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.

Previous Showing 11-17 of 17 results.

A325951 G.f.: 1/(1-x)^3 * Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 4, 10, 21, 39, 67, 109, 170, 256, 375, 537, 754, 1041, 1416, 1901, 2523, 3314, 4312, 5563, 7121, 9050, 11426, 14338, 17890, 22204, 27422, 33709, 41257, 50288, 61058, 73863, 89043, 106988, 128146, 153029, 182222, 216393, 256302, 302813, 356908, 419700
Offset: 0

Views

Author

Vaclav Kotesovec, May 28 2019

Keywords

Crossrefs

Cf. A000009, A014160, A036469, A095944, A120477, A292508.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[(1/(1-x)^3 * Product[1+x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

a(n) ~ 2 * 3^(5/4) * n^(3/4) * exp(Pi*sqrt(n/3)) / Pi^3.

A360487 Convolution of A000009 and A000290.

Original entry on oeis.org

0, 1, 5, 14, 31, 60, 106, 176, 279, 426, 631, 912, 1291, 1795, 2457, 3317, 4424, 5837, 7626, 9875, 12684, 16171, 20476, 25764, 32228, 40094, 49626, 61131, 74966, 91545, 111346, 134921, 162906, 196031, 235134, 281175, 335251, 398615, 472695, 559115, 659721, 776608
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 09 2023

Keywords

Crossrefs

Cf. A000009, A000290, A095944, A360486.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> add(b(n-j)*j^2, j=0..n):
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 09 2023
  • Mathematica
    Table[Sum[PartitionsQ[k]*(n-k)^2, {k, 0, n}], {n, 0, 60}]
CoefficientList[Series[x*(1+x)*QPochhammer[-1, x] / (2*(1-x)^3), {x, 0, 60}], x]

Formula

a(n) = Sum_{k=0..n} A000009(k) * (n-k)^2.
G.f.: x*(1+x)/(1-x)^3 * Product_{k>=1} (1 + x^k).
a(n) ~ 4 * 3^(5/4) * n^(3/4) * exp(sqrt(n/3)*Pi) / Pi^3.

A178684 Partial sums of cardinalities of coalition sets A095941.

Original entry on oeis.org

0, 0, 1, 5, 18, 53, 138, 332, 757, 1661, 3546, 7424, 15328, 31336, 63618, 128531, 258811, 519956, 1042992, 2090009, 4185231, 8377158, 16762853, 33536516, 67086633, 134190278, 268401718, 536829625, 1073691505, 2147422558
Offset: 1

Views

Author

Jonathan Vos Post, Dec 25 2010

Keywords

Comments

Partial sums of number of subsets of {1,2,...,n} such that every number in the set is no larger than the sum of the other numbers in the set. See formula in A095944. The subsequence of primes begins: 5, 53, 757, 2090009, 16762853.

Examples

			a(9) = 0 + 0 + 1 + 4 + 13 + 35 + 85 + 194 + 425 = 757 is prime.

Crossrefs

Cf. A000009, A002623, A095941, A095944.

A270105 a(n) = Sum_{k=0..n} k*A000009(k).

Original entry on oeis.org

0, 1, 3, 9, 17, 32, 56, 91, 139, 211, 311, 443, 623, 857, 1165, 1570, 2082, 2728, 3556, 4582, 5862, 7458, 9416, 11808, 14736, 18286, 22576, 27760, 33976, 41400, 50280, 60820, 73300, 88084, 105492, 125967, 150015, 178135, 210967, 249265, 293785, 345445, 405337
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 12 2016

Keywords

Crossrefs

Cf. A000009, A000070, A014153, A036469, A095944, A182738.
Partial sums of A066189.

Programs

  • Mathematica
    Table[Sum[PartitionsQ[k]*k, {k, 0, n}], {n, 0, 50}]

Formula

a(n) ~ 3^(1/4) * n^(3/4) * exp(sqrt(n/3)*Pi) / (2*Pi).
G.f.: x*f'(x)/(1 - x), where f(x) = Product_{k>=1} (1 + x^k). - Ilya Gutkovskiy, Apr 13 2017

A304781 a(n) = [x^n] (1/(1 - x)^n)*Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 2, 6, 21, 75, 274, 1016, 3807, 14377, 54627, 208584, 799669, 3076167, 11867511, 45897145, 177888715, 690770763, 2686879415, 10466761637, 40828165464, 159453481037, 623427464093, 2439907421914, 9557831470082, 37472409664888, 147028505564603, 577302980976146
Offset: 0

Views

Author

Ilya Gutkovskiy, May 18 2018

Keywords

Comments

Number of partitions of n into odd parts with n + 1 kinds of 1.

Links

Crossrefs

Cf. A000009, A036469, A095944, A128566, A128593, A292463, A292613, A293467.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 - x)^n Product[(1 + x^k), {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x)^n Product[1/(1 - x^(2 k - 1)), {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x)^n Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[QPochhammer[-1, x]/(2 (1 - x)^n), {x, 0, n}], {n, 0, 26}]

Formula

a(n) = [x^n] (1/(1 - x)^n)*Product_{k>=1} 1/(1 - x^(2*k-1)).
a(n) = [x^n] (1/(1 - x)^n)*exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))).
a(n) ~ QPochhammer[-1, 1/2] * 4^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, May 18 2018

A325952 G.f.: 1/(1-x)^4 * Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 5, 15, 36, 75, 142, 251, 421, 677, 1052, 1589, 2343, 3384, 4800, 6701, 9224, 12538, 16850, 22413, 29534, 38584, 50010, 64348, 82238, 104442, 131864, 165573, 206830, 257118, 318176, 392039, 481082, 588070, 716216, 869245, 1051467, 1267860, 1524162, 1826975
Offset: 0

Views

Author

Vaclav Kotesovec, May 28 2019

Keywords

Comments

In general, if g.f. = 1/(1-x)^m * Product_{k>=1} (1 + x^k), then a(n) ~ 2^(m - 2) * 3^(m/2 - 1/4) * n^(m/2 - 3/4) * exp(Pi*sqrt(n/3)) / Pi^m.

Crossrefs

Cf. A000009, A014161, A036469, A095944, A120477, A292508, A325951.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[(1/(1-x)^4 * Product[1+x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

a(n) ~ 4 * 3^(7/4) * n^(5/4) * exp(Pi*sqrt(n/3)) / Pi^4.

A360489 Convolution of A000219 and A001477.

Original entry on oeis.org

0, 1, 3, 8, 19, 43, 91, 187, 369, 711, 1335, 2459, 4442, 7904, 13851, 23965, 40958, 69248, 115872, 192097, 315652, 514485, 832112, 1336214, 2131099, 3377178, 5319290, 8330147, 12973662, 20100411, 30986772, 47542096, 72609729, 110410791, 167186826, 252138816, 378781852
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 09 2023

Keywords

Comments

In general, for 0 < p < 1, delta > 1, beta > -1, the convolution of (delta^(n^p) * n^alfa) and n^beta is asymptotic to delta^(n^p) * n^(alfa + (1-p)*(beta+1)) * Gamma(beta+1) / (p^(beta+1) * log(delta)^(beta+1)).
For p = 1 is the convolution of (delta^(n^p) * n^alfa) and n^beta asymptotic to delta^n * n^alfa * polylog(-beta, 1/delta).

Crossrefs

Cf. A000219, A001477, A014153, A095944, A161870.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> add(b(n-j)*j, j=0..n):
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 09 2023
  • Mathematica
    nmax = 50; CoefficientList[Series[x/(1-x)^2 * Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) = Sum_{k=0..n} A000219(k) * (n-k).
G.f.: x/(1-x)^2 * Product_{k>=1} 1/(1 - x^k)^k.
a(n) ~ exp(1/12 + 3*zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * sqrt(3*Pi) * 2^(35/36) * zeta(3)^(17/36) * n^(1/36)), where A is the Glaisher-Kinkelin constant A074962.
Previous Showing 11-17 of 17 results.

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