A295543 G.f. A(x) satisfies: x = 1 - A(x) - A(x)^3 + A(x)^5.

1, 1, -7, 89, -1405, 24805, -468958, 9285898, -190113771, 3991746445, -85484876191, 1859994631209, -41001506969059, 913746936612503, -20552899913260580, 465989150086168796, -10638491148102729290, 244351953995837766480, -5642578772157047006865, 130921289778460781274315, -3050692106182384617622200, 71360604327670212927839280, -1675070936638578659774532300

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

G.f. A(x) satisfies:

(1) A(x) = 1 + Series_Reversion( x + 7*x^2 + 9*x^3 + 5*x^4 + x^5 ).

(2) 1/A(x) = 1 - Series_Reversion( x*(1 + 3*x - 6*x^2 + 4*x^3 - x^4)/(1-x)^5 ).

(3) 1 + x = A( x + 7*x^2 + 9*x^3 + 5*x^4 + x^5 ).

(4) 1/(1-x) = A ( x*(1 + 3*x - 6*x^2 + 4*x^3 - x^4)/(1-x)^5 ).

(5) x = (1 - A(x)) * (1 - A(x)^3 - A(x)^4).

Example

G.f.: A(x) = 1 + x - 7*x^2 + 89*x^3 - 1405*x^4 + 24805*x^5 - 468958*x^6 + 9285898*x^7 - 190113771*x^8 + 3991746445*x^9 - 85484876191*x^10 +...
such that x = 1 - A(x) - A(x)^3 + A(x)^5.
RELATED SERIES.
x/(1 - A(x)) = -1 - 7*x + 40*x^2 - 502*x^3 + 7896*x^4 - 139173*x^5 + 2628727*x^6 - 52020689*x^7 + 1064597341*x^8 - 22346116664*x^9 +...
which equals 1 - A(x)^3 - A(x)^4.
1/A(x) = 1 - x + 8*x^2 - 104*x^3 + 1654*x^4 - 29304*x^5 + 555141*x^6 - 11006891*x^7 + 225555857*x^8 - 4739130265*x^9 +...
which equals 1 - Series_Reversion( x*(2-x)/(1-x)^5 - x/(1-x) ).
A(x)^2 = 1 + 2*x - 13*x^2 + 164*x^3 - 2583*x^4 + 45554*x^5 - 860715*x^6 + 17036520*x^7 - 348701019*x^8 + 7320086202*x^9 +...
A(x)^3 = 1 + 3*x - 18*x^2 + 226*x^3 - 3555*x^4 + 62661*x^5 - 1183567*x^6 + 23422137*x^7 - 479333574*x^8 + 10061344340*x^9 +...
A(x)^4 = 1 + 4*x - 22*x^2 + 276*x^3 - 4341*x^4 + 76512*x^5 - 1445160*x^6 + 28598552*x^7 - 585263767*x^8 + 12284772324*x^9 +...
A(x)^5 = 1 + 5*x - 25*x^2 + 315*x^3 - 4960*x^4 + 87466*x^5 - 1652525*x^6 + 32708035*x^7 - 669447345*x^8 + 14053090785*x^9 +...
where A(x) = 1 - x - A(x)^3 + A(x)^5.

Prog

(PARI) {a(n) = my(A=1); A = 1 - serreverse( subst(1 - x - x^3 + x^5, x, 1-x) +x^2*O(x^n) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); A = 1/(1 - serreverse( x*(2-x)/(1-x)^5 - x/(1-x) +x^2*O(x^n)) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A295544.

Sequence in context: A069662 A359646 A371386 * A062747 A099719 A142995

Adjacent sequences: A295540 A295541 A295542 * A295544 A295545 A295546

Keyword

sign

Author

Paul D. Hanna, Nov 24 2017

Status

approved