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 1-10 of 14 results. Next

A297328 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 6, 0, 1, 4, 12, 18, 14, 0, 1, 5, 18, 37, 49, 25, 0, 1, 6, 25, 64, 114, 114, 56, 0, 1, 7, 33, 100, 219, 312, 282, 97, 0, 1, 8, 42, 146, 375, 676, 855, 624, 198, 0, 1, 9, 52, 203, 594, 1276, 2030, 2178, 1422, 354, 0, 1, 10, 63, 272, 889, 2196, 4155, 5736, 5496, 3058, 672, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 28 2017

Keywords

Examples

			G.f. of column k: A_k(x) = 1 + k*x + (1/2)*k*(k + 5)*x^2 + (1/6)*k*(k^2 + 15*k + 20)*x^3 + (1/24)*k*(k^3 + 30*k^2 + 155*k + 150)*x^4 + (1/120)*k*(k^4 + 50*k^3 + 575*k^2 + 1750*k + 624)*x^5 + ...
Square array begins:
  1,   1,    1,    1,    1,     1,  ...
  0,   1,    2,    3,    4,     5,  ...
  0,   3,    7,   12,   18,    25,  ...
  0,   6,   18,   37,   64,   100,  ...
  0,  14,   49,  114,  219,   375,  ...
  0,  25,  114,  312,  676,  1276,  ...

Links

Crossrefs

Columns k=0..32 give A000007, A006906, A022726, A022727, A022728, A022729, A022730, A022731, A022732, A022733, A022734, A022735, A022736, A022737, A022738, A022739, A022740, A022741, A022742, A022743, A022744, A022745, A022746, A022747, A022748, A022749, A022750, A022751, A022752, A022753, A022754, A022755, A022756.
Main diagonal gives A297329.
Antidiagonal sums give A299162.
Cf. A078308, A266941, A297321, A297323, A297325.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Product[1/(1 - i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
  • PARI
    first(n, k) = my(res = matrix(n, k)); for(u=1, k, my(col = Vec(prod(j=1, n, 1/(1 - j*x^j)^(u-1)) + O(x^n))); for(v=1, n, res[v, u] = col[v])); res \\ Iain Fox, Dec 28 2017

Formula

G.f. of column k: Product_{j>=1} 1/(1 - j*x^j)^k.
A(0,k) = 1; A(n,k) = (k/n) * Sum_{j=1..n} A078308(j) * A(n-j,k). - Seiichi Manyama, Aug 16 2023

A297321 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j*x^j)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 5, 0, 1, 4, 9, 14, 7, 0, 1, 5, 14, 28, 28, 15, 0, 1, 6, 20, 48, 69, 64, 25, 0, 1, 7, 27, 75, 137, 174, 133, 43, 0, 1, 8, 35, 110, 240, 380, 413, 266, 64, 0, 1, 9, 44, 154, 387, 726, 998, 933, 513, 120, 0, 1, 10, 54, 208, 588, 1266, 2075, 2488, 2046, 1000, 186, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 28 2017

Keywords

Examples

			G.f. of column k: A_k(x) = 1 + k*x + (1/2)*k*(k + 3)*x^2 + (1/6)*k*(k^2 + 9*k + 20)*x^3 + (1/24)*k*(k^3 + 18*k^2 + 107*k + 42)*x^4 + (1/120)*k*(k^4 + 30*k^3 + 335*k^2 + 810*k + 624)*x^5 + ...
Square array begins:
1,   1,   1,    1,    1,    1,  ...
0,   1,   2,    3,    4,    5,  ...
0,   2,   5,    9,   14,   20,  ...
0,   5,  14,   28,   48,   75,  ...
0,   7,  28,   69,  137,  240,  ...
0,  15,  64,  174,  380,  726,  ...

Links

Crossrefs

Columns k=0..32 give A000007, A022629, A022630, A022631, A022632, A022633, A022634, A022635, A022636, A022637, A022638, A022639, A022640, A022641, A022642, A022643, A022644, A022645, A022646, A022647, A022648, A022649, A022650, A022651, A022652, A022653, A022654, A022655, A022656, A022657, A022658, A022659, A022660.
Main diagonal gives A297322.
Antidiagonal sums give A299164.
Cf. A266891, A297323, A297325, A297328.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Product[(1 + i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

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

A297325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + j*x^j)^k.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -1, -2, 0, 1, -4, 0, -2, 2, 0, 1, -5, 2, -1, 9, -1, 0, 1, -6, 5, 0, 18, -2, 4, 0, 1, -7, 9, 0, 27, -12, 10, -1, 0, 1, -8, 14, -2, 35, -36, 11, -16, 18, 0, 1, -9, 20, -7, 42, -76, 14, -54, 38, -22, 0, 1, -10, 27, -16, 49, -132, 35, -104, 84, -98, 12, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 28 2017

Keywords

Examples

			G.f. of column k: A_k(x) = 1 - k*x + (1/2)*k*(k - 3)*x^2 - (1/6)*k*(k^2 - 9*k + 20)*x^3 + (1/24)*k*(k^3 - 18*k^2 + 107*k - 42)*x^4 - (1/120)*k*(k^4 - 30*k^3 + 335*k^2 - 810*k + 624)*x^5 + ...
Square array begins:
  1,  1,  1,   1,   1,   1,  ...
  0, -1, -2,  -3,  -4,  -5,  ...
  0, -1, -1,   0,   2,   5,  ...
  0, -2, -2,  -1,   0,   0,  ...
  0,  2,  9,  18,  27,  35,  ...
  0, -1, -2, -12, -36, -76,  ...

Links

Crossrefs

Columns k=0..32 give A000007, A022693, A022694, A022695, A022696, A022697, A022698, A022699, A022700, A022701, A022702, A022703, A022704, A022705, A022706, A022707, A022708, A022709, A022710, A022711, A022712, A022713, A022714, A022715, A022716, A022717, A022718, A022719, A022720, A022721, A022722, A022723, A022724.
Main diagonal gives A297326.
Antidiagonal sums give A299210.
Cf. A266971, A297321, A297323, A297328.

Programs

  • Maple
    with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, -k*add(add(
      (-d)^(1+j/d), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Apr 20 2018
  • Mathematica
    Table[Function[k, SeriesCoefficient[Product[1/(1 + i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

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

A297324 a(n) = [x^n] Product_{k>=1} (1 - k*x^k)^n.

Original entry on oeis.org

1, -1, -3, 8, 9, -26, -168, 489, 1041, -5599, 12, 27103, 23436, -222912, -435473, 3177433, 375569, -24956018, 6931209, 181844002, 57372644, -2158209675, 853739235, 20642183588, -25063980804, -148768035501, 224915906836, 1322267927471, -2337343745721, -12604818831294
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 28 2017

Keywords

Links

Crossrefs

Main diagonal of A297323.
Cf. A297322, A297326, A297329.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 - k x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 29}]

Formula

a(n) = A297323(n,n).

A299209 Expansion of 1/(1 - x*Product_{k>=1} (1 - k*x^k)).

Original entry on oeis.org

1, 1, 0, -3, -6, -5, 11, 37, 59, 13, -155, -402, -415, 263, 1981, 3748, 2289, -6643, -22642, -31322, -187, 99040, 229410, 216823, -230029, -1223267, -2097812, -955237, 4468902, 13393758, 16752461, -3891704, -62382597, -131974181, -106680562, 173622424, 741553622, 1163057561, 329176545
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A297323.
Cf. A022661, A067687, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299210, A299211, A299212.

Programs

  • Mathematica
    nmax = 38; CoefficientList[Series[1/(1 - x Product[1 - k x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 - k*x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A022661(k-1)*a(n-k).

A022662 Expansion of Product_{m>=1} (1 - m*q^m)^2.

Original entry on oeis.org

1, -2, -3, 2, 4, 16, -3, 6, -31, -72, -15, -44, 9, 154, 521, 48, 426, 66, 2, -1618, -3782, -210, -3882, -1282, 1119, 3940, 10867, 37208, 11647, 20574, 6256, 534, -1915, -120006, -161755, -312622, -93923, -271850, -25782, -197026, 1112303, 574604, 209604, 3038822, 4187500, 1398330
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A022662, A022726.
Column k=2 of A297323.

Programs

  • Magma
    Coefficients(&*[(1-m*x^m)^2:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 18 2018
  • Mathematica
    nmax = 50; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = -2; poly[[3]] = 1; Do[Do[Do[poly[[j+1]] -= k*poly[[j-k+1]], {j, nmax, k, -1}];, {p, 1, 2}], {k, 2, nmax}]; poly  (* Vaclav Kotesovec, Jan 07 2016 *)
With[{nmax = 50}, CoefficientList[Series[Product[(1 - k*q^k)^2, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 18 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n)^2)) \\ G. C. Greubel, Feb 18 2018

Formula

G.f.: exp(-2*Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 07 2018

A022663 Expansion of Product_{m>=1} (1 - m*q^m)^3.

Original entry on oeis.org

1, -3, -3, 8, 9, 18, -35, -33, -66, -91, 216, 189, 386, 315, 333, -1483, -2268, -2214, -1883, -456, -801, 23032, 12186, 22665, 18622, -20328, -39549, -78834, -146838, -249342, -146662, 15678, 564771, 238159, 1274913, 1398063, 1572593, 1423266, -833778, -3484732, -5261736, -9671502
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -3, g(n) = n. - Seiichi Manyama, Dec 29 2017

Links

Crossrefs

Column k=3 of A297323.

Programs

  • Magma
    Coefficients(&*[(1-m*x^m)^3:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 23 2018
  • Mathematica
    With[{nmax=34}, CoefficientList[Series[Product[(1-k*q^k)^3, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 23 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n)^3)) \\ G. C. Greubel, Feb 23 2018

Formula

G.f.: exp(-3*Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 07 2018

Extensions

More terms added by G. C. Greubel, Feb 23 2018

A022664 Expansion of Product_{m>=1} (1 - m*q^m)^4.

Original entry on oeis.org

1, -4, -2, 16, 9, 4, -90, -56, 12, 60, 700, 232, -51, -1128, -2006, -3648, -2999, 6292, 12004, 19192, 8829, 35024, -43368, -92480, -113859, -227356, -33906, 55072, 569221, 631620, 1193412, 1593152, 1178350, -2589588, -4131366, -6312376, -12864282, -6891608, -10022026, 10270984
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -4, g(n) = n. - Seiichi Manyama, Dec 29 2017

Links

Crossrefs

Column k=4 of A297323.

Programs

  • Magma
    Coefficients(&*[(1-m*x^m)^4:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 23 2018
  • Mathematica
    With[{nmax=34}, CoefficientList[Series[Product[(1-k*q^k)^4, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 23 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n)^4)) \\ G. C. Greubel, Feb 23 2018

Formula

G.f.: exp(-4*Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 07 2018

Extensions

More terms added by G. C. Greubel, Feb 23 2018

A022665 Expansion of Product_{m>=1} (1 - m*q^m)^5.

Original entry on oeis.org

1, -5, 0, 25, 0, -26, -145, 0, 265, 265, 993, -825, -2070, -3190, -2335, 2739, 7890, 29570, 21085, -5250, -73006, -71945, -191140, -176805, 185045, 295675, 1204590, 1067375, 1353655, -910885, -3688009, -4645850, -9409195, -12021485, -4296815, 19981183, 28942560, 76843230, 70996895
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=5 of A297323.

Programs

  • Magma
    Coefficients(&*[(1-m*x^m)^5:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 23 2018
  • Mathematica
    With[{nmax=50}, CoefficientList[Series[Product[(1-k*q^k)^5, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 23 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n)^5)) \\ G. C. Greubel, Feb 23 2018

Formula

G.f.: exp(-5*Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 07 2018

Extensions

Terms a(32) onward added by G. C. Greubel, Feb 23 2018

A022666 Expansion of Product_{m>=1} (1 - m*q^m)^6.

Original entry on oeis.org

1, -6, 3, 34, -21, -66, -168, 180, 645, 176, 540, -3282, -4265, -1068, 5805, 21226, 16398, 27498, -42993, -139110, -199998, -46374, 127917, 467016, 1424954, 881958, 895899, -2559102, -5166543, -9792708, -11899179, 5560560, 13493076, 39293062, 65560674, 94059054, 14988615
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=6 of A297323.

Programs

  • Magma
    Coefficients(&*[(1-m*x^m)^6:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 24 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 - k*q^k)^6, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 24 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n)^6)) \\ G. C. Greubel, Feb 24 2018

Formula

G.f.: exp(-6*Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 07 2018

Extensions

Terms a(31) onward added by G. C. Greubel, Feb 24 2018
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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