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.

A382125 G.f. A(x) = exp( Sum_{n>=1} sigma(n)*sigma(2*n) * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Original entry on oeis.org

1, 3, 15, 52, 180, 555, 1696, 4809, 13410, 35844, 93771, 238305, 594403, 1449441, 3476607, 8190824, 19015548, 43492230, 98197506, 218885763, 482337864, 1051051262, 2266904481, 4840955055, 10242621395, 21479302368, 44666897613, 92139573135, 188617118541, 383280793962, 773395096907
Offset: 0

Views

Author

Paul D. Hanna, Apr 06 2025

Keywords

Comments

Compare with g.f. for partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
Equals the self-convolution cube of A382124.
Conjectures: a(3*n) == A382124(n) (mod 3) for n >= 0; a(3*n+1) == 0 (mod 3) and a(3*n+2) == 0 (mod 3) for n >= 0.

Examples

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 52*x^3 + 180*x^4 + 555*x^5 + 1696*x^6 + 4809*x^7 + 13410*x^8 + 35844*x^9 + 93771*x^10 + ...
where
A(x) = exp(3*x + 21*x^2/2 + 48*x^3/3 + 105*x^4/4 + 108*x^5/5 + 336*x^6/6 + 192*x^7/7 + 465*x^8/8 + 507*x^9/9 + 756*x^10/10 + ... + sigma(n)*sigma(2*n)*x^n/n + ...).
RELATED SERIES.
A(x)^(1/3) = 1 + x + 4*x^2 + 9*x^3 + 22*x^4 + 44*x^5 + 105*x^6 + 200*x^7 + 425*x^8 + 825*x^9 + 1634*x^10 + ... + A382124(n)*x^n + ...

Links

Crossrefs

Cf. A382124, A382123, A156302, A347108, A000203 (sigma), A000041 (partitions).
Cf. A087943, A329963.

Programs

  • Mathematica
    nmax=30; CoefficientList[Series[Exp[Sum[DivisorSigma[1,n]DivisorSigma[1,2*n] * x^n/n ,{n,nmax}]],{x,0,nmax}],x] (* Stefano Spezia, Apr 06 2025 *)
  • PARI
    {a(n) = my(A = exp( sum(m=1,n, sigma(m)*sigma(2*m)*x^m/m ) +x*O(x^n) ));
polcoef(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = exp( Sum_{n>=1} sigma(n)*sigma(2*n) * x^n/n ).
(2) A(x) = exp( Sum_{n>=1} Sum_{k>=1} sigma(2*n*k) * x^(n*k) / n ).
(3) a(n) = (1/n) * Sum_{k=1..n} sigma(k)*sigma(2*k) * a(n-k) for n>0, with a(0) = 1.

A382124 G.f. A(x) = exp( Sum_{n>=1} sigma(n)*sigma(2*n)/3 * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Original entry on oeis.org

1, 1, 4, 9, 22, 44, 105, 200, 425, 825, 1634, 3072, 5917, 10846, 20153, 36436, 65882, 116831, 207293, 361502, 629539, 1083068, 1856251, 3150554, 5328137, 8933266, 14920357, 24745481, 40869317, 67089425, 109697089, 178379353, 288953043, 465805681, 748079686, 1196148976, 1905801579, 3024212984
Offset: 0

Views

Author

Paul D. Hanna, Apr 06 2025

Keywords

Comments

Compare with g.f. for partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
Equals the self-convolution cube root of A382125.
Conjecture: a(n) == A382125(3*n) (mod 3) for n >= 0.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 22*x^4 + 44*x^5 + 105*x^6 + 200*x^7 + 425*x^8 + 825*x^9 + 1634*x^10 + 3072*x^11 + 5917*x^12 + ...
RELATED SERIES.
A(x)^3 = 1 + 3*x + 15*x^2 + 52*x^3 + 180*x^4 + 555*x^5 + 1696*x^6 + 4809*x^7 + 13410*x^8 + ... + A382125(n)*x^n + ...

Links

Crossrefs

Cf. A382125, A382123, A156302, A347108, A000203 (sigma), A000041 (partitions).
Cf. A087943, A329963.

Programs

  • Mathematica
    nmax=37; CoefficientList[Series[Exp[Sum[DivisorSigma[1,n]DivisorSigma[1,2*n] * x^n/(3n) ,{n,nmax}]],{x,0,nmax}],x] (* Stefano Spezia, Apr 06 2025 *)
  • PARI
    {a(n) = my(A = exp( sum(m=1,n, sigma(m)*sigma(2*m)/3*x^m/m ) +x*O(x^n) ));
polcoef(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = exp( (1/3) * Sum_{n>=1} sigma(n)*sigma(2*n) * x^n/n ).
(2) A(x) = exp( (1/3) * Sum_{n>=1} (1/n) * Sum_{k>=1} sigma(2*n*k) * x^(n*k) ).
(3) a(n) = (1/n) * Sum_{k=1..n} sigma(k)*sigma(2*k)/3 * a(n-k) for n > 0, with a(0) = 1.

A382122 G.f. satisfies Sum_{n>=0} x^n * abs(1/A(x)^n) = C(x), where C(x) = 1 + x*C(x)^2 and abs(F(x)) equals the series expansion formed by the unsigned coefficients in F(x).

Original entry on oeis.org

1, 1, 3, 12, 49, 202, 838, 3486, 14575, 60820, 254406, 1061438, 4444802, 18602018, 78066384, 326985608, 1365996909, 5697914836, 23752394338, 99027785702, 413203462516, 1726164299990, 7219911692522, 30228722494504, 126658682953328, 530772842793396, 2224199143900798, 9319843329508200, 39051457052597480
Offset: 0

Views

Author

Paul D. Hanna, Mar 16 2025

Keywords

Comments

Compare to Sum_{n>=0} x^n * C(x)^n = C(x), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
Conjecture: for n > 0, a(n) is odd iff n = 2^k for k >= 0.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 49*x^4 + 202*x^5 + 838*x^6 + 3486*x^7 + 14575*x^8 + 60820*x^9 + 254406*x^10 + 1061438*x^11 + 4444802*x^12 + ...
Below we illustrate the defining property of this sequence.
The coefficients in 1/A(x)^n begin
 1: [1,  -1, -2,  -7, -24, -84, -298, -1063, ...];
 2: [1,  -2, -3, -10, -30, -92, -283,  -858, ...];
 3: [1,  -3, -3, -10, -24, -57, -119,  -156, ...];
 4: [1,  -4, -2,  -8, -11,  -4,   82,   568, ...];
 5: [1,  -5,  0,  -5,   5,  49,  250,  1060, ...];
 6: [1,  -6,  3,  -2,  21,  90,  348,  1224, ...];
 7: [1,  -7,  7,   0,  35, 112,  364,  1070, ...];
 8: [1,  -8, 12,   0,  46, 112,  304,   672, ...];
 9: [1,  -9, 18,  -3,  54,  90,  186,   135, ...];
10: [1, -10, 25, -10,  60,  48,   35,  -430, ...];
...
The table of unsigned coefficients that form the series abs(1/A(x)^n) begins
 0: [1,  0,  0,  0,  0,   0,   0,    0,    0, ...];
 1: [1,  1,  2,  7, 24,  84, 298, 1063, 3858, ...];
 2: [1,  2,  3, 10, 30,  92, 283,  858, 2646, ...];
 3: [1,  3,  3, 10, 24,  57, 119,  156,  144, ...];
 4: [1,  4,  2,  8, 11,   4,  82,  568, 2578, ...];
 5: [1,  5,  0,  5,  5,  49, 250, 1060, 3800, ...];
 6: [1,  6,  3,  2, 21,  90, 348, 1224, 3654, ...];
 7: [1,  7,  7,  0, 35, 112, 364, 1070, 2394, ...];
 8: [1,  8, 12,  0, 46, 112, 304,  672,  469, ...];
 9: [1,  9, 18,  3, 54,  90, 186,  135, 1629, ...];
10: [1, 10, 25, 10, 60,  48,  35,  430, 3465, ...];
...
the antidiagonals of which add to the Catalan numbers (A000108):
  1 = 1;
  0 + 1 = 1;
  0 + 1 + 1 = 2;
  0 + 2 + 2 + 1 = 5;
  0 + 7 + 3 + 3 + 1 = 14;
  0 + 24 + 10 + 3 + 4 + 1 = 42;
  0 + 84 + 30 + 10 + 2 + 5 + 1 = 132;
  0 + 298 + 92 + 24 + 8 + 0 + 6 + 1 = 429;
  0 + 1063 + 283 + 57 + 11 + 5 + 3 + 7 + 1 = 1430;
  0 + 3858 + 858 + 119 + 4 + 5 + 2 + 7 + 8 + 1 = 4862;
  ...

Links

Crossrefs

Cf. A000108, A382123.

Programs

  • PARI
    {a(n) = my(V=[1,1], A, C = (1/x)*serreverse(x - x^2 +x^4*O(x^n)));
for(i=1,n, V = concat(V,'t); A = Ser(V);
V[#V] = 't + polcoef(C - sum(m=1,#V+1, x^m * Ser(abs(Vec( 1/A^m ))) ),#V) );V[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) Sum_{n>=0} x^n * abs(1/A(x)^n) = C(x), where C(x) = 1 + x*C(x)^2.
(2) Sum_{k=0..n} abs( [x^k] 1/A(x)^(n-k) ) = binomial(2*n+1,n)/(2*n+1) for n >= 0.
a(n) ~ c * d^n, where d = 4.1935797816358..., c = 0.142779... - Vaclav Kotesovec, Mar 28 2025
Showing 1-3 of 3 results.

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

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