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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A086718 Convolution of sequence of primes with sequence sigma(n).

Original entry on oeis.org

2, 9, 22, 48, 85, 151, 231, 355, 500, 709, 937, 1267, 1617, 2069, 2575, 3193, 3860, 4686, 5549, 6593, 7725, 8985, 10337, 11961, 13591, 15464, 17498, 19714, 22036, 24690, 27378, 30382, 33603, 37023, 40597, 44733, 48720, 53152, 57950, 62978, 68074, 73898, 79558
Offset: 1

Views

Author

Jon Perry, Jul 29 2003

Keywords

Comments

From Omar E. Pol, Dec 06 2021: (Start)
Antidiagonal sums of A272214.
Convolution of A000040 and A000203.
Convolution of A054541 and A024916.
Convolution of the nonzero terms of A007504 and A340793.
a(n) is also the volume of a tower or polycube in which the successive terraces are the symmetric representation of sigma(k), k = 1..n starting from the top, and the successive heights of the terraces are the prime numbers starting from the base. (End)

Links

Crossrefs

Cf. A000040, A000203.
Cf. A007504, A024916, A054541, A066186, A175254, A191831, A237593, A272214, A340793.

Programs

  • Maple
    N:= 100: # to get a(1)..a(N)
P:= [seq(ithprime(i),i=1..N+1)]:
S:= [seq(numtheory:-sigma(i),i=1..N+1)]:
seq(add(P[i]*S[n-i],i=1..n-1),n=2..N+1); # Robert Israel, Sep 09 2020
  • PARI
    p=primes(30); s=vector(30,i, sigma(i)); conv(u,v)=local(w); w=vector(length(u),i,sum(j=1,i,u[j]*v[i+1-j])); w;
conv(p,s)

Extensions

More terms from Robert Israel, Sep 09 2020

A277029 Convolution of A000203 and A000009.

Original entry on oeis.org

0, 1, 4, 8, 16, 25, 42, 61, 90, 130, 178, 242, 332, 436, 566, 747, 952, 1210, 1540, 1926, 2400, 2994, 3674, 4506, 5526, 6708, 8108, 9808, 11768, 14080, 16850, 20022, 23738, 28128, 33152, 39015, 45854, 53662, 62696, 73166, 85118, 98826, 114636, 132586, 153102
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 25 2016

Keywords

Comments

Apart from initial zero this is the convolution of A340793 and A036469. - Omar E. Pol, Feb 16 2021

Links

Crossrefs

Cf. A066186 (convolution of A000203 and A000041).
Cf. A276432 (convolution of A000203 and A000219).
Cf. A066189, A036469, A340793.

Programs

  • Mathematica
    Table[Sum[DivisorSigma[1, k] * PartitionsQ[n-k], {k,1,n}], {n,0,50}]
nmax = 50; CoefficientList[Series[Sum[j*x^j/(1-x^j), {j, 1, nmax}]*Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

A276432 Sum of the traces of all plane partitions of n.

Original entry on oeis.org

1, 4, 10, 26, 56, 126, 252, 512, 980, 1866, 3427, 6258, 11121, 19618, 33975, 58328, 98732, 165804, 275246, 453544, 740338, 1200088, 1929897, 3083898, 4893775, 7720826, 12106814, 18883104, 29291740, 45215386, 69451631, 106197524, 161656759, 245050410, 369935066
Offset: 1

Views

Author

Emeric Deutsch, Sep 24 2016

Keywords

Comments

Convolution of A000203 and A000219. - Vaclav Kotesovec, Sep 25 2016
Convolution of A340793 and A091360. - Omar E. Pol, Feb 16 2021

Examples

			a(3) = 10 because the 6 (=A000219(3)) planar partitions of 3 are [3], [2,1], [2;1], [1,1,1], [1;1;1], [1,1;1] (; indicates a new row); the sum of their traces is 3+2+2+1+1+1 = 10.

References

  • G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, pp. 179-201.

Links

Crossrefs

Cf. A000219, A089353, A277029.
Cf. A091360, A340793.

Programs

  • Maple
    g:= (sum(j*x^j/(1-x^j),j = 1..100))/(product((1-x^k)^k,k = 1..100)): gser := series(g, x = 0,40): seq(coeff(gser, x, m), m = 1 .. 35);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0, add((p
      ->p+[0, j*p[1]])(b(n-i*j, i-1))*binomial(i+j-1, j), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Sep 24 2018
  • Mathematica
    nmax = 50; Rest[CoefficientList[Series[Sum[j*x^j/(1-x^j), {j, 1, nmax}]*Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 25 2016 *)

Formula

G.f.: g(x) = Sum_{j>=1} (j*x^j/(1-x^j))/Product_{k>=1} (1-x^k)^k.
a(n) = Sum(k*A089353(n,k), k>=1).

A086733 Convolution of sigma(n) with phi(n).

Original entry on oeis.org

1, 4, 9, 19, 31, 54, 74, 117, 148, 217, 252, 366, 408, 562, 612, 833, 853, 1171, 1203, 1566, 1606, 2104, 2030, 2718, 2655, 3347, 3332, 4262, 3954, 5226, 4984, 6161, 5971, 7566, 6874, 8961, 8361, 10194, 9732, 12210, 10912, 14122, 13012, 15654, 14858, 18494
Offset: 1

Views

Author

Jon Perry, Jul 29 2003

Keywords

Comments

Convolution of A340793 and the nonzero terms of A002088. - Omar E. Pol, Feb 17 2021

Crossrefs

Cf. A000010, A000203, A002088, A340793.

Programs

  • Mathematica
    Table[ListConvolve[DivisorSigma[1,Range[n]],EulerPhi[Range[n]]],{n,50}]// Flatten (* Harvey P. Dale, Jul 18 2021 *)
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