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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A360486 Convolution of A000041 and A000290.

Original entry on oeis.org

0, 1, 5, 15, 36, 76, 147, 267, 462, 769, 1240, 1947, 2988, 4496, 6649, 9683, 13909, 19734, 27686, 38447, 52892, 72138, 97604, 131084, 174835, 231687, 305173, 399687, 520675, 674865, 870540, 1117869, 1429298, 1820018, 2308521, 2917260, 3673428, 4609885, 5766245
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 09 2023

Keywords

Crossrefs

Cf. A000041, A000290, A014153, A360487.

Programs

  • Maple
    a:= n-> add(combinat[numbpart](n-j)*j^2, j=0..n):
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 09 2023
  • Mathematica
    Table[Sum[PartitionsP[k]*(n-k)^2, {k, 0, n}], {n, 0, 60}]
CoefficientList[Series[x*(1+x) / ((1-x)^3 * QPochhammer[x]), {x, 0, 60}], x]
  • PARI
    a(n) = sum(k=0, n, numbpart(k)*(n-k)^2); \\ Michel Marcus, Feb 09 2023

Formula

a(n) = Sum_{k=0..n} A000041(k) * (n-k)^2.
G.f.: x*(1+x)/(1-x)^3 * Product_{k>=1} 1/(1 - x^k).
a(n) ~ 3 * sqrt(2*n) * exp(sqrt(2*n/3)*Pi) / Pi^3.

A263982 Number of partitions of n with a palindromicity of 3.

Original entry on oeis.org

1, 1, 3, 4, 8, 10, 18, 22, 36, 44, 67, 81, 119, 142
Offset: 6

Views

Author

Gregory L. Simay, Nov 01 2015

Keywords

Comments

A non-strict partition of n can be arranged into two consecutive sequences that are mirror images of each other, separated by a central sequence of k distinct summands, k>=0. Strict partitions (ref. A000009) only have the central sequence. In both cases, the palindromicity of the partition is k.
Palindromic partitions (ref. A025065) have palindromicity 0 (no central summand) or 1 (central summand). Non-palindromic partitions have palindromicities >=2.

Examples

			If considered unordered rather than a nonincreasing sequence, the partition 9,7,7,5,4,4,2,2,1,1,1 can be arranged as 7,4,2,1 [central sequence of 9,5,1] 1,2,4,7. Therefore the palindromicity of this particular partition is 3.
w(14,3) = w(11,2) + w(11,3) = A014153(4) + 10 = 26 + 10 = 36.
w(15,3) = w(2*8-1,3) = A014153(4) + A014153(3) + A014153(1) + A014153(0) = 26 + 14 + 3 + 1 = 44.

Crossrefs

Cf. A000070, A014153, A025065.

Formula

p(n,k) = number of partitions of n with palindromicity k.
If k*(k+1)/2 <= p(n) < (k+1)*(k+2)/2, then p(n) = p(n,0) + .. + p(n,k)
Let q(n,k)= number of strict partitions of n (ref. A000009) with exactly k parts. Then p(n,k) = Sum_{j>=0} q(n-2j,k)*p(j), which affords another way to demonstrate that the convolution of q(2n-j) with p(j) equals p(2n).
p(2n,0) = p(n) and p(2n+1,0) = 0 (ref. A025065).
p(2n,1) = p(2n+1,1) = A000070(n-1), the first partial sum of A000041 (ref. A025065).
p(2n,2) = p(2n-1,2) = A014153(n-2), the second partial sum of A000041.
p(2n,3) = A014153(n-3) + A014153(n-5) + A014153(n-6) + A014153(n-8) + A014153(n-9) + A014153(n-11) + A014153(n-12) + ...
G.f. for p(2n,3): p(x)* x^3*(1+x+x^2+x^3)/(1-x)*(1-x^2)*(1-x^3) where p(x) is the g.f. for A000041.
p(2n-1,3) = A014153(n-4) + A014153(n-5) + A014153(n-7) + A014153(n-8) + A014153(n-10) + A014153(n-11) + A014153(n-13) + A014153(n-14) + ...
G.f. for p(2n-1,3): p(x)* x^3*(1+2x+x^2)/(1-x)*(1-x^2)*(1-x^3) where p(x) is the g.f. for A000041.
More generally, p(n,k>=3) = p(n-k,k-1) + p(n-2k, k-1) + p(n-3k,n-1) + ... for k>=3 = p(n-k, k-1) + p(n-k,k).

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

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 31-34 of 34 results.

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