A341967 G.f. D(x) satisfies: D(x) = (1 - x*D(x))*(1 - 2*x*D(x))*(1 - 3*x*D(x)) / (1 - 4*x*D(x))^3.

1, 6, 71, 1036, 16850, 292974, 5330003, 100198252, 1930974350, 37944361084, 757400034534, 15314501499440, 313016784476500, 6456745853475198, 134240505281434443, 2810148747648315484, 59181433880505401750, 1252997327856618663540

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

G.f.: D(x) = (1/x) * Series_Reversion( x*(1 - 4*x)^3 / ((1 - x)*(1 - 2*x)*(1 - 3*x)) ).

G.f. D = D(x) and related functions A = A(x), B = B(x), C = C(x), E = E(x), satisfy:

(1.a) A = 1/((1 - 2*x*B)*(1 - 3*x*C)*(1 - 4*x*D)).

(1.b) B = 1/((1 - x*A)*(1 - 3*x*C)*(1 - 4*x*D)).

(1.c) C = 1/((1 - x*A)*(1 - 2*x*B)*(1 - 4*x*D)).

(1.d) D = 1/((1 - x*A)*(1 - 2*x*B)*(1 - 3*x*C)).

(1.e) E = 1/((1 - x*A)*(1 - 2*x*B)*(1 - 3*x*C)*(1 - 4*x*D)).

(1.f) E = (A*B*C*D)^(1/3).

(2.a) A = (1 + 2*x*E)*(1 + 3*x*E)*(1 + 4*x*E).

(2.b) B = (1 + x*E)*(1 + 3*x*E)*(1 + 4*x*E).

(2.c) C = (1 + x*E)*(1 + 2*x*E)*(1 + 4*x*E).

(2.d) D = (1 + x*E)*(1 + 2*x*E)*(1 + 3*x*E).

(2.e) E = (1 + x*E)*(1 + 2*x*E)*(1 + 3*x*E)*(1 + 4*x*E).

(3.a) A = B/(1 - x*B) = C/(1 - 2*x*C) = D/(1 - 3*x*D) = E/(1 + x*E).

(3.b) B = A/(1 + x*A) = C/(1 - x*C) = D/(1 - 2*x*D) = E/(1 + 2*x*E).

(3.c) C = A/(1 + 2*x*A) = B/(1 + x*B) = D/(1 - x*D) = E/(1 + 3*x*E).

(3.d) D = A/(1 + 3*x*A) = B/(1 + 2*x*B) = C/(1 + x*C) = E/(1 + 4*x*E).

(3.e) E = A/(1 - x*A) = B/(1 - 2*x*B) = C/(1 - 3*x*C) = D/(1 - 4*x*D).

(3.f) 1 = (1 + x*A)*(1 - x*B) = (1 + 2*x*A)*(1 - 2*x*C) = (1 + 3*x*A)*(1 - 3*x*D) = (1 + x*B)*(1 - x*C) = (1 + 2*x*B)*(1 - 2*x*D) = (1 + x*C)*(1 - x*D).

(3.g) 1 = (1 - x*A)*(1 + x*E) = (1 - 2*x*B)*(1 + 2*x*E) = (1 - 3*x*C)*(1 + 3*x*E) = (1 - 4*x*D)*(1 + 4*x*E).

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

(4.b) B = (1 - x^2*B^2)*(1 + 2*x*B)/(1 - 2*x*B)^3.

(4.c) C = (1 - x^2*C^2)*(1 - 2*x*C)/(1 - 3*x*C)^3.

(4.d) D = (1 - x*D)*(1 - 2*x*D)*(1 - 3*x*D)/(1 - 4*x*D)^3.

(4.e) E = (1 + x*E)*(1 + 2*x*E)*(1 + 3*x*E)*(1 + 4*x*E).

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

(5.b) B = (1/x)*Series_Reversion( x*(1 - 2*x)^3 / ((1 - x^2)*(1 + 2*x)) ).

(5.c) C = (1/x)*Series_Reversion( x*(1 - 3*x)^3 / ((1 - x^2)*(1 - 2*x)) ).

(5.d) D = (1/x)*Series_Reversion( x*(1 - 4*x)^3 / ((1 - x)*(1 - 2*x)*(1 - 3*x)) ).

(5.e) E = (1/x)*Series_Reversion( x / ((1 + x)*(1 + 2*x)*(1 + 3*x)*(1 + 4*x)) ).

Example

G.f. D(x) = 1 + 6*x + 71*x^2 + 1036*x^3 + 16850*x^4 + 292974*x^5 + 5330003*x^6 + 100198252*x^7 + 1930974350*x^8 + 37944361084*x^9 + ...
such that D(x) = 1/((1 - x*A(x))*(1 - 2*x*B(x))*(1 - 3*x*C(x))) where
A(x) = 1 + 9*x + 116*x^2 + 1759*x^3 + 29240*x^4 + 515586*x^5 + 9472148*x^6 + 179354443*x^7 + 3475611320*x^8 + 68596806526*x^9 + ...
B(x) = 1 + 8*x + 99*x^2 + 1472*x^3 + 24190*x^4 + 423352*x^5 + 7736687*x^6 + 145920704*x^7 + 2819185470*x^8 + 55507755152*x^9 + ...
C(x) = 1 + 7*x + 84*x^2 + 1233*x^3 + 20120*x^4 + 350558*x^5 + 6386772*x^6 + 120190501*x^7 + 2318113560*x^8 + 45580597858*x^9 + ...
RELATED SERIES.
E(x) = (A(x)*B(x)*C(x)*D(x))^(1/3) = 1 + 10*x + 135*x^2 + 2100*x^3 + 35474*x^4 + 632450*x^5 + 11712915*x^6 + 223143700*x^7 + 4345018254*x^8 + ...
E(x)^2 = 1 + 20*x + 370*x^2 + 6900*x^3 + 131173*x^4 + 2541380*x^5 + 50062810*x^6 + 1000298000*x^7 + 20230092234*x^8 + 413392833400*x^9 + ...
E(x)^3 = A(x)*B(x)*C(x)*D(x) = 1 + 30*x + 705*x^2 + 15400*x^3 + 327597*x^4 + 6903540*x^5 + 145162260*x^6 + 3055654800*x^7 + 64487256390*x^8 + ...
B(x)*C(x)*D(x) = E(x)^2 + x*E(x)^3 = 1 + 21*x + 400*x^2 + 7605*x^3 + 146573*x^4 + 2868977*x^5 + 56966350*x^6 + 1145460260*x^7 + ...
A(x)*C(x)*D(x) = E(x)^2 + 2*x*E(x)^3 = 1 + 22*x + 430*x^2 + 8310*x^3 + 161973*x^4 + 3196574*x^5 + 63869890*x^6 + 1290622520*x^7 + ...
A(x)*B(x)*D(x) = E(x)^2 + 3*x*E(x)^3 = 1 + 23*x + 460*x^2 + 9015*x^3 + 177373*x^4 + 3524171*x^5 + 70773430*x^6 + 1435784780*x^7 + ...
A(x)*B(x)*C(x) = E(x)^2 + 4*x*E(x)^3 = 1 + 24*x + 490*x^2 + 9720*x^3 + 192773*x^4 + 3851768*x^5 + 77676970*x^6 + 1580947040*x^7 + ...

Prog

(PARI) {d(n) = my(A=1, B=1, C=1, D=1, E=1); for(i=1, n,
A = 1/((1-2*x*B)*(1-3*x*C)*(1-4*x*D) +x*O(x^n));
B = 1/((1-1*x*A)*(1-3*x*C)*(1-4*x*D) +x*O(x^n));
C = 1/((1-1*x*A)*(1-2*x*B)*(1-4*x*D) +x*O(x^n));
D = 1/((1-1*x*A)*(1-2*x*B)*(1-3*x*C) +x*O(x^n)); );
E = (A*B*C*D)^(1/3); polcoeff(D, n)}
for(n=0, 20, print1(d(n), ", "))
(PARI) /* By Series Reversion: */
{d(n) = my(D = 1/x*serreverse( x*(1 - 4*x)^3 / ((1 - x)*(1 - 2*x)*(1 - 3*x) +x*O(x^n)) )); polcoeff(D, n)}
for(n=0, 20, print1(d(n), ", "))

Crossrefs

Cf. A341964 (A(x)), A341965 (B(x)), A341966 (C(x)), A341968 (E(x)).

Sequence in context: A023038 A092660 A186658 * A371366 A092085 A028844

Adjacent sequences: A341964 A341965 A341966 * A341968 A341969 A341970

Keyword

nonn

Author

Paul D. Hanna, Mar 03 2021

Status

approved