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 1-10 of 16 results. Next

A333198 Decimal expansion of a constant related to the asymptotics of A306734 and A333179.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 11 2020

Keywords

Examples

			1.86429525435844065924747599856112246877295244507368421574403...

Crossrefs

Cf. A306734, A333179, A333180, A333181, A357471, A376621.

Programs

  • Mathematica
    RealDigits[E^Sqrt[4*Log[r]^2/3 + 4*PolyLog[2, 1-r] - Pi^2/3] /. r -> (2 - 5*(2/(-11 + 3*Sqrt[69]))^(1/3) + ((-11 + 3*Sqrt[69])/2)^(1/3))/3, 10, 120][[1]] (* Vaclav Kotesovec, Oct 07 2024 *)

Formula

Equals limit_{n->infinity} A306734(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A333179(n)^(1/sqrt(n)).
Equals exp(sqrt(4*log(r)^2/3 + 4*polylog(2, 1-r) - Pi^2/3)), where r = 1 - A357471 = 0.4301597090019467340886... is the real root of the equation r^2 = (1-r)^3. - Vaclav Kotesovec, Oct 07 2024

Extensions

More digits from Vaclav Kotesovec, Oct 07 2024

A333179 G.f.: Sum_{k>=0} (x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 3, 2, 2, 3, 2, 3, 4, 4, 4, 5, 5, 5, 5, 4, 5, 5, 4, 4, 4, 4, 5, 6, 5, 6, 7, 7, 8, 8, 8, 8, 9, 8, 8, 8, 7, 8, 8, 8, 8, 9, 9, 10, 11, 11
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 10 2020

Keywords

Links

Crossrefs

Cf. A003106, A053261, A306734.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[1+x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*x^(2*k)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

a(n) ~ c * A333198^sqrt(n) / sqrt(n), where c = 0.3207396095989103757477946185... = sqrt((1 - (2/(23*(23 + 3*sqrt(69))))^(1/3) + ((1/2)*(23 + 3*sqrt(69)))^(1/3)/23^(2/3))/3)/2, c = sqrt(s)/2, where s is the real root of the equation -1 + 8*s - 23*s^2 + 23*s^3 = 0.
Limit_{n->infinity} A306734(n) / a(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885...

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.

A066447 Number of basis partitions (or basic partitions) of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 8, 10, 13, 16, 20, 26, 32, 40, 50, 61, 74, 90, 108, 130, 156, 186, 222, 264, 313, 370, 436, 512, 600, 702, 818, 952, 1106, 1282, 1484, 1715, 1978, 2278, 2620, 3008, 3448, 3948, 4512, 5150, 5872, 6684, 7600, 8632, 9791, 11094, 12558, 14198, 16036, 18096, 20398
Offset: 0

Views

Author

Herbert S. Wilf, Dec 29 2001

Keywords

Comments

The k-th successive rank of a partition pi = (pi_1, pi_2, ..., pi_s) of the integer n is r_k = pi_k - pi'_k, where pi' denotes the conjugate partition. A partition pi is basic if the number of dots in its Ferrers diagram is the least among all the Ferrers diagrams of partitions with the same rank vector.
The g.f. Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 - x^k) = Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 + x^k - 2*x^k) == Sum_{n >= 0} x^(n^2) (mod 2). It follows that a(n) is odd iff n is a square (Nolan et al., equation 6, p. 282). - Peter Bala, Jan 08 2025

Links

Crossrefs

Row sums of A066448.
Cf. A001130, A001935, A003114, A306734, A333374.
Cf. A192918, A376841.

Programs

  • Maple
    b := proc(n,d); option remember; if n=0 and d=0 then RETURN(1) elif n<=0 or d<=0 then RETURN(0) else RETURN(b(n-d,d)+b(n-2*d+1,d-1)+b(n-3*d+1,d-1)) fi: end: A066447 := n->add(b(n,d),d=0..n);
  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[x^(n^2)*Product[(1 + x^k)/(1 - x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 17 2020 *)
nmax = 60; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k - 1), {x, 0, nmax}]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Mar 17 2020 *)
  • PARI
    N=66; x='x+O('x^N); s=sum(n=0,N,x^(n^2)*prod(k=1,n,(1+x^k)/(1-x^k))); Vec(s) /* Joerg Arndt, Apr 07 2011 */

Formula

G.f.: Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 - x^k) [Given in Nolan et al. reference]. - Joerg Arndt, Apr 07 2011
Limit_{n->infinity} a(n) / A333374(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... - Vaclav Kotesovec, Mar 17 2020
a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.193340468476900308848561788251945... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 + cosh(arccosh(53*sqrt(11/2)/64)/3) / (3*sqrt(22))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - Vaclav Kotesovec, Mar 19 2020, updated Oct 10 2024

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).

A216222 Counting a set of restricted partitions.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 4, 3, 3, 3, 3, 6, 7, 8, 10, 9, 9, 9, 9, 11, 13, 16, 20, 22, 25, 28, 27, 28, 29, 30, 32, 35, 40, 45, 53, 60, 67, 73, 79, 85, 87, 92, 95, 98, 105, 111, 120, 132, 145, 160, 178, 196, 212, 231, 247, 263, 280, 291, 305, 319, 334, 352, 371, 393
Offset: 0

Views

Author

David S. Newman, Mar 13 2013

Keywords

Links

Crossrefs

Cf. A001622, A306734, A376542, A376580, A376812, A376813.

Programs

  • Mathematica
    Take[CoefficientList[Sum[x^(k^2)*Product[1 + x^i, {i, k}]^2, {k, 0, 7}], x], 63] (* Giovanni Resta, Mar 13 2013 *)
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*(1 + x^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 09 2024 *)

Formula

G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^j)^2 = 1 +x^1*(1+x)^2 +x^4*(1+x)^2*(1+x^2)^2 +...+ x^k^2*(1+x)^2*(1+x^2)^2*(1+x^3)^2*...*(1+x^k)^2+...
a(n) ~ phi^(3/2) * exp(Pi*sqrt(2*n/15)) / (4*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 29 2024

Extensions

a(14)-a(62) from Giovanni Resta, Mar 13 2013

A333374 G.f.: Sum_{k>=1} (x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)/(1 - x^j)).

Original entry on oeis.org

1, 0, 1, 2, 2, 2, 3, 4, 6, 8, 10, 12, 15, 18, 22, 28, 34, 42, 52, 62, 75, 90, 106, 126, 150, 176, 208, 246, 288, 338, 397, 462, 538, 626, 724, 838, 968, 1114, 1282, 1474, 1690, 1936, 2217, 2532, 2890, 3296, 3750, 4264, 4844, 5492, 6222, 7042, 7958, 8986, 10138
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 17 2020

Keywords

Comments

The g.f. Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j) = Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j + 2*x^j) == Sum_{k >= 1} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - Peter Bala, Jan 04 2025

Links

Crossrefs

Cf. A001935, A001936, A003106, A053261, A066447, A003114, A306734, A333179.
Cf. A192918, A376841.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[(1+x^k)/(1-x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k), {x, 0, nmax}]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

Limit_{n->infinity} A066447(n) / a(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... (the tribonacci constant).
Compare with: A306734(n) / A333179(n) -> A060006 (the plastic constant) and A003114(n) / A003106(n) -> A001622 (golden ratio).
a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.10511708841962944170826735560432... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 - sinh(arcsinh(sqrt(11)/4)/3) / (12*sqrt(11))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - Vaclav Kotesovec, Mar 17 2020, updated Oct 10 2024

A333180 G.f.: Sum_{k>=1} (k * x^(k^2) * Product_{j=1..k} (1 + x^j)).

Original entry on oeis.org

0, 1, 1, 0, 2, 2, 2, 2, 0, 3, 3, 3, 6, 3, 3, 3, 4, 4, 4, 8, 8, 8, 8, 8, 4, 9, 9, 5, 10, 10, 15, 15, 15, 15, 15, 15, 16, 16, 11, 17, 17, 18, 24, 24, 24, 30, 30, 30, 30, 31, 31, 31, 32, 26, 33, 34, 41, 41, 42, 49, 49, 56, 56, 56, 64, 64, 57, 65, 58, 59, 67, 68
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 10 2020

Keywords

Links

Crossrefs

Cf. A268188, A306734, A333181.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[n*x^(n^2)*Product[1+x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 0; Do[p = Expand[p*(1 + x^k)*x^(2*k - 1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += k*p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

a(n) ~ c * A333198^sqrt(n), where c = 0.3836313809149103736315...
Limit_{n->infinity} a(n) / A333181(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885...

A376581 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, 3, 5, 7, 9, 13, 17, 22, 30, 38, 48, 62, 78, 97, 122, 151, 184, 228, 278, 335, 408, 491, 588, 707, 843, 1000, 1189, 1407, 1658, 1955, 2295, 2686, 3145, 3670, 4270, 4968, 5763, 6671, 7720, 8909, 10263, 11816, 13577, 15574, 17850, 20424, 23333, 26638, 30365
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 29 2024

Keywords

Links

Crossrefs

Cf. A053264, A306734, A340647, A376542, A376580.

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) ~ exp(Pi*sqrt(2*n/5)) / (4*5^(1/4)*sqrt(n)).

A376945 G.f.: Sum_{k>=0} 2^k * x^(k^2) * Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

1, 2, 2, 0, 4, 4, 4, 4, 0, 8, 8, 8, 16, 8, 8, 8, 16, 16, 16, 32, 32, 32, 32, 32, 16, 48, 48, 32, 64, 64, 96, 96, 96, 96, 96, 96, 128, 128, 96, 160, 160, 192, 256, 256, 256, 320, 320, 320, 320, 384, 384, 384, 448, 384, 512, 576, 704, 704, 768, 896, 896, 1024, 1024, 1024
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 10 2024

Keywords

Links

Crossrefs

Cf. A273065, A306734, A376943, A376944, A376948.

Programs

  • Mathematica
    nmax = 80; CoefficientList[Series[Sum[2^k * x^(k^2) * Product[1+x^j, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 80; p = 1; s = 1; Do[p = Normal[Series[2*p*(1 + x^k) * x^(2*k - 1), {x, 0, nmax}]]; s += p; , {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

a(n) ~ (1+r) * exp(sqrt((2*log(2)^2 + 8*log(2)*log(r) + 12*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((3*r + 2)*n)), where r = ((46 - 6*sqrt(57))^(1/3) + (46 + 6*sqrt(57))^(1/3) - 2)/6 is the real root of the equation 2*r^2*(1+r) = 1 (A273065).
Showing 1-10 of 16 results. Next

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