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 1-5 of 5 results.

A276554 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, -3, -1, 0, 1, -4, -3, 2, 0, 0, 1, -5, -2, 8, 6, 4, 0, 1, -6, 0, 16, 12, 12, 4, 0, 1, -7, 3, 25, 13, 9, 1, 7, 0, 1, -8, 7, 34, 5, -12, -29, -10, 3, 0, 1, -9, 12, 42, -15, -51, -78, -54, -32, -2, 0, 1, -10, 18, 48, -49, -102
Offset: 0

Views

Author

Seiichi Manyama, Apr 10 2017

Keywords

Examples

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -2, -3, -4, ...
   0, -2, -3, -3, -2, ...
   0, -1,  2,  8, 16, ...
   0,  0,  6, 12, 13, ...

Links

Crossrefs

Columns k=0-5 give: A000007, A073592, A276551, A276552, A316463, A316464.
Main diagonal gives A281267.
Antidiagonal sums give A299211.
Cf. A255961, A277938.

Formula

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

A301624 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k)^k.

Original entry on oeis.org

1, -1, -1, 4, 1, -17, -6, 118, -8, -876, 625, 5966, -7486, -41937, 75969, 306312, -768637, -2164992, 7487063, 14461466, -70259884, -89410774, 646971980, 459817892, -5861484630, -1128608133, 52082250637, -15894742662, -453574650852, 366848121166, 3866670213663, -5215687717614
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 24 2018

Keywords

Examples

			G.f. A(x) = 1 - x - x^2 + 4*x^3 + x^4 - 17*x^5 - 6*x^6 + 118*x^7 - 8*x^8 - 876*x^9 + 625*x^10 + ...
G.f. A(x) satisfies: A(x) = (1 - x*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * (1 - x^4*A(x)^4)^4 * ...
log(A(x)) = -x - 3*x^2/2 + 8*x^3/3 + 13*x^4/4 - 51*x^5/5 - 120*x^6/6 + 538*x^7/7 + 781*x^8/8 - 5419*x^9/9 - 3053*x^10/10 + ... + A281267(n)*x^n/n + ...

Crossrefs

Cf. A000219, A006195, A066398, A073592, A109085, A181315, A278428, A281267, A301455, A301456, A301625.

Programs

  • Maple
    with(numtheory):
Order := 33:
Gser := solve(series(x*exp(add(sigma[2](n)*x^n/n, n = 1..32)), x) = y, x):
seq(coeff(Gser, y^k), k = 1..32); # Peter Bala, Feb 09 2020

Formula

From Peter Bala, Feb 09 2020: (Start)
A(x) = 1/x * series reversion of ( exp( Sum_{n >= 1} sigma_2(n)*x^n/n ) ), where sigma_2(n) = A001157(n).
Equivalently, the o.g.f. A(x) satisfies [x^n](1/A(x))^n = sigma_2(n) for n >= 1. Cf. A066398. (End)
A(x) equals (1/x) * series reversion of (x * the o.g.f. for the sequence of planar partitions A000219). - Peter Bala, Feb 11 2020

A299033 a(n) = n! * [x^n] Product_{k>=1} (1 - x^k)^(n/k).

Original entry on oeis.org

1, -1, 0, 15, -136, 885, -4896, 43085, -787200, 7775271, 326355200, -22138191801, 781498160640, -18924340012435, 239123351330304, 5915023788331125, -568462201562300416, 25327272129182225295, -795994018378027868160, 15538852668590468027711
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 01 2018

Keywords

Examples

			The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} (1 - x^k)^(n/k) begins:
n = 0: (1),  0,    0,    0,     0,      0,      0,  ...
n = 1:  1, (-1),  -1,    1,    -1,     41,   -131,  ...
n = 2:  1,  -2,   (0),   8,    -4,     72,   -704,  ...
n = 3:  1,  -3,    3,  (15),  -45,     63,  -1539,  ...
n = 4:  1,  -4,    8,   16, (-136),   224,  -1856,  ...
n = 5:  1,  -5,   15,    5,  -265,   (885), -2075,  ...
n = 6:  1,  -6,   24,  -24,  -396,   2376, (-4896), ...

Crossrefs

Cf. A000005, A028343, A281267, A299034.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Product[(1 - x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

Formula

a(n) = n! * [x^n] exp(-n*Sum_{k>=1} d(k)*x^k/k), where d(k) is the number of divisors of k (A000005).

A300457 a(n) = [x^n] Product_{k=1..n} (1 - x^k)^(n^k).

Original entry on oeis.org

1, -1, -3, -1, 25, 624, 9871, 170470, 3027249, 55077245, 979330606, 15079702923, 94670678245, -7958168036625, -626145997536240, -34564907982551791, -1733699815491494303, -84294315853736719077, -4067859614343931897505, -196552300464314521511610, -9519733465269825759734169
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 06 2018

Keywords

Examples

			The table of coefficients of x^k in expansion of Product_{k>=1} (1 - x^k)^(n^k) begins:
n = 0: (1),  0,    0,    0,   0,     0,  ...
n = 1:  1, (-1),  -1,    0,   0,     1,  ...
n = 2:  1,  -2,  (-3),   0,   2,    12,  ...
n = 3:  1,  -3,   -6,  (-1),  9,    63,  ...
n = 4:  1,  -4,  -10,   -4, (25),  224,  ...
n = 5:  1,  -5,  -15,  -10,  55,  (624), ...

Crossrefs

Cf. A010815, A008705, A252654, A252782, A255672, A270917, A270922, A281266, A281267, A281268, A283333, A292805, A300456, A300458.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 - x^k)^(n^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

A300458 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n^k).

Original entry on oeis.org

1, -1, -1, -10, 11, 374, 9792, 183847, 3469427, 65038049, 1195396233, 19667738452, 189089161562, -6219720781782, -606316892131934, -35104997710496175, -1795953382595105853, -88223902016631657740, -4283800987347611165184, -207864171877269042498096, -10102590396625592962089500
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 06 2018

Keywords

Examples

			The table of coefficients of x^k in expansion of Product_{k>=1} 1/(1 + x^k)^(n^k) begins:
n = 0: (1),  0,    0,    0,   0,     0,  ...
n = 1:  1, (-1),   0,   -1,   1,    -1,  ...
n = 2:  1,  -2,  (-1),  -4,   3,    -2,  ...
n = 3:  1,  -3,   -3, (-10),  6,    15,  ...
n = 4:  1,  -4,   -6,  -20, (11),  104,  ...
n = 5:  1,  -5,  -10,  -35,  20,  (374), ...

Crossrefs

Cf. A081362, A252654, A255526, A252782, A255672, A270917, A270922, A281266, A281267, A281268, A283333, A292805, A300456, A300457.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 + x^k)^(n^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Showing 1-5 of 5 results.

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