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.

A279928 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1+x^j)^(j*k) in powers of x.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -1, -2, 0, 1, -4, 0, -2, 1, 0, 1, -5, 2, -1, 7, 0, 0, 1, -6, 5, 0, 15, 2, 4, 0, 1, -7, 9, 0, 23, -3, 10, 2, 0, 1, -8, 14, -2, 30, -20, 8, -8, 8, 0, 1, -9, 20, -7, 36, -51, 2, -42, 5, -2, 0, 1, -10, 27, -16, 42, -96, 5, -88, 6
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, -1, -1,  0,  2, ...
   0, -2, -2, -1,  0, ...
   0,  1,  7, 15, 23, ...

Links

Crossrefs

Columns k=0-5 give: A000007, A255528, A278710, A279031, A279411, A279932.
Main diagonal gives A281266.
Antidiagonal sums give A299212.
Cf. A255961, A277938.

Formula

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

A300188 a(n) = n! * [x^n] Product_{k>=1} 1/(1 + x^k)^(n/k).

Original entry on oeis.org

1, -1, 4, -39, 536, -9115, 185904, -4461877, 123647488, -3886461081, 136538590400, -5300491027711, 225313697972736, -10409021924850211, 519298241645107456, -27824560148201248125, 1593597443825288904704, -97153909607626767338353, 6281720886474120790582272
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 28 2018

Keywords

Examples

			The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} 1/(1 + x^k)^(n/k) begins:
n = 0: (1),  0,   0,     0,     0,       0,        0,  ...
n = 1:  1, (-1),  1,    -5,    23,    -119,      619,  ...
n = 2:  1,  -2,  (4),  -16,    92,    -568,     3856,  ...
n = 3:  1,  -3,   9,  (-39),  243,   -1737,    13671,  ...
n = 4:  1,  -4,  16,   -80,  (536),  -4256,    37504,  ...
n = 5:  1,  -5,  25,  -145,  1055,  (-9115),   88075,  ...
n = 6:  1,  -6,  36,  -240,  1908,  -17784,  (185904), ...

Links

Crossrefs

Cf. A048272, A281266, A294356, A299033, A299034, A300187.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Product[1/(1 + x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 18}]

Formula

a(n) = n! * [x^n] exp(-n*Sum_{k>=1} A048272(k)*x^k/k).
a(n) ~ (-1)^n * c * d^n * n^n, where d = 1.3587950730244927060955... and c = 0.6449711831436950784... - Vaclav Kotesovec, Sep 08 2018

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}]

A301831 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 + x^k*A(x)^k)^k.

Original entry on oeis.org

1, -1, 0, 0, 6, -16, 16, -34, 217, -681, 1343, -3466, 13370, -42380, 109477, -312448, 1040248, -3267138, 9447529, -28367596, 90504001, -283611105, 861087913, -2654231074, 8386506600, -26359974392, 81902319183, -256179313766, 809890745232, -2557697524240, 8046530976599
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 27 2018

Keywords

Examples

			G.f. A(x) = 1 - x + 6*x^4 - 16*x^5 + 16*x^6 - 34*x^7 + 217*x^8 - 681*x^9 + 1343*x^10 - 3466*x^11 + ...
log(A(x)) = -x - x^2/2 - x^3/3 + 23*x^4/4 - 51*x^5/5 + 35*x^6/6 - 197*x^7/7 + ... + A281266(n)*x^n/n + ...

Crossrefs

Cf. A109085, A181315, A255528, A281266, A301455, A301456, A301624.

Formula

G.f. satisfies: A(x) = exp(Sum_{k>=1} (-1)^k*x^k*A(x)^k/(k*(1 - x^k*A(x)^k)^2)).
a(n) = [x^n] (Sum_{k>=0} A255528(k)*x^k)^(n+1)/(n + 1).
Showing 1-5 of 5 results.

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