A301312 -id:A301312 - cp's OEIS frontend

A301307 G.f.: Sum_{n>=0} (1 + (1+x)^n)^n / 3^(n+1).

1, 5, 98, 3239, 150176, 8958473, 653364947, 56325265925, 5603297711741, 631787569243643, 79620187792726844, 11090608163844996365, 1692024644610151317068, 280593919265423518611017, 50255068227934275890880470, 9667645123441963396364779439, 1988058929295585346059732920903, 435204469378969786061222253686549, 101044871217450582545711556498557285

Offset: 0

Paul D. Hanna, Mar 21 2018

nonn

			G.f.: A(x) = 1 + 5*x + 98*x^2 + 3239*x^3 + 150176*x^4 + 8958473*x^5 + 653364947*x^6 + 56325265925*x^7 + 5603297711741*x^8 + ...
such that
A(x) = 1/3 + (1 + (1+x))/3^2 + (1 + (1+x)^2)^2/3^3 + (1 + (1+x)^3)^3/3^4 + (1 + (1+x)^4)^4/3^5 + (1 + (1+x)^5)^5/3^6 + (1 + (1+x)^6)^6/3^7 + ...
Also,
A(x) = 1/2 + (1+x)/(3 - (1+x))^2 + (1+x)^4/(3 - (1+x)^2)^3 + (1+x)^9/(3 - (1+x)^3)^4 + (1+x)^16/(3 - (1+x)^4)^5 + (1+x)^25/(3 - (1+x)^5)^6 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..195

Cf. A301312.

G.f.: Sum_{n>=0} (1+x)^(n^2) / (3 - (1+x)^n)^(n+1).
G.f.: Sum_{n>=0} Sum_{k=0..n} binomial(n,k) * (1 + x)^(n*k) / 3^(n+1).
a(n) = Sum_{j>=0} Sum_{k=0..j} binomial(j, k) * binomial(j*k, n) / 3^(j+1).
a(n) ~ c * d^n * n^n, where d = 4.88100884940898277361223446294548499145552953621086588549015342712172151... and c = 1.0401387348267211789387929284813380774183533880659572052994951... - Vaclav Kotesovec, Mar 22 2018

A301465 G.f.: Sum_{n>=0} ( (1+x)^n + (1+2x)^n )^n x^n.

Original entry on oeis.org

1, 2, 7, 32, 180, 1142, 8162, 63807, 541202, 4926969, 47799429, 491254999, 5322191407, 60533874755, 720300053902, 8939841040157, 115427887156492, 1546880293065700, 21472465304481214, 308173758866549496, 4565492159926897329, 69713352146246150541, 1095715822838490323551, 17705238547423980566108, 293792275476690070452598

Offset: 0

Paul D. Hanna, Mar 24 2018

nonn

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 32*x^3 + 180*x^4 + 1142*x^5 + 8162*x^6 + 63807*x^7 + 541202*x^8 + 4926969*x^9 + 47799429*x^10 + ...
such that
A(x) = 1 + ((1+x) + (1+2*x))*x + ((1+x)^2 + (1+2*x)^2)^2*x^2 + ((1+x)^3 + (1+2*x)^3)^3*x^3 + ((1+x)^4 + (1+2*x)^4)^4*x^4 + ((1+x)^5 + (1+2*x)^5)^5*x^5 + ...

Cf. A301306, A301312, A301308.

{a(n) = my(A=1); A = sum(k=0, n, ((1+x)^k + (1+2*x)^k +x*O(x^n))^k * x^k ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (1+x)^(n*(n-k)) * (1+2*x)^(n*k).

cp's OEIS Frontend

A301307 G.f.: Sum_{n>=0} (1 + (1+x)^n)^n / 3^(n+1).

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A301465 G.f.: Sum_{n>=0} ( (1+x)^n + (1+2x)^n )^n x^n.

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cp's OEIS Frontend

A301307 G.f.: Sum_{n>=0} (1 + (1+x)^n)^n / 3^(n+1).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A301465 G.f.: Sum_{n>=0} ( (1+x)^n + (1+2*x)^n )^n * x^n.

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Author

Keywords

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Programs

PARI

Formula

A301465 G.f.: Sum_{n>=0} ( (1+x)^n + (1+2x)^n )^n x^n.