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.

Showing 1-9 of 9 results.

A376621 Decimal expansion of a constant related to the asymptotics of A369557 and A376580.

Original entry on oeis.org

2, 7, 5, 1, 0, 8, 5, 0, 9, 0, 8, 8, 8, 9, 1, 9, 9, 3, 9, 4, 3, 4, 2, 0, 4, 9, 6, 2, 0, 4, 8, 9, 4, 7, 0, 3, 6, 4, 1, 8, 1, 7, 8, 6, 0, 2, 6, 3, 7, 1, 7, 5, 0, 9, 8, 2, 8, 1, 1, 3, 2, 5, 9, 3, 9, 3, 2, 9, 1, 3, 8, 2, 2, 8, 4, 0, 1, 1, 7, 9, 3, 5, 6, 5, 7, 6, 2, 5, 2, 6, 2, 6, 0, 8, 7, 8, 2, 8, 0, 4, 9, 2, 4
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Examples

			2.75108509088891993943420496204894703641817860263717...

Crossrefs

Cf. A333198, A369557, A376542, A376580, A376623, A376658, A376659, A376660.
Cf. A088559.

Programs

  • Mathematica
    RealDigits[E^Sqrt[3*Log[r]^2/2 + 4*PolyLog[2, r^(1/2)] - Pi^2/3] /. r -> (-2 + ((29 - 3*Sqrt[93])/2)^(1/3) + ((29 + 3*Sqrt[93])/2)^(1/3))/3, 10, 120][[1]] (* Vaclav Kotesovec, Oct 07 2024 *)

Formula

Equals limit_{n->infinity} A369557(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376580(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376542(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376623(n)^(1/sqrt(n)).
Equals exp(sqrt(3*log(r)^2/2 + 4*polylog(2, r^(1/2)) - Pi^2/3)), where r = A088559 = 0.465571231876768026656731... is the real root of the equation r*(1+r)^2 = 1. - Vaclav Kotesovec, Oct 07 2024

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)

A369676 a(n) = Product_{k=0..n} (5^k + 5^(n-k)).

Original entry on oeis.org

2, 36, 6760, 14288400, 331135220000, 87265295649000000, 252668462115852250000000, 8322480168806663555062500000000, 3012058207750727786980181328125000000000, 12401474551899042876552569922821191406250000000000, 561039675887726306551826113078284190093383789062500000000000
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2024

Keywords

Comments

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. (End)

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 5)*(5 + 1) = 36;
a(2) = (1 + 5^2)*(5 + 5)*(5^2 + 1) = 6760;
a(3) = (1 + 5^3)*(5 + 5^2)*(5^2 + 5)*(5^3 + 1) = 14288400;
a(4) = (1 + 5^4)*(5 + 5^3)*(5^2 + 5^2)*(5^3 + 5)*(5^4 + 1) = 331135220000;
a(5) = (1 + 5^5)*(5 + 5^4)*(5^2 + 5^3)*(5^3 + 5^2)*(5^4 + 5)*(5^5 + 1) = 87265295649000000;
...
RELATED SERIES.
Let F(x) be the g.f. of A369557, then
F(1/5) = 2 + 36/5^2 + 6760/5^6 + 14288400/5^12 + 331135220000/5^20 + 87265295649000000/5^30 + ... + a(n)/5^(n*(n+1)) + ... = 3.934732308501055907377639201049737298238369356...

Crossrefs

Cf. A369673, A369674, A369675, A369557.

Programs

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

Formula

a(n) = Product_{k=0..n} (5^k + 5^(n-k)).
a(n) = 5^(n*(n+1)) * Product_{k=0..n} (1/5^k + 1/5^(n-k)).
a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 1/5^(n-2*k)).
From Vaclav Kotesovec, Feb 07 2024: (Start)
a(n) ~ c * 5^(3*n^2/4 + n), where
c = 2.170417138549358... = QPochhammer(-1, 1/25)^2/2 if n is even and
c = 2.189351749288445... = 5^(1/4) * QPochhammer(-5, 1/25)^2 / 36 if n is odd. (End)

A376542 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j))^2.

Original entry on oeis.org

1, 1, 0, 2, 1, 1, 2, 0, 3, 1, 4, 2, 3, 3, 2, 6, 2, 7, 2, 8, 3, 10, 6, 8, 9, 8, 12, 8, 16, 6, 20, 8, 22, 10, 24, 14, 27, 20, 26, 26, 25, 34, 26, 42, 25, 51, 26, 58, 31, 66, 36, 72, 43, 76, 56, 82, 70, 82, 86, 84, 106, 87, 124, 90, 145, 95, 168, 102, 187, 115, 206
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 28 2024

Keywords

Links

Crossrefs

Cf. A216222, A306734, A333179, A340647, A369557, A376530.

Programs

  • Mathematica
    nmax=100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

Formula

a(n) ~ A369557(n) / 4.

A376580 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j-1))^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 3, 3, 3, 4, 4, 5, 5, 7, 9, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 18, 17, 19, 24, 23, 25, 27, 28, 31, 32, 33, 37, 40, 42, 44, 47, 52, 54, 59, 62, 67, 75, 75, 80, 87, 90, 95, 102, 109, 114, 119, 127, 134, 142, 150, 159, 171, 178, 187, 199, 211
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 29 2024

Keywords

Links

Crossrefs

Cf. A053258, A216222, A306734, A369557, A376542, A376581, A376621.

Programs

  • Mathematica
    nmax=100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

Formula

a(n) ~ c * A376621^sqrt(n) / sqrt(n), where c = 1/(2*sqrt(3 - 4*sinh(arcsinh(3^(3/2)/2) / 3) / sqrt(3))) = 0.390989767113799449629...
a(n) ~ c * A376542(n), where c = (108 + 12*sqrt(93))^(1/3)/3 - 4/(108 + 12*sqrt(93))^(1/3) = 1.364655607... is the real root of the equation c*(4 + c^2) = 8.
a(n) ~ c * A369557(n), where c = A347178 = -sinh(log((-3*sqrt(3) + sqrt(31))/2)/3) / sqrt(3) = 0.3411639019... is the real root of the equation 2*c*(1 + 4*c^2) = 1.
a(n) ~ A376631(n) * (A376621/A376660)^sqrt(n).

A376530 G.f. A(x) = (1/3) * Sum_{n>=0} Product_{k=0..2*n} (x^k + x^n + x^(2*n-k)).

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 5, 6, 10, 11, 15, 15, 18, 20, 25, 30, 38, 47, 57, 67, 78, 89, 100, 111, 128, 144, 168, 191, 227, 260, 305, 347, 403, 451, 514, 571, 644, 710, 795, 881, 989, 1099, 1237, 1384, 1559, 1746, 1963, 2196, 2457, 2733, 3044, 3369, 3729, 4107, 4529, 4975, 5473, 6003, 6605, 7243, 7973
Offset: 0

Views

Author

Paul D. Hanna, Sep 27 2024

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 10*x^8 + 11*x^9 + 15*x^10 + 15*x^11 + 18*x^12 + 20*x^13 + 25*x^14 + 30*x^15 + 38*x^16 + 47*x^17 + 57*x^18 + 67*x^19 + 78*x^20 + ...
where
A(x) = 1  +  (1 + x + x^2)*(x)*(x^2 + x + 1)  +  (1 + x^2 + x^4)*(x + x^2 + x^3)*(x^2)*(x^3 + x^2 + x)*(x^4 + x^2 + 1)  +  (1 + x^3 + x^6)*(x + x^3 + x^5)*(x^2 + x^3 + x^4)*(x^3)*(x^4 + x^3 + x^2)*(x^5 + x^3 + x)*(x^6 + x^3 + 1)  +  (1 + x^4 + x^8)*(x + x^4 + x^7)*(x^2 + x^4 + x^6)*(x^3 + x^4 + x^5)*(x^4)*(x^5 + x^4 + x^3)*(x^6 + x^4 + x^2)*(x^7 + x^4 + x)*(x^8 + x^4 + 1)  +  (1 + x^5 + x^10)*(x + x^5 + x^9)*(x^2 + x^5 + x^8)*(x^3 + x^5 + x^7)*(x^4 + x^5 + x^6)*(x^5)*(x^6 + x^5 + x^4)*(x^7 + x^5 + x^3)*(x^8 + x^5 + x^2)*(x^9 + x^5 + x)*(x^10 + x^5 + 1) + ...
Also,
A(x) = 1  +  x*(1 + x + x^2)^2  +  x^2*(1 + x^2 + x^4)^2*(x + x^2 + x^3)^2  +  x^3*(1 + x^3 + x^6)^2*(x + x^3 + x^5)^2*(x^2 + x^3 + x^4)^2  +  x^4*(1 + x^4 + x^8)^2*(x + x^4 + x^7)^2*(x^2 + x^4 + x^6)^2*(x^3 + x^4 + x^5)^2  +  x^5*(1 + x^5 + x^10)^2*(x + x^5 + x^9)^2*(x^2 + x^5 + x^8)^2*(x^3 + x^5 + x^7)^2*(x^4 + x^5 + x^6)^2 + ...
SPECIFIC VALUES.
A(t) = 7 at t = 0.66668704736936585046859672241821389017558257705439339...
A(t) = 6 at t = 0.64513809385910573788372368634772347751697803188552164...
A(t) = 5 at t = 0.61639010238633204213526430692013003520814209008383800...
A(t) = 4 at t = 0.57545188136244196253678514659912022278976129786049251...
A(t) = 3 at t = 0.51093469574142600352566002004049869356160992832828805...
A(t) = 2 at t = 0.38925040919555545279428903616909363335667114006118874...
A(4/5) = 33.86295094486999840248628061724081807284197309832190750...
A(3/4) = 15.71390570183068296805142809300098703963996686273128437...
A(2/3) = 6.998922814611911009050207691553160959950411531472265898...
A(3/5) = 4.551745873136373778485262039216993578932737039944687958...
A(1/2) = 2.87450225671651109577680741009657439874438592581613285485257...
where A(1/2) = 1 + 7^2/2^5 + 147^2/2^16 + 10731^2/2^33 + 2929563^2/2^56 + 3096548091^2/2^85 + 12884736606651^2/2^120 + 212765655585627963^2/2^161 + 13998490777945220569659^2/2^208 + ... + A376227(n)^2/2^(n*(3*n+2)) + ..., where A376227(n) = Product_{k=1..n} (1 + 2^k + 2^(2*k)).
A(2/5) = 2.062036845797808963480254546496778756663866279595140073...
A(1/3) = 1.728128514830894263417956669231253604769749542061786338...
A(1/4) = 1.438324287250845860310741641820056491309903730120221376...
A(1/5) = 1.310189970721194144762503370434773514855060963388422496...

Links

Crossrefs

Cf. A000726, A369557, A376152, A376542, A376227.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(k^2)*Product[1 + x^j + x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 28 2024 *)
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k + x^(2*k))*(1 + x^k + x^(2*k)) * x^(2*k-1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Oct 08 2024 *)
  • PARI
    {a(n) = my(A = (1/3)*sum(m=0,n, prod(k=0,2*m, x^k + x^m + x^(2*m-k) +x*O(x^n)))); polcoeff(A,n)}
for(n=0,60, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
(1) A(x) = (1/3) * Sum_{n>=0} Product_{k=0..2*n} (x^k + x^n + x^(2*n-k)).
(2) A(x) = Sum_{n>=0} x^n * Product_{k=0..n-1} (x^k + x^n + x^(2*n-k))^2.
(3) A(x) = (1/3) * Sum_{n>=0} x^(n*(2*n+1)) * Product_{k=0..2*n} (1 + x^(n-k) + x^(2*n-2*k)).
(4) A(x) = (1/3) * Sum_{n>=0} x^(n*(2*n+1)) * Product_{k=0..2*n} (1/x^(n-k) + 1 + x^(n-k)).
(5) A(x) = (1/3) * Sum_{n>=0} x^(n*(2*n+1)) * Product_{k=0..2*n} (1 + 1/x^(n-k) + 1/x^(2*n-2*k)).
(6) A(x) = (1/3) * Sum_{n>=0} x^(2*n*(2*n+1)) * Product_{k=0..2*n} (1/x^k + 1/x^n + 1/x^(2*n-k)).
From Paul D. Hanna, Oct 09 2024: (Start)
(7) A(x) = Sum_{n>=0} x^(n^2) * Product_{k=1..n} (1 - x^(3*k))^2 / (1 - x^k)^2.
(8) A(x) = Sum_{n>=0} x^(n^2) * Product_{k=1..n} (1 + x^k + x^(2*k))^2.
(End)
a(n) ~ c * d^sqrt(n) / sqrt(n), where d = A376152 = 4.9880208766009... and c = sqrt(1/54 + 5*cosh(arccosh(7*sqrt(11/2)/16)/3)/(27*sqrt(22))) = 0.241068202175... - Vaclav Kotesovec, Sep 28 2024, updated Oct 09 2024

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

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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)).
Showing 1-9 of 9 results.

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