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 27 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

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.

A297323 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, -3, -1, 0, 1, -4, -3, 2, -1, 0, 1, -5, -2, 8, 4, 5, 0, 1, -6, 0, 16, 9, 16, 1, 0, 1, -7, 3, 25, 9, 18, -3, 13, 0, 1, -8, 7, 34, 0, 4, -35, 6, 4, 0, 1, -9, 12, 42, -21, -26, -90, -33, -31, 0, 0, 1, -10, 18, 48, -56, -66, -145, -56, -66, -72, 2, 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, -2,  -3,  -3,  -2,   0,  ...
  0, -1,   2,   8,  16,  25,  ...
  0, -1,   4,   9,   9,   0,  ...
  0,  5,  16,  18,   4, -26,  ...

Crossrefs

Columns k=0..32 give A000007, A022661, A022662, A022663, A022664, A022665, A022666, A022667, A022668, A022669, A022670, A022671, A022672, A022673, A022674, A022675, A022676, A022677, A022678, A022679, A022680, A022681, A022682, A022683, A022684, A022685, A022686, A022687, A022688, A022689, A022690, A022691, A022692.
Main diagonal gives A297324.
Antidiagonal sums give A299209.
Cf. A266964, A297321, 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
  • PARI
    first(n, k) = my(res = matrix(n, k)); for(u=1, k, my(col = Vec(prod(j=1, n, (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 - j*x^j)^k.

A299164 Expansion of 1/(1 - x*Product_{k>=1} (1 + k*x^k)).

Original entry on oeis.org

1, 1, 2, 5, 14, 35, 91, 233, 597, 1517, 3885, 9922, 25333, 64683, 165181, 421828, 1077277, 2750993, 7025168, 17940298, 45814165, 116996152, 298774246, 762982615, 1948434235, 4975732669, 12706571546, 32448880807, 82864981016, 211613009498, 540397935771, 1380018797044, 3524165721799
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A297321.
Cf. A022629, A067687, A299105, A299106, A299108, A299162, A299166, A299167.

Programs

  • Mathematica
    nmax = 32; 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} A022629(k-1)*a(n-k).

A297322 a(n) = [x^n] Product_{k>=1} (1 + k*x^k)^n.

Original entry on oeis.org

1, 1, 5, 28, 137, 726, 3896, 21071, 115089, 633007, 3500740, 19448573, 108458924, 606787572, 3404112479, 19142919543, 107874784017, 609021410570, 3443952349385, 19503777943838, 110599636109572, 627924447630011, 3568885868192419, 20304321490356084
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 28 2017

Keywords

Links

Crossrefs

Main diagonal of A297321.
Cf. A297324, A297326, A297329.

Programs

  • Maple
    f:= proc(n) local k;
coeff(series(mul((1+k*x^k)^n,k=1..n),x,n+1),x,n);
end proc:
map(f, [$0..30]); # Robert Israel, Dec 28 2017
  • Mathematica
    Table[SeriesCoefficient[Product[(1 + k x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]

Formula

a(n) = A297321(n,n).
a(n) ~ c * d^n / sqrt(n), where d = 5.814548482877687529318372516965305077397562... and c = 0.2563102401728134539247148322678842806264... - Vaclav Kotesovec, Aug 01 2019

A022643 Expansion of Product_{m>=1} (1 + m*q^m)^15.

Original entry on oeis.org

1, 15, 135, 950, 5670, 30003, 144680, 647055, 2717760, 10820640, 41128374, 150073470, 528074655, 1798537380, 5947216050, 19142919543, 60113026305, 184513760775, 554517086825, 1634047143090, 4727605374594, 13444544485435, 37620762642885, 103678546403985, 281639925782930
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=15 of A297321.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^15:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
  • Maple
    [seq(coeff(series(mul((1+m*q^m)^(15), m=1..100),q,101),q,j),j=0..25)]; # Muniru A Asiru, Feb 18 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^15, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^15)) \\ G. C. Greubel, Feb 17 2018

A022644 Expansion of Product_{m>=1} (1 + m*q^m)^16.

Original entry on oeis.org

1, 16, 152, 1120, 6972, 38368, 191968, 889184, 3862214, 15881616, 62275840, 234205472, 848652120, 2974133152, 10112507808, 33448941824, 107874784017, 339879773648, 1047953793136, 3166817754880, 9391718326404, 27366626142688, 78435144301696, 221322772710464, 615375631077094
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=16 of A297321.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^16:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
  • Maple
    [seq(coeff(series(mul((1+m*q^m)^(16), m=1..100),q,101),q,j),j=0..25)]; # Muniru A Asiru, Feb 18 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^16, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^16)) \\ G. C. Greubel, Feb 17 2018

A022630 Expansion of Product_{m>=1} (1 + m*q^m)^2.

Original entry on oeis.org

1, 2, 5, 14, 28, 64, 133, 266, 513, 1000, 1873, 3420, 6257, 11078, 19585, 34192, 58714, 99870, 168858, 281666, 467082, 768994, 1253038, 2030658, 3269551, 5227868, 8304467, 13133256, 20630535, 32250274, 50181624, 77653530, 119634925, 183532470, 280245365
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=2 of A297321.
Cf. A022629, A022694.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^2:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 16 2018
  • Mathematica
    nn=34; CoefficientList [Series[ Product[(1 + m*q^m)^2, {m, nn}], {q, 0, nn}],q] (* Robert G. Wilson v, Feb 08 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^2)) \\ G. C. Greubel, Feb 16 2018

Formula

Self-convolution of A022629. - Alois P. Heinz, Dec 28 2017
G.f.: exp(2*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018

A022631 Expansion of Product_{m>=1} (1 + m*q^m)^3.

Original entry on oeis.org

1, 3, 9, 28, 69, 174, 413, 933, 2046, 4391, 9168, 18675, 37522, 73725, 142893, 273159, 514512, 957666, 1762837, 3208884, 5783727, 10330732, 18280590, 32086827, 55880614, 96579240, 165733335, 282513246, 478419366, 805196022, 1347288750, 2241377166, 3708721887
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 A297321.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^3:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 16 2018
  • Mathematica
    With[{nmax=34}, CoefficientList[Series[Product[(1+k*q^k)^3, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 16 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^3)) \\ G. C. Greubel, Feb 16 2018

Formula

G.f.: exp(3*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018

A022632 Expansion of Product_{m>=1} (1 + m*q^m)^4.

Original entry on oeis.org

1, 4, 14, 48, 137, 380, 998, 2488, 5996, 14020, 31868, 70616, 153389, 326248, 681914, 1402880, 2841769, 5678316, 11201956, 21833480, 42081245, 80264752, 151572328, 283577152, 525894397, 967100700, 1764378830, 3194682272, 5742739237, 10252117308, 18182247316
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 A297321.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m)^4:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // Vincenzo Librandi, Jan 24 2018
  • Mathematica
    CoefficientList[Take[Expand[Product[(1 + k x^k)^4, {k, 40}]], 40], x] (* Vincenzo Librandi, Jan 24 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^4)) \\ G. C. Greubel, Feb 16 2018

Formula

G.f.: exp(4*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018
Showing 1-10 of 27 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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