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 1-7 of 7 results.

A281267 Main diagonal of A276554.

Original entry on oeis.org

1, -1, -3, 8, 13, -51, -120, 538, 781, -5419, -3053, 47673, 5080, -427740, 136462, 3922383, -3278067, -34819588, 48561567, 299316651, -603368637, -2509708844, 6948730643, 20210062532, -76150197416, -152569240051, 801154765564, 1039352472008, -8158396721266
Offset: 0

Views

Author

Seiichi Manyama, Apr 13 2017

Keywords

Comments

From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 3 and all positive integers n and k. (End)

Links

Crossrefs

Cf. A255672, A270922, A276554.

Programs

  • Mathematica
    nmax = 40; Table[SeriesCoefficient[Product[(1 - x^k)^(n*k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Apr 17 2017 *)

Formula

a(n) = [x^n] exp(-n*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018

A277938 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1+x^j)^(j*k) in powers of x.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 5, 0, 1, 4, 9, 14, 8, 0, 1, 5, 14, 28, 30, 16, 0, 1, 6, 20, 48, 72, 68, 28, 0, 1, 7, 27, 75, 141, 183, 145, 49, 0, 1, 8, 35, 110, 245, 396, 443, 298, 83, 0, 1, 9, 44, 154, 393, 751, 1058, 1026, 600, 142, 0, 1, 10, 54, 208
Offset: 0

Views

Author

Seiichi Manyama, Apr 11 2017

Keywords

Examples

			Square array begins:
   1, 1,  1,  1,   1, ...
   0, 1,  2,  3,   4, ...
   0, 2,  5,  9,  14, ...
   0, 5, 14, 28,  48, ...
   0, 8, 30, 72, 141, ...

Links

Crossrefs

Columns k=0-4 give: A000007, A026007, A026011, A027346, A027906.
Rows n=0-3 give: A000012, A001477, A000096, A005586.
Main diagonal gives A270922.
Antidiagonal sums give A299167.
Cf. A276554, A279928.

Formula

G.f. of column k: Product_{j>=1} (1+x^j)^(j*k).

A299211 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)^k).

Original entry on oeis.org

1, 1, 0, -3, -6, -4, 12, 39, 52, -9, -186, -392, -285, 610, 2291, 3200, -150, -10626, -23487, -18841, 32957, 134848, 198246, 13961, -605248, -1409604, -1234474, 1744213, 7898753, 12209679, 2161666, -34344627, -84393284, -79993042, 90692470, 461463974, 749309529, 207447895, -1939084232
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A276554.
Cf. A067687, A073592, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299212.

Programs

  • Maple
    N:= 100: # for a(0)..a(N)
S:= series(1/(1-x*mul((1-x^k)^k,k=1..N)),x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Feb 05 2023
  • Mathematica
    nmax = 38; CoefficientList[Series[1/(1 - x Product[(1 - x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 - x^k)^k).
a(0) = 1; a(n) = Sum_{k=1..n} A073592(k-1)*a(n-k).

A276551 Convolution square of A073592.

Original entry on oeis.org

1, -2, -3, 2, 6, 12, 1, -10, -32, -46, -24, 18, 96, 168, 213, 124, -61, -386, -734, -992, -957, -386, 685, 2288, 3939, 5158, 5012, 2930, -1853, -8888, -17283, -24782, -28312, -24422, -9825, 16674, 54197, 96584, 134729, 153718, 138624, 73438, -49526, -228614
Offset: 0

Views

Author

Seiichi Manyama, Apr 10 2017

Keywords

Links

Crossrefs

Column k=2 of A276554.
Product_{k>0} (1-x^k)^(k*m): A161870 (m=-2), A073592 (m=1), this sequence (m=2), A276552 (m=3).

Formula

G.f.: Product_{k>0} (1-x^k)^(k*2).
G.f.: exp(-2*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018

A276552 Expansion of Product_{k>0} (1 - x^k)^(k*3).

Original entry on oeis.org

1, -3, -3, 8, 12, 9, -29, -54, -51, 8, 168, 273, 270, -18, -546, -1220, -1539, -969, 796, 3693, 6591, 8098, 5412, -2568, -16053, -31524, -42195, -38684, -11868, 41994, 117630, 193365, 235497, 197758, 42852, -247224, -639547, -1041432, -1291425, -1184100, -520650
Offset: 0

Views

Author

Seiichi Manyama, Apr 10 2017

Keywords

Links

Crossrefs

Column k=3 of A276554.
Product_{k>0} (1-x^k)^(k*m): A255610 (m=-3), A073592 (m=1), A276551 (m=2), this sequence (m=3).

A316463 Expansion of Product_{k>0} (1 - x^k)^(4*k).

Original entry on oeis.org

1, -4, -2, 16, 13, -12, -78, -72, 54, 244, 444, 184, -543, -1592, -2106, -1040, 2256, 7360, 11010, 9096, -3000, -24484, -49358, -60888, -38432, 32956, 150792, 275608, 335573, 231568, -109460, -689560, -1365517, -1832404, -1649570, -386008, 2148258, 5640240
Offset: 0

Views

Author

Seiichi Manyama, Jul 04 2018

Keywords

Links

Crossrefs

Column k=4 of A276554.
Cf. A255611.

Formula

Convolution inverse of A255611.

A316464 Expansion of Product_{k>0} (1 - x^k)^(5*k).

Original entry on oeis.org

1, -5, 0, 25, 5, -51, -120, 0, 325, 500, 273, -1000, -2475, -2575, 610, 7449, 13885, 13275, -3650, -37005, -73075, -78475, -16450, 130600, 326860, 452704, 339375, -174900, -1082100, -2092990, -2553196, -1602400, 1426925, 6411775, 11803520, 14448032, 10108055
Offset: 0

Views

Author

Seiichi Manyama, Jul 04 2018

Keywords

Links

Crossrefs

Column k=5 of A276554.
Cf. A255612.

Formula

Convolution inverse of A255612.
Showing 1-7 of 7 results.

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