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 1-3 of 3 results.

A077335 Sum of products of squares of parts in all partitions of n.

Original entry on oeis.org

1, 1, 5, 14, 46, 107, 352, 789, 2314, 5596, 14734, 34572, 92715, 210638, 531342, 1250635, 3042596, 6973974, 16973478, 38399806, 91301956, 207992892, 483244305, 1089029008, 2533640066, 5642905974, 12912848789, 28893132440, 65342580250, 144803524640
Offset: 0

Views

Author

Vladeta Jovovic, Nov 30 2002

Keywords

Examples

			The partitions of 4 are 4, 1+3, 2+2, 2+1+1, 1+1+1+1, the corresponding products of squares of parts are 16,9,16,4,1 and their sum is a(4) = 46.

Links

Crossrefs

Cf. A006906, A074141, A092484, A265844, A292164.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, i^2*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 07 2014
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, i^2*b[n-i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 02 2015, after Alois P. Heinz *)
Table[Total[Times@@(#^2)&/@IntegerPartitions[n]],{n,0,30}] (* Harvey P. Dale, Apr 29 2018 *)
Table[Total[Times@@@(IntegerPartitions[n]^2)],{n,0,30}] (* Harvey P. Dale, Sep 07 2023 *)
  • Maxima
    S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */
  • PARI
    N=22;q='q+O('q^N); Vec(1/prod(n=1,N,1-n^2*q^n)) \\ Joerg Arndt, Aug 31 2015

Formula

G.f.: 1/Product_{m>0} (1 - m^2*x^m).
Recurrence: a(n) = (1/n)*Sum_{k=1..n} b(k)*a(n-k), where b(k) = Sum_{d divides k} d^(2*k/d + 1).
a(n) = S(n,1), where S(n,m) = n^2 + Sum_{k=m..n/2} k^2*S(n-k,k), S(n,n) = n^2, S(n,m) = 0 for m > n. - Vladimir Kruchinin, Sep 07 2014
From Vaclav Kotesovec, Mar 16 2015: (Start)
a(n) ~ c * 3^(2*n/3), where
c = 668.1486183948153029651700839617715291485899132694809388646986235... if n=3k
c = 667.8494657534167286226227360927068283390090685342574808235616845... if n=3k+1
c = 667.8481656987523944806949678900876994934226621916594805916358627... if n=3k+2
(End)
In closed form, a(n) ~ (Product_{k>=4}(1/(1 - k^2/3^(2*k/3))) / ((1 - 3^(-2/3)) * (1 - 4*3^(-4/3))) + Product_{k>=4}(1/(1 - (-1)^(2*k/3)*k^2/3^(2*k/3))) / ((-1)^(2*n/3) * (1 + 4/3*(-1/3)^(1/3)) * (1 - (-1/3)^(2/3))) + Product_{k>=4}(1/(1 - (-1)^(4*k/3)*k^2/3^(2*k/3))) / ((-1)^(4*n/3) * (1 + (-1)^(1/3)*3^(-2/3)) * (1 - 4*(-1)^(2/3)*3^(-4/3)))) * 3^(2*n/3 - 1). - Vaclav Kotesovec, Apr 25 2017
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

A292166 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j^k*x^j).

Original entry on oeis.org

1, 1, -1, 1, -1, -1, 1, -1, -2, 0, 1, -1, -4, -1, 0, 1, -1, -8, -5, -1, 1, 1, -1, -16, -19, -7, 5, 0, 1, -1, -32, -65, -37, 27, 1, 1, 1, -1, -64, -211, -175, 155, 17, 13, 0, 1, -1, -128, -665, -781, 927, 205, 167, 4, 0, 1, -1, -256, -2059, -3367, 5675, 2129, 2089, 110, 0, 0
Offset: 0

Views

Author

Seiichi Manyama, Sep 10 2017

Keywords

Examples

			Square array begins:
    1,  1,  1,   1,    1, ...
   -1, -1, -1,  -1,   -1, ...
   -1, -2, -4,  -8,  -16, ...
    0, -1, -5, -19,  -65, ...
    0, -1, -7, -37, -175, ...

Links

Crossrefs

Columns k=0..2 give A010815, A022661, A292164.
Rows n=0..2 give A000012, (-1)*A000012, (-1)*A000079.
Main diagonal gives A292167.

Programs

  • Mathematica
    A[n_, k_] := A[n, k] = If[n == 0, 1, -(1/n)*Sum[Sum[d^(1+k*j/d), {d, Divisors[j]}]*A[n-j, k], {j, 1, n}]];
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 04 2022 *)

Formula

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(1+k*j/d)) * A(n-j,k) for n > 0. - Seiichi Manyama, Nov 02 2017

A294567 a(n) = Sum_{d|n} d^(1 + 2*n/d).

Original entry on oeis.org

1, 9, 28, 97, 126, 588, 344, 2049, 2917, 6174, 1332, 53764, 2198, 52320, 258648, 430081, 4914, 2463429, 6860, 8352582, 15181712, 8560308, 12168, 242240964, 48843751, 134606598, 1167064120, 1651526120, 24390, 14202123408, 29792, 25905102849, 94162701936
Offset: 1

Views

Author

Seiichi Manyama, Nov 02 2017

Keywords

Comments

If p is prime, a(p) = 1 + p^3. - Robert Israel, Nov 03 2017

Links

Crossrefs

Column k=2 of A294579.
Cf. A292164.

Programs

  • Maple
    f:= n -> add(d^(1+2*n/d),d=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Nov 03 2017
  • Mathematica
    sd[n_] := Module[{d = Divisors[n]}, Total[d^(1 + (2 n)/d)]]; Array[sd,40] (* Harvey P. Dale, Mar 17 2020 *)
a[n_] := DivisorSum[n, #^(1 + 2*n/#) &]; Array[a, 33] (* Amiram Eldar, Oct 04 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d^(1+2*n/d)); \\ Michel Marcus, Nov 02 2017
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^3*x^k/(1-k^2*x^k))) \\ Seiichi Manyama, Jan 14 2023

Formula

L.g.f.: -log(Product_{k>=1} (1 - k^2*x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 12 2018
G.f.: Sum_{k>0} k^3 * x^k / (1 - k^2 * x^k). - Seiichi Manyama, Jan 14 2023
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.