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.

A383034 Inverse Weigh transform of 2^(n-1).

Original entry on oeis.org

1, 2, 2, 5, 6, 11, 18, 35, 56, 105, 186, 346, 630, 1179, 2182, 4115, 7710, 14588, 27594, 52482, 99858, 190743, 364722, 699216, 1342176, 2581425, 4971008, 9587574, 18512790, 35792449, 69273666, 134219795, 260300986, 505294125, 981706806, 1908881548, 3714566310
Offset: 1

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Author

Seiichi Manyama, Apr 13 2025

Keywords

Crossrefs

Column k=2 of A383033.

Formula

a(n) = (1/n) * (2^n - 1 + Sum_{d
a(n) = A306156(n) - A209229(n).
Product_{k>=1} (1 + x^k)^a(k) = (1 - x)/(1 - 2*x).

A383035 Inverse Weigh transform of 3^(n-1).

Original entry on oeis.org

1, 3, 6, 18, 42, 113, 294, 798, 2128, 5823, 15918, 43998, 122010, 340617, 954394, 2686728, 7588770, 21509824, 61144062, 174289710, 498012094, 1426229109, 4092816966, 11767220068, 33890202192, 97761550215, 282424564744, 817018885362, 2366546223930, 6863002420335
Offset: 1

Author

Seiichi Manyama, Apr 13 2025

Keywords

Crossrefs

Column k=3 of A383033.

Formula

a(n) = (1/n) * (3^n - 2^n + Sum_{d
a(n) = A306157(n) - A306156(n).
Product_{k>=1} (1 + x^k)^a(k) = (1 - 2*x)/(1 - 3*x).

A383042 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Euler transform of j-> k^(j-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 15, 6, 0, 1, 5, 20, 42, 42, 9, 0, 1, 6, 30, 90, 156, 107, 18, 0, 1, 7, 42, 165, 420, 554, 294, 30, 0, 1, 8, 56, 273, 930, 1910, 2028, 780, 56, 0, 1, 9, 72, 420, 1806, 5155, 8820, 7350, 2128, 99, 0
Offset: 1

Author

Seiichi Manyama, Apr 13 2025

Keywords

Examples

			Square array begins:
  1,  1,   1,    1,    1,     1,     1, ...
  0,  1,   2,    3,    4,     5,     6, ...
  0,  2,   6,   12,   20,    30,    42, ...
  0,  3,  15,   42,   90,   165,   273, ...
  0,  6,  42,  156,  420,   930,  1806, ...
  0,  9, 107,  554, 1910,  5155, 11809, ...
  0, 18, 294, 2028, 8820, 28830, 77658, ...
  ...
		

Crossrefs

Columns k=1..5 give A000007, A059966, A065178, A065179, A065180.
Main diagonal gives A306173.
Cf. A065177 (another version).

Programs

  • PARI
    a(n, k) = sumdiv(n, d, moebius(n/d)*(k^d-(k-1)^d))/n;

Formula

A(n,k) = (1/n) * Sum_{d|n} mu(n/d) * (k^d - (k-1)^d).
A(n,k) = (1/n) * (k^n - (k-1)^n - Sum_{d
A(n,k) = A074650(n,k) - A074650(n,k-1).
Product_{n>=1} 1/(1 - x^n)^A(n,k) = (1 - (k-1)*x)/(1 - k*x).
G.f. of column k: Sum_{j>=1} mu(j) * log(1 + x^j/(1 - k*x^j)) / j.
Showing 1-3 of 3 results.