cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A369674 a(n) = Product_{k=0..n} (3^k + 3^(n-k)).

Original entry on oeis.org

2, 16, 600, 112896, 108928800, 544431476736, 14105702277360000, 1900051576637594075136, 1328360485647389567734080000, 4830166933124609654538067824869376, 91168969237139220357818392868757600000000, 8950497893393998236587417126220897399198550327296
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2024

Keywords

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 3)*(3 + 1) = 16;
a(2) = (1 + 3^2)*(3 + 3)*(3^2 + 1) = 600;
a(3) = (1 + 3^3)*(3 + 3^2)*(3^2 + 3)*(3^3 + 1) = 112896;
a(4) = (1 + 3^4)*(3 + 3^3)*(3^2 + 3^2)*(3^3 + 3)*(3^4 + 1) = 108928800;
a(5) = (1 + 3^5)*(3 + 3^4)*(3^2 + 3^3)*(3^3 + 3^2)*(3^4 + 3)*(3^5 + 1) = 544431476736;
...
RELATED SERIES.
Let F(x) be the g.f. of A369557, then
F(1/3) = 2 + 16/3^2 + 600/3^6 + 112896/3^12 + 108928800/3^20 + 544431476736/3^30 + 14105702277360000/3^42 + ... + a(n)/3^(n*(n+1)) + ... = 4.847274134844057155467506697748724715389597193...

Crossrefs

Cf. A369673, A369675, A369676, A369557.

Programs

  • PARI
    {a(n) = prod(k=0, n, 3^k + 3^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (3^k + 3^(n-k)).
a(n) = 3^(n*(n+1)) * Product_{k=0..n} (1/3^k + 1/3^(n-k)).
a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 1/3^(n-2*k)).
From Vaclav Kotesovec, Feb 07 2024: (Start)
a(n) ~ c * 3^(3*n^2/4 + n), where
c = 2.538295806020848... = QPochhammer(-1, 1/9)^2/2 if n is even and
c = 2.539569717896307... = 3^(1/4) * QPochhammer(-3, 1/9)^2 / 16 if n is odd. (End)

A369673 a(n) = Product_{k=0..n} (2^k + 2^(n-k)).

Original entry on oeis.org

2, 9, 100, 2916, 231200, 50808384, 31258240000, 54112148361216, 264265663201280000, 3645603832850650497024, 142153785549232537600000000, 15673043740102659990892604030976, 4886752115388739132874502963200000000, 4309225323078788454199311474023086952546304, 10747393363422494556085100202291563069440000000000
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2024

Keywords

Comments

Conjectures:
(C.1) a(n) is a square iff n is not divisible by 4.
(C.2) a(2*n+1) is not divisible by 5 for n >= 0.
(C.3) exponent of highest power of 5 dividing a(4*n) = 2*A127428(n).
(C.4) exponent of highest power of 5 dividing a(4*n+2) = 2*A127428(n+1).
From Vaclav Kotesovec, Feb 07 2024: (Start)
For q > 1, Product_{k=0..n} (q^k + q^(n-k)) ~ c * q^(3*n^2/4 + n), where
c = QPochhammer(-1, 1/q^2)^2/2 if n is even and
c = q^(1/4) * QPochhammer(-q, 1/q^2)^2 / (q + 1)^2 if n is odd.
c_even / c_odd = EllipticTheta[2, 0, 1/q] / EllipticTheta[3, 0, 1/q] = JacobiTheta2(0, 1/q) / JacobiTheta3(0, 1/q). (End)

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 2)*(2 + 1) = 9;
a(2) = (1 + 2^2)*(2 + 2)*(2^2 + 1) = 100;
a(3) = (1 + 2^3)*(2 + 2^2)*(2^2 + 2)*(2^3 + 1) = 2916;
a(4) = (1 + 2^4)*(2 + 2^3)*(2^2 + 2^2)*(2^3 + 2)*(2^4 + 1) = 231200;
a(5) = (1 + 2^5)*(2 + 2^4)*(2^2 + 2^3)*(2^3 + 2^2)*(2^4 + 2)*(2^5 + 1) = 50808384;
a(6) = (1 + 2^6)*(2 + 2^5)*(2^2 + 2^4)*(2^3 + 2^3)*(2^4 + 2^2)*(2^5 + 2)*(2^6 + 1) = 31258240000;
...
RELATED SERIES.
Let F(x) be the g.f. of A369557, then
F(1/2) = 2 + 9/2^2 + 100/2^6 + 2916/2^12 + 231200/2^20 + 50808384/2^30 + 31258240000/2^42 + 54112148361216/2^56 + ... + a(n)/2^(n*(n+1)) + ... = 6.800139835051923542641455169580774467247971025...

Crossrefs

Cf. A369674, A369675, A369676, A369557, A127428.

Programs

  • PARI
    {a(n) = prod(k=0,n, 2^k + 2^(n-k))}
for(n=0,15, print1(a(n),", "))

Formula

a(n) = Product_{k=0..n} (2^k + 2^(n-k)).
a(n) = 2^(n*(n+1)) * Product_{k=0..n} (1/2^k + 1/2^(n-k)).
a(n) = 2^(n*(n+1)/2)*QPochhammer(-2^n, 1/4, 1 + n). - Stefano Spezia, Feb 06 2024
From Vaclav Kotesovec, Feb 07 2024: (Start)
a(n) ~ c * 2^(3*n^2/4 + n), where
c = 3.676982087353134... = QPochhammer(-1, 1/4)^2/2 if n is even and
c = 3.676991719144565... = 2^(1/4) * QPochhammer(-2, 1/4)^2 / 9 if n is odd.
c_even / c_odd = EllipticTheta[2, 0, 1/2] / EllipticTheta[3, 0, 1/2] = JacobiTheta2(0, 1/2) / JacobiTheta3(0, 1/2) = 0.9999973805240351337720926619... (End)

A369675 a(n) = Product_{k=0..n} (4^k + 4^(n-k)).

Original entry on oeis.org

2, 25, 2312, 1690000, 9773138432, 454542400000000, 167983232813812416512, 499835663627223040000000000, 11821129880009981801801971612516352, 2251076882713432721110048178176000000000000, 3407215210591493267547957182357614317126952945713152, 41525058946342607360045945411073338768005424742400000000000000
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2024

Keywords

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 4)*(4 + 1) = 25;
a(2) = (1 + 4^2)*(4 + 4)*(4^2 + 1) = 2312;
a(3) = (1 + 4^3)*(4 + 4^2)*(4^2 + 4)*(4^3 + 1) = 1690000;
a(4) = (1 + 4^4)*(4 + 4^3)*(4^2 + 4^2)*(4^3 + 4)*(4^4 + 1) = 9773138432;
a(5) = (1 + 4^5)*(4 + 4^4)*(4^2 + 4^3)*(4^3 + 4^2)*(4^4 + 4)*(4^5 + 1) = 454542400000000;
...
RELATED SERIES.
Let F(x) be the g.f. of A369557, then
F(1/4) = 2 + 25/4^2 + 2312/4^6 + 1690000/4^12 + 9773138432/4^20 + 454542400000000/4^30 + ... + a(n)/4^(n*(n+1)) + ... = 4.236976626306045459467696438142250301516563681...

Crossrefs

Cf. A369673, A369674, A369676, A369557.

Programs

  • PARI
    {a(n) = prod(k=0, n, 4^k + 4^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (4^k + 4^(n-k)).
a(n) = 4^(n*(n+1)) * Product_{k=0..n} (1/4^k + 1/4^(n-k)).
a(n) = 4^(n*(n+1)/2) * Product_{k=0..n} (1 + 1/4^(n-2*k)).
From Vaclav Kotesovec, Feb 07 2024: (Start)
a(n) ~ c * 4^(3*n^2/4 + n), where
c = 2.276671433133289... = QPochhammer(-1, 1/16)^2/2 if n is even and
c = 2.284052876870834... = sqrt(2) * QPochhammer(-4, 1/16)^2 / 25 if n is odd. (End)

A369557 Expansion of Sum_{n>=0} Product_{k=0..n} (x^k + x^(n-k)).

Original entry on oeis.org

3, 4, 2, 6, 3, 4, 9, 4, 8, 6, 13, 8, 12, 12, 10, 22, 13, 22, 14, 26, 20, 34, 23, 32, 36, 34, 42, 36, 59, 38, 67, 46, 75, 56, 82, 66, 98, 84, 100, 102, 105, 126, 116, 152, 119, 184, 136, 202, 154, 230, 181, 256, 203, 276, 250, 306, 285, 326, 342, 348, 398, 374, 463, 404, 525, 438, 610, 486, 666, 542, 744, 610
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2024

Keywords

Examples

			G.f.: A(x) = 3 + 4*x + 2*x^2 + 6*x^3 + 3*x^4 + 4*x^5 + 9*x^6 + 4*x^7 + 8*x^8 + 6*x^9 + 13*x^10 + 8*x^11 + 12*x^12 + ...
where
A(x) = (1 + 1) + (1 + x)*(x + 1) + (1 + x^2)*(x + x)*(x^2 + 1) + (1 + x^3)*(x + x^2)*(x^2 + x)*(x^3 + 1) + (1 + x^4)*(x + x^3)*(x^2 + x^2)*(x^3 + x)*(x^4 + 1) + (1 + x^5)*(x + x^4)*(x^2 + x^3)*(x^3 + x^2)*(x^4 + x)*(x^5 + 1) + ...
Also,
A(1/x) = (1 + 1) + (1 + x)*(x + 1)/x^2 + (1 + x^2)*(x + x)*(x^2 + 1)/x^6 + (1 + x^3)*(x + x^2)*(x^2 + x)*(x^3 + 1)/x^12 + (1 + x^4)*(x + x^3)*(x^2 + x^2)*(x^3 + x)*(x^4 + 1)/x^20 + (1 + x^5)*(x + x^4)*(x^2 + x^3)*(x^3 + x^2)*(x^4 + x)*(x^5 + 1)/x^30 + ...
For example, at x = 1/2,
A(1/2) = 2 + 9/2^2 + 100/2^6 + 2916/2^12 + 231200/2^20 + 50808384/2^30 + 31258240000/2^42 + 54112148361216/2^56 + 264265663201280000/2^72 + ... + A369673(n)/2^(n*(n+1)) + ... = 6.80013983505192354264...
SPECIFIC VALUES.
A(t) = 4 at t = 0.21135479438007733067820905390237206358880...
A(t) = 5 at t = 0.35111207737762337157349938790010474080253...
A(t) = 6 at t = 0.44509902476179757380223857309576063477813...
A(3/4) = 18.04139246037655138841324835985762487898724341...
A(2/3) = 11.59103511448176661974748662249737201844158309...
A(Phi) = 9.595623356758087506923478384122062088751068609...
A(1/2) = 6.800139835051923542641455169580774467247971025...
A(1/3) = 4.847274134844057155467506697748724715389597193...
A(1/4) = 4.236976626306045459467696438142250301516563681...
A(1/5) = 3.934732308501055907377639201049737298238369356...

Links

Crossrefs

Cf. A369673, A369674, A369675, A369676, A376542, A376621.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[Product[x^j + x^(k - j), {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 08 2024 *)
nmax = 100; CoefficientList[Series[-1 + 2*Sum[x^(k^2) * Product[1 + x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}] + Sum[x^((k-1)*k) * Product[1 + x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 27 2024 *)
  • PARI
    {a(n) = my(A = sum(m=0,n+1, prod(k=0,m, x^k + x^(m-k)) +x*O(x^n) )); polcoeff(A,n)}
for(n=0,70, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = Sum_{n>=0} Product_{k=0..n} (x^k + x^(n-k)).
(2) A(x) = Sum_{n>=0} x^(n*(n+1)) * Product_{k=0..n} (1/x^k + 1/x^(n-k)).
(3) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * Product_{k=0..n} (1 + x^(n-2*k)).
From Vaclav Kotesovec, Sep 29 2024: (Start)
a(n) ~ c * d^sqrt(n) / sqrt(n), where d = A376621 = 2.7510850908889199... and c = sqrt((1 + ((197 - sqrt(27/31)) / 62)^(1/3) + ((197 + sqrt(27/31)) / 62)^(1/3))/3) = 1.146046709280363...
a(n) ~ 4*A376542(n). (End)

A369680 a(n) = Product_{k=0..n} (2^k + 3^(n-k)).

Original entry on oeis.org

2, 12, 250, 19404, 5780918, 6691848108, 30261978906250, 535757771934053916, 37171553237849766044342, 10113067879819381109893992732, 10789224041146220828897229003906250, 45150513047221188662211059385153001179564, 741117672560101894851755994230829254062662140918
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2024

Keywords

Comments

For p > 1, q > 1, limit_{n->oo} (Product_{k=0..n} (p^k + q^(n-k)) )^(1/n^2) = q^(1/2)*p^(1/(2*(1 + log(q)/log(p)))) = p^(1/2)*q^(1/(2*(1 + log(p)/log(q)))). - Vaclav Kotesovec, Feb 07 2024
Equivalently, limit_{n->oo} ( Product_{k=0..n} (p^k + q^(n-k)) )^(1/n^2) = exp((1/2) * (log(p)^2 + log(p)*log(q) + log(q)^2) / log(p*q)). - Paul D. Hanna, Feb 08 2024

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 3)*(2 + 1) = 12;
a(2) = (1 + 3^2)*(2 + 3)*(2^2 + 1) = 250;
a(3) = (1 + 3^3)*(2 + 3^2)*(2^2 + 3)*(2^3 + 1) = 19404;
a(4) = (1 + 3^4)*(2 + 3^3)*(2^2 + 3^2)*(2^3 + 3)*(2^4 + 1) = 5780918;
a(5) = (1 + 3^5)*(2 + 3^4)*(2^2 + 3^3)*(2^3 + 3^2)*(2^4 + 3)*(2^5 + 1) = 6691848108;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/2^k + 1/3^(n-k)) = 2 + 12/6 + 250/6^3 + 19404/6^6 + 5780918/6^10 + 6691848108/6^15 + ... + a(n)/6^(n*(n+1)/2) + ... = 5.6846111010137973166832330595516662115250385271...

Crossrefs

Cf. A369673, A369674, A369675, A369676, A369677, A369678, A369679.
Cf. A323715.

Programs

  • Mathematica
    Table[Product[2^k+3^(n-k),{k,0,n}],{n,0,12}] (* James C. McMahon, Feb 07 2024 *)
  • PARI
    {a(n) = prod(k=0, n, 2^k + 3^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (2^k + 3^(n-k)).
a(n) = 6^(n*(n+1)/2) * Product_{k=0..n} (1/2^k + 1/3^(n-k)).
a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 2^n/6^k).
a(n) = 2^(n*(n+1)/2) * Product_{k=0..n} (1 + 3^n/6^k).
a(n) = 2^(-n*(n+1)/2) * Product_{k=0..n} (2^n + 6^k).
a(n) = 3^(-n*(n+1)/2) * Product_{k=0..n} (3^n + 6^k).
a(n) = 2^(n*(n+1)/2)*QPochhammer(-3^n, 1/6, n + 1). - Stefano Spezia, Feb 07 2024
Limit_{n->oo} a(n)^(1/n^2) = 2^(1/(2*(1 + log(3)/log(2)))) * sqrt(3) = 3^(1/(2*(1 + log(2)/log(3)))) * sqrt(2) = 1.9805589654474717155611061670180902111915926... - Vaclav Kotesovec, Feb 07 2024
Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(2)^2 + log(2)*log(3) + log(3)^2) / log(6)). - Paul D. Hanna, Feb 08 2024

A369677 a(n) = Product_{k=0..n} (2^k + 5^(n-k)).

Original entry on oeis.org

2, 18, 910, 275562, 509528318, 5782203860202, 403066704971309470, 172986911139059942455818, 457494980583771669025834718462, 7445459859979605380607238308201858858, 746155118699551878624986638597659812003763550, 461066589238234272286243169377378506495126815749310922
Offset: 0

Views

Author

Paul D. Hanna, Feb 07 2024

Keywords

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 5)*(2 + 1) = 18;
a(2) = (1 + 5^2)*(2 + 5)*(2^2 + 1) = 910;
a(3) = (1 + 5^3)*(2 + 5^2)*(2^2 + 5)*(2^3 + 1) = 275562;
a(4) = (1 + 5^4)*(2 + 5^3)*(2^2 + 5^2)*(2^3 + 5)*(2^4 + 1) = 509528318;
a(5) = (1 + 5^5)*(2 + 5^4)*(2^2 + 5^3)*(2^3 + 5^2)*(2^4 + 5)*(2^5 + 1) = 5782203860202;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/2^k + 1/5^(n-k)) = 2 + 18/10 + 910/10^3 + 275562/10^6 + 509528318/10^10 + 5782203860202/10^15 + ... + a(n)/10^(n*(n+1)/2) + ... = 5.0427178660718059961260933841217518099...

Crossrefs

Cf. A369673, A369674, A369675, A369676, A369678, A369679, A369680.

Programs

  • PARI
    {a(n) = prod(k=0, n, 2^k + 5^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (2^k + 5^(n-k)).
a(n) = 10^(n*(n+1)/2) * Product_{k=0..n} (1/2^k + 1/5^(n-k)).
a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 2^n/10^k).
a(n) = 2^(n*(n+1)/2) * Product_{k=0..n} (1 + 5^n/10^k).
a(n) = 2^(-n*(n+1)/2) * Product_{k=0..n} (2^n + 10^k).
a(n) = 5^(-n*(n+1)/2) * Product_{k=0..n} (5^n + 10^k).
a(n) = 2^(n*(n+1)/2)*QPochhammer(-5^n, 1/10, n + 1). - Stefano Spezia, Feb 07 2024
Limit_{n->oo} a(n)^(1/n^2) = 2^(1/(2*(1 + log(5)/log(2)))) * sqrt(5) = 5^(1/(2*(1 + log(2)/log(5)))) * sqrt(2) = 2.481958590195459039209137154563963236753327... - Vaclav Kotesovec, Feb 07 2024
Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(2)^2 + log(2)*log(5) + log(5)^2) / log(10)). - Paul D. Hanna, Feb 08 2024

A369678 a(n) = Product_{k=0..n} (3^k + 5^(n-k)).

Original entry on oeis.org

2, 24, 2080, 1382976, 7148699648, 287041728769536, 90391546425391144960, 221202979125273147766738944, 4237647337376998325597017538035712, 633933934421036224259931934460116571357184, 738292285249623417870561091674252758908330993254400
Offset: 0

Views

Author

Paul D. Hanna, Feb 07 2024

Keywords

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 5)*(3 + 1) = 24;
a(2) = (1 + 5^2)*(3 + 5)*(3^2 + 1) = 2080;
a(3) = (1 + 5^3)*(3 + 5^2)*(3^2 + 5)*(3^3 + 1) = 1382976;
a(4) = (1 + 5^4)*(3 + 5^3)*(3^2 + 5^2)*(3^3 + 5)*(3^4 + 1) = 7148699648;
a(5) = (1 + 5^5)*(3 + 5^4)*(3^2 + 5^3)*(3^3 + 5^2)*(3^4 + 5)*(3^5 + 1) = 287041728769536;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/3^k + 1/5^(n-k)) = 2 + 24/15 + 2080/15^3 + 1382976/15^6 + 7148699648/15^10 + 287041728769536/15^15 + ... + a(n)/15^(n*(n+1)/2) + ... = 4.3507806549816093424129450104392682482776...

Crossrefs

Cf. A369673, A369674, A369675, A369676, A369677, A369679, A369680.

Programs

  • PARI
    {a(n) = prod(k=0, n, 3^k + 5^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (3^k + 5^(n-k)).
a(n) = 15^(n*(n+1)/2) * Product_{k=0..n} (1/3^k + 1/5^(n-k)).
a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 3^n/15^k).
a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 5^n/15^k).
a(n) = 3^(-n*(n+1)/2) * Product_{k=0..n} (3^n + 15^k).
a(n) = 5^(-n*(n+1)/2) * Product_{k=0..n} (5^n + 15^k).
a(n) = 3^(n*(n+1)/2)*QPochhammer(-5^n, 1/15, n + 1). - Stefano Spezia, Feb 07 2024
Limit_{n->oo} a(n)^(1/n^2) = 3^(1/(2*(1 + log(5)/log(3)))) * sqrt(5) = 5^(1/(2*(1 + log(3)/log(5)))) * sqrt(3) = 2.794249622709633938040980858655052416325961... - Vaclav Kotesovec, Feb 07 2024
Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(3)^2 + log(3)*log(5) + log(5)^2) / log(15)). - Paul D. Hanna, Feb 08 2024

A369679 a(n) = Product_{k=0..n} (3^k + 4^(n-k)).

Original entry on oeis.org

2, 20, 1190, 449540, 1094267450, 17283181758500, 1774660248586902950, 1182579046508766038251700, 5134581376819479940742838299450, 144547890423248529154421336209389168500, 26527720524980501045637796065988864058001683750, 31574745853363739268697794406696294745967395732336262500
Offset: 0

Views

Author

Paul D. Hanna, Feb 07 2024

Keywords

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 4)*(3 + 1) = 20;
a(2) = (1 + 4^2)*(3 + 4)*(3^2 + 1) = 1190;
a(3) = (1 + 4^3)*(3 + 4^2)*(3^2 + 4)*(3^3 + 1) = 449540;
a(4) = (1 + 4^4)*(3 + 4^3)*(3^2 + 4^2)*(3^3 + 4)*(3^4 + 1) = 1094267450;
a(5) = (1 + 4^5)*(3 + 4^4)*(3^2 + 4^3)*(3^3 + 4^2)*(3^4 + 4)*(3^5 + 1) = 17283181758500;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/3^k + 1/4^(n-k)) = 2 + 20/12 + 1190/12^3 + 449540/12^6 + 1094267450/12^10 + 17283181758500/12^15 + ... + a(n)/12^(n*(n+1)/2) + ... = 4.5247082137580440222914164418070212438323...

Crossrefs

Cf. A369673, A369674, A369675, A369676, A369677, A369678, A369680.

Programs

  • PARI
    {a(n) = prod(k=0, n, 3^k + 4^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (3^k + 4^(n-k)).
a(n) = 12^(n*(n+1)/2) * Product_{k=0..n} (1/3^k + 1/4^(n-k)).
a(n) = 4^(n*(n+1)/2) * Product_{k=0..n} (1 + 3^n/12^k).
a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 4^n/12^k).
a(n) = 3^(-n*(n+1)/2) * Product_{k=0..n} (3^n + 12^k).
a(n) = 4^(-n*(n+1)/2) * Product_{k=0..n} (4^n + 12^k).
a(n) = 3^(n*(n+1)/2)*QPochhammer(-4^n, 1/12, n + 1). - Stefano Spezia, Feb 07 2024
Limit_{n->oo} a(n)^(1/n^2) = 3^(1/(2*(1 + log(4)/log(3)))) * 2 = 2^(1/(1 + log(3)/log(4))) * sqrt(3) = 2.54977004574388327607102436919328599299374003... - Vaclav Kotesovec, Feb 07 2024
Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(3)^2 + log(3)*log(4) + log(4)^2) / log(12)). - Paul D. Hanna, Feb 08 2024

A369681 a(n) = Product_{k=0..n} (4^k + 5^(n-k)).

Original entry on oeis.org

2, 30, 3978, 4987710, 58712437962, 6601051349841150, 7017151861981535193738, 70966047508527496843460412990, 6820716704126571481897874317127918922, 6205644698427009393117687864650447521113942270, 53916867047490616763228279441645027173409633988839675658
Offset: 0

Views

Author

Paul D. Hanna, Feb 07 2024

Keywords

Comments

For p > 1, q > 1, limit_{n->oo} ( Product_{k=0..n} (p^k + q^(n-k)) )^(1/n^2) = exp((1/2) * (log(p)^2 + log(p)*log(q) + log(q)^2) / log(p*q)); formula due to Vaclav Kotesovec (cf. A369680).

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 5)*(4 + 1) = 30;
a(2) = (1 + 5^2)*(4 + 5)*(4^2 + 1) = 3978;
a(3) = (1 + 5^3)*(4 + 5^2)*(4^2 + 5)*(4^3 + 1) = 4987710;
a(4) = (1 + 5^4)*(4 + 5^3)*(4^2 + 5^2)*(4^3 + 5)*(4^4 + 1) = 58712437962;
a(5) = (1 + 5^5)*(4 + 5^4)*(4^2 + 5^3)*(4^3 + 5^2)*(4^4 + 5)*(4^5 + 1) = 6601051349841150;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/4^k + 1/5^(n-k)) = 2 + 30/20 + 3978/20^3 + 4987710/20^6 + 58712437962/20^10 + 6601051349841150/20^15 + ... + a(n)/20^(n*(n+1)/2) + ... = 4.0811214259450988699292249336017494522520...

Crossrefs

Cf. A369673, A369674, A369675, A369676, A369677, A369678, A369679, A369680.

Programs

  • PARI
    {a(n) = prod(k=0, n, 4^k + 5^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (4^k + 5^(n-k)).
a(n) = 20^(n*(n+1)/2) * Product_{k=0..n} (1/4^k + 1/5^(n-k)).
a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 4^n/20^k).
a(n) = 4^(n*(n+1)/2) * Product_{k=0..n} (1 + 5^n/20^k).
a(n) = 4^(-n*(n+1)/2) * Product_{k=0..n} (4^n + 20^k).
a(n) = 5^(-n*(n+1)/2) * Product_{k=0..n} (5^n + 20^k).
Limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(4)^2 + log(4)*log(5) + log(5)^2) / log(20)) = 3.0816872899745614612763875038173884057052077... [from a formula by Vaclav Kotesovec].

A370241 Expansion of Sum_{n>=0} Product_{k=0..n} (x^k*(1+x)^(n-k) + x^(n-k)*(1+x)^k).

Original entry on oeis.org

3, 6, 15, 36, 98, 258, 677, 1830, 5006, 13340, 35215, 95702, 264851, 717760, 1894473, 5031846, 13788409, 38375030, 105005017, 279236168, 734728565, 1967715202, 5416631023, 15061949148, 41271428388, 110250824636, 289840310574, 766277436248, 2072808806434, 5730605191220
Offset: 0

Views

Author

Paul D. Hanna, Feb 13 2024

Keywords

Examples

			G.f.: A(x) = 3 + 6*x + 15*x^2 + 36*x^3 + 98*x^4 + 258*x^5 + 677*x^6 + 1830*x^7 + 5006*x^8 + 13340*x^9 + 35215*x^10 + 95702*x^11 + 264851*x^12 + ...
where
A(x) = (1 + 1) + ((1+x) + x)*(x + (1+x)) + ((1+x)^2 + x^2)*(x*(1+x) + x*(1+x))*(x^2 + (1+x)^2) + ((1+x)^3 + x^3)*(x*(1+x)^2 + x^2*(1+x))*(x^2*(1+x) + x*(1+x)^2)*(x^3 + (1+x)^3) + ((1+x)^4 + x^4)*(x*(1+x)^3 + x^3*(1+x))*(x^2*(1+x)^2 + x^2*(1+x)^2)*(x^3*(1+x) + x*(1+x)^3)*(x^4 + (1+x)^4) + ...
SPECIFIC VALUES.
A(1/5) = 5.4216712041652671338354486...
A(1/4) = Sum_{n>=0} A369676(n)/4^(n*(n+1)) = 7.1437109433775269577074586...
A(1/3) = Sum_{n>=0} A369675(n)/3^(n*(n+1)) = 19.589361786409617133535937...
A(-1/3) = 1.9743720303058511269360725...
Although the g.f. A(x) diverges at x = -1/2, it may be evaluated formally as
A(-1/2) = Sum_{n>=0} (-1)^n * 2 / 16^(n^2) = 1.875030517549021169...

Links

Crossrefs

Cf. A369557, A369674, A369675, A369676.

Programs

  • PARI
    {a(n) = my(A = sum(m=0, n+1, prod(k=0, m, x^k*(1+x)^(m-k) + x^(m-k)*(1+x)^k +x*O(x^n)) )); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = Sum_{n>=0} Product_{k=0..n} (x^k*(1+x)^(n-k) + x^(n-k)*(1+x)^k).
(2) A(x) = Sum_{n>=0} (1+x)^(n*(n+1)) * Product_{k=0..n} ((x/(1+x))^k + (x/(1+x))^(n-k)).
(3) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * (1+x)^(n*(n+1)/2) * Product_{k=0..n} (1 + (x/(1+x))^(n-2*k)).
(4) A(x/(1-x)) = Sum_{n>=0} 1/(1-x)^(n*(n+1)) * Product_{k=0..n} (x^k + x^(n-k)).
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