C-S)).

Sequence in context: A209123 A139003 A264881 * A211880 A365862 A209868

Adjacent sequences: A280619 A280620 A280621 * A280623 A280624 A280625

Keyword

nonn

Author

Paul D. Hanna, Jan 06 2017

Status

approved