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

1, 0, 1, 4, 1, 100, 161, 1764, 22001, 49700, 1649921, 10057124, 105372001, 2044251300, 12879413281, 315936586084, 3892292034001, 49987743460900, 1185027040858241, 13878406361764644, 321536491629592001, 6033371812540110500, 100320731761080176801, 2657253524511363224804, 47170890863193411630001, 1155602674484930034008100, 28284869824153625118984961

Offset

1,4

Links

Table of n, a(n) for n=1..27.

Formula

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

E.g.f. S(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.: 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! +...
such that
(1) C(x)^2 - S(x)^2 = 1,
(2) D(x)^3 - S(x)^3 = 1,
where functions C(x) and D(x) are illustrated below.
RELATED SERIES.
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! +...
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^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(S, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* Explicit formula for the e.g.f. S(x) */
{a(n) = n!*polcoeff( serreverse( intformal( 1/((1+x^2 +x*O(x^n))^(1/2)*(1+x^3 +x*O(x^n))^(2/3)) )), n)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A280621 (C), A280622 (D), A280623 (C+S), A280624 (1/(C-S)).

Sequence in context: A255192 A299582 A211341 * A262405 A152841 A125083

Adjacent sequences: A280617 A280618 A280619 * A280621 A280622 A280623

Keyword

nonn

Author

Paul D. Hanna, Jan 06 2017

Status

approved