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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A385912 A diagonal (unsigned) of triangle A385910; a(n) = -A385910(n+2,n) for n >= 1.

Original entry on oeis.org

1, 8, 49, 296, 1815, 11284, 70924, 449616, 2869779, 18418400, 118749345, 768537120, 4990021764, 32489701776, 212048505160, 1386886206112, 9087724409547, 59646983740680, 392071446052195, 2580601721867400, 17005938279649935, 112190574812699460, 740878216459158960, 4897062582469861440, 32395964187696107700
Offset: 1

Views

Author

Paul D. Hanna, Jul 14 2025

Keywords

Comments

Triangle A385910 has g.f. A(x,y) where A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y).
The main diagonal of A385910 has g.f. G(x) = 1 + x*G(x)^3 (A001764).
Conjectures:
(C.1) a(n) is odd iff n = 2*A003714(k) + 1 for k >= 0, where A003714 is the Fibbinary numbers.
(C.2) a(n) == 1 (mod 3) iff n = 3^k + 1 for k >= 0.
(C.3) a(n) == 2 (mod 3) iff n = A038464(k)/2 + 1 for k >= 1 or n = 1, where A038464 lists sums of 2 distinct powers of 3.

Examples

			G.f. A(x) = x + 8*x^2 + 49*x^3 + 296*x^4 + 1815*x^5 + 11284*x^6 + 70924*x^7 + 449616*x^8 + 2869779*x^9 + 18418400*x^10 + 118749345*x^11 + 768537120*x^12 + ...

Links

Crossrefs

Cf. A385910, A003714, A118113, A038464.

Programs

  • PARI
    \\ a(n) = -A385910(n+2,n) for n >= 1
{A385910(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A] = polcoeff( subst(Ax, x, x^3 + 3*y*x*Ax^3 ) - Ax*subst(Ax, x, x^2 + 2*y*x*Ax^2 ), #A+1)); A[n+1]}
for(n=1, 25, print1(polcoef(-A385910(n+2),n),", "))

A385911 If n = 2^b*3^c, then a(n) = (-1)^b * binomial(b+c, b), else a(n) = 0, for n >= 1.

Original entry on oeis.org

1, -1, 1, 1, 0, -2, 0, -1, 1, 0, 0, 3, 0, 0, 0, 1, 0, -3, 0, 0, 0, 0, 0, -4, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0
Offset: 1

Views

Author

Paul D. Hanna, Jul 12 2025

Keywords

Comments

See comment by David Wasserman in related sequence A061984.
Triangle A385910 has g.f. A(x,y) where A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y).
a(n) = A385910(n+1, 1) for n >= 1.

Examples

			G.f. A(x) = x - x^2 + x^3 + x^4 - 2*x^6 - x^8 + x^9 + 3*x^12 + x^16 - 3*x^18 - 4*x^24 + x^27 - x^32 + 6*x^36 + 5*x^48 - 4*x^54 + x^64 - 10*x^72 + x^81 - 6*x^96 + ...
where A(x) equals the sum of the following series
A(x) = (x + x^3 + x^9 + x^27 + ... + x^(3^k) + ...)
  - (x^2 + 2*x^6 + 3*x^18 + 4*x^54 + ... + (k+1)*x^(2*3^k) + ...)
  + (x^4 + 3*x^12 + 6*x^36 + 10*x^108 + ... + C(2+k,k)*x^(2^2*3^k) + ...)
  - (x^8 + 4*x^24 + 10*x^72 + 20*x^216 + ... + C(3+k,k)*x^(2^3*3^k) + ...)
  + (x^16 + 5*x^48 + 15*x^144 + 35*x^432 + ... + C(4+k,k)*x^(2^4*3^k) + ...)
  + ... + (-1)^n * Sum_{k>=0} binomial(n+k,k) * x^(2^n*3^k) + ...

Crossrefs

Cf. A385910, A061984.

Programs

  • PARI
    {a(n) = my(p2,p3); if(n<1,0, p2 = valuation(n,2); p3 = valuation(n,3);
if(n/(2^p2*3^p3)>1,0, (-1)^p2 * binomial(p2 + p3, p2) ))}
for(n=1,100,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = Sum_{n>=0} (-1)^n * Sum_{k>=0} binomial(n+k,k) * x^(2^n*3^k).
(2) If n = 2^b*3^c, then a(n) = (-1)^b * binomial(b+c, b) else a(n) = 0, for n >= 1.
Showing 1-2 of 2 results.

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