A340942 Row k = 1 of rectangular table A340940.

1, 1, 2, 9, 46, 253, 1467, 8842, 54878, 348489, 2254007, 14799922, 98405915, 661315560, 4485060016, 30660329139, 211061833430, 1461899228702, 10181435691655, 71258997482752, 500958491353507, 3536041471033681, 25051475034240297

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