A318644 -id:A318644 - cp's OEIS frontend

A320951 G.f.: A(x) satisfies: A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)) / A(x)^n.

1, 1, 2, 3, 9, 28, 110, 485, 2358, 12486, 70726, 425747, 2702837, 18004835, 125337381, 908737863, 6843536374, 53407750147, 431075414218, 3592384229312, 30862831600689, 272976843937138, 2482698463801148, 23192576636266041, 222310388884578760, 2184486850658804107, 21985733344615744620, 226455749821063728474, 2385331864619907236147, 25676170688883138634306, 282253492062060457638824

Offset: 0

Paul D. Hanna, Nov 20 2018

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 9*x^4 + 28*x^5 + 110*x^6 + 485*x^7 + 2358*x^8 + 12486*x^9 + 70726*x^10 + 425747*x^11 + 2702837*x^12 + ...
such that
A(x) = 1 + x*(1+x)^2/A(x) + x^2*(1+x)^6/A(x)^2 + x^3*(1+x)^12/A(x)^3 + x^4*(1+x)^20/A(x)^4 + x^5*(1+x)^30/A(x)^5 + ...
Also
1 + x = 1 + x/A(x) + x^2*(1+x)^2/A(x)^2 + x^3*(1+x)^6/A(x)^3 + x^4*(1+x)^12/A(x)^4 + x^5*(1+x)^20/A(x)^5 + x^6*(1+x)^30/A(x)^6 + ...
RELATED SERIES.
Sum_{n>=0} x^n * (1+x)^(n^2) / A(x)^n = 1 + x + x^2 + x^3 + 3*x^4 + 8*x^5 + 32*x^6 + 135*x^7 + 649*x^8 + 3381*x^9 + 18894*x^10 + 112382*x^11 + 705174*x^12 + ...
A(A(x)-1) = 1 + x + 4*x^2 + 14*x^3 + 56*x^4 + 251*x^5 + 1239*x^6 + 6627*x^7 + 38112*x^8 + 233692*x^9 + 1517788*x^10 + 10384824*x^11 + ...
where A(A(x)-1) = Sum_{n>=0} (A(x)-1)^n * A(x)^(n*(n+1)) / A(A(x)-1)^n.

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A318644, A303058.

{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(n=0, #A, ((1+x)^n +x*O(x^#A))^(n+1) * x^n/Ser(A)^n ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)) / A(x)^n.
(2) 1 + x = Sum_{n>=0} x^n * (1+x)^(n*(n-1)) / A(x)^n.
(3) A(x) = Sum_{n>=0} (A(x)-1)^n * A(x)^(n*(n-1)) / A(A(x)-1)^n.

A325155 G.f. A(x) satisfies: x = Sum_{n>=1} x^n * (1+x)^(n^2/2) / A(x)^(n/2).

Original entry on oeis.org

1, 3, 5, 7, 11, 21, 49, 133, 408, 1376, 5020, 19564, 80741, 350551, 1593066, 7547792, 37163568, 189662934, 1001046684, 5453972462, 30622950955, 176942133603, 1050773432990, 6405898358012, 40048848677954, 256521565555908, 1681897617101795, 11278819380424173, 77301464920178158, 541084956406886214, 3865540113371340736, 28167799470180443028, 209238063076396838375, 1583562040116769584091, 12204247180832799551059, 95731651337427271893873

Offset: 0

Paul D. Hanna, Apr 06 2019

nonn

			G.f.: A(x) = 1 + 3*x + 5*x^2 + 7*x^3 + 11*x^4 + 21*x^5 + 49*x^6 + 133*x^7 + 408*x^8 + 1376*x^9 + 5020*x^10 + 19564*x^11 + 80741*x^12 + ...
such that
A(x) = 1 + x*(1+x)^(1/2)/A(x)^(1/2) + x^2*(1+x)^2/A(x) + x^3*(1+x)^(9/2)/A(x)^(3/2) + x^4*(1+x)^8/A(x)^2 + x^5*(1+x)^(25/2)/A(x)^(5/2) + x^6*(1+x)^18/A(x)^3 + x^7*(1+x)^(49/2)/A(x)^(7/2) + x^8*(1+x)^32/A(x)^4 + ...
Note that
sqrt(A(x))*sqrt(1+x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 11*x^6 + 32*x^7 + 106*x^8 + 376*x^9 + 1433*x^10 + 5782*x^11 + ... + A318644(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A318644, A303058.

{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = 2*polcoeff( sum(n=0,#A+1, x^n*(1+x +x*O(x^#A))^(n^2/2) / Ser(A)^(n/2) ),#A)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) satisfies the following identities.
(1) 1 + x = Sum_{n>=0} x^n * (1+x)^(n^2/2) / A(x)^(n/2).
(2) 1 + x = 1/(1 - q*x/(sqrt(A(x)) - q*(q^2-1)*x/(1 - q^5*x/(sqrt(A(x)) - q^3*(q^4-1)*x/(1 - q^9*x/(sqrt(A(x)) - q^5*(q^6-1)*x/(1 - q^13*x/(sqrt(A(x)) - q^7*(q^8-1)*x/(1 - ...))))))))), where q = sqrt(1+x), a continued fraction due to a partial elliptic theta function identity.
(3) 1 + x = Sum_{n>=0} x^n * (1+x)^(n/2) / A(x)^(n/2) * Product_{k=1..n} (sqrt(A(x)) - x*sqrt(1+x)^(4*k-3)) / (sqrt(A(x)) - x*sqrt(1+x)^(4*k-1)), due to a q-series identity.
(4) A(x) = (1+x)*G(x)^2 where G(x) is the g.f. of A318644.

cp's OEIS Frontend

A320951 G.f.: A(x) satisfies: A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)) / A(x)^n.

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