login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A340942 Row k = 1 of rectangular table A340940. 10
1, 1, 2, 9, 46, 253, 1467, 8842, 54878, 348489, 2254007, 14799922, 98405915, 661315560, 4485060016, 30660329139, 211061833430, 1461899228702, 10181435691655, 71258997482752, 500958491353507, 3536041471033681, 25051475034240297 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The g.f. A(x) of this sequence employs the related identities:

(1) Sum_{n>=0} (n+1) * p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n)^2,

(2) Sum_{n>=0} (n+1) * p^n/(1 - q*r^n)^2 = Sum_{n>=0} (n+1) * q^n/(1 - p*r^n)^2.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

FORMULA

G.f. A(x) satisfies:

(1) A(x) = P(x)/Q(x) where

P(x) = Sum_{n>=0} (n+1)*x^n * A(x)^(2*n) / (1 - x*A(x)^n)^2,

Q(x) = Sum_{n>=0} (n+1)*x^n * A(x)^n / (1 - x*A(x)^(n+1)).

(2) A(x) = P(x)/Q(x) where

P(x) = Sum_{n>=0} (n+1)*x^n / (1 - x*A(x)^(n+2))^2,

Q(x) = Sum_{n>=0} x^n * A(x)^n / (1 - x*A(x)^(n+1))^2.

EXAMPLE

G.f. A(x) = 1 + x + 2*x^2 + 9*x^3 + 46*x^4 + 253*x^5 + 1467*x^6 + 8842*x^7 + 54878*x^8 + 348489*x^9 + 2254007*x^10 + 14799922*x^11 + 98405915*x^12 + ...

such that A(x) = P(x)/Q(x) where

P(x) = 1/(1 - x)^2 + 2*x*A(x)^2/(1 - x*A(x))^2 + 3*x^2*A(x)^4/(1 - x*A(x)^2)^2 + 4*x^3*A(x)^6/(1 - x*A(x)^3)^2 + 5*x^4*A(x)^8/(1 - x*A(x)^4)^2 + ...

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

equivalently,

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

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

explicitly,

P(x) = 1 + 4*x + 14*x^2 + 54*x^3 + 241*x^4 + 1214*x^5 + 6651*x^6 + 38566*x^7 + 232727*x^8 + 1446432*x^9 + 9196742*x^10 + 59545914*x^11 + 391304285*x^12 + ...

Q(x) = 1 + 3*x + 9*x^2 + 30*x^3 + 120*x^4 + 562*x^5 + 2939*x^6 + 16523*x^7 + 97551*x^8 + 596461*x^9 + 3744416*x^10 + 23996814*x^11 + 156370334*x^12 + ...

PROG

(PARI) {a(n) = my(A=1+x+x*O(x^n), P=1, Q=1);

for(i=0, n,

P = sum(m=0, n, (m+1)*x^m*A^(2*m)/(1 - x*A^m + x*O(x^n))^2 );

Q = sum(m=0, n, (m+1)*x^m*A^m/(1 - x*A^(m+1) + x*O(x^n)) );

A = P/Q); polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n) = my(A=1+x+x*O(x^n), P=1, Q=1);

for(i=0, n,

P = sum(m=0, n, (m+1)*x^m/(1 - x*A^(m+2) + x*O(x^n))^2 );

Q = sum(m=0, n, x^m*A^m/(1 - x*A^(m+1) + x*O(x^n))^2 );

A = P/Q); polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) /* As generated by Finite Differences in the columns of table A340940 */

{A340940(k, n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); H=A; A=concat(A, 0);

H[#A-1] = -polcoeff( sum(m=0, #A, x^m/(1 - x*Ser(A)^(m+k)) ) - sum(m=0, #A, x^m*Ser(A)^m/(1 - x*Ser(A)^(k*m+k-1)) ), #A)/(k-1); A=H); A[n+1] }

for(n=0, 20, if(n<2, print1("1, "), print1( Vec(-(x-1)^n*Ser(vector(n+1, k, A340940(k+1, n))))[n], ", ") ))

CROSSREFS

Cf. A340940, A340941, A340894, A340895.

Sequence in context: A168431 A036726 A219197 * A270386 A181997 A020053

Adjacent sequences:  A340939 A340940 A340941 * A340943 A340944 A340945

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 03 2021

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 25 15:22 EDT 2021. Contains 346290 sequences. (Running on oeis4.)