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.

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.

A297326 a(n) = [x^n] Product_{k>=1} 1/(1 + k*x^k)^n.

Original entry on oeis.org

1, -1, -1, -1, 27, -76, 95, -295, 2035, -8119, 22714, -66793, 254223, -988651, 3444055, -11402626, 39248691, -141740051, 511583207, -1798826901, 6256648862, -22054706773, 78889160635, -281698897727, 996551999479, -3520566280801, 12522382445455, -44731559517301
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 28 2017

Keywords

Links

Crossrefs

Main diagonal of A297325.
Cf. A297322, A297324, A297329.

Programs

  • Maple
    f:= proc(n) local k;
coeff(series(mul(1/(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/(1 + k x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

Formula

a(n) = A297325(n,n).

A299210 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + k*x^k)).

Original entry on oeis.org

1, 1, 0, -2, -5, -3, 5, 20, 27, 17, -53, -152, -192, 31, 576, 1110, 694, -1297, -4519, -6160, -1107, 13665, 31914, 30643, -19339, -119260, -196142, -103318, 289543, 859631, 1062684, 13710, -2690348, -5675946, -4940757, 4167527, 21343918, 33874107, 16524162, -51704908, -150454546
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A297325.
Cf. A022693, A067687, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299211, A299212.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[1/(1 - x Product[1/(1 + k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + k*x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A022693(k-1)*a(n-k).

A022694 Expansion of Product_{m>=1} (1 + m*q^m)^-2.

Original entry on oeis.org

1, -2, -1, -2, 9, -2, 10, -16, 38, -98, 53, -116, 340, -434, 463, -990, 2378, -2792, 3660, -7058, 11454, -18900, 24104, -36206, 81623, -119400, 128194, -248062, 447066, -576154, 880401, -1415926, 2297516, -3724290, 4854450, -7299306, 13411402, -19129752, 25135890, -42841396, 71321016
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) = 2, g(n) = -n. - Seiichi Manyama, Dec 30 2017

Links

Crossrefs

Column k=2 of A297325.

Programs

  • Magma
    Coefficients(&*[1/(1+m*x^m)^2:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018
  • Mathematica
    With[{nmax=50}, CoefficientList[Series[Product[1/(1+k*q^k)^2, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 22 2018 *)
  • PARI
    apply(x->round(x), Vec(prodinf(m=1, 1/(1+m*q^m)^2+O(q^50)))) \\ Michel Marcus, Dec 30 2017
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,1/(1+n*q^n)^2)) \\ G. C. Greubel, Feb 25 2018

Formula

G.f.: exp(-2*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018

A022695 Expansion of Product_{m>=1} (1 + m*q^m)^-3.

Original entry on oeis.org

1, -3, 0, -1, 18, -12, 11, -54, 84, -218, 243, -270, 1046, -1524, 1692, -3547, 7722, -11868, 15364, -29130, 52416, -83467, 125514, -190716, 380406, -628290, 808218, -1394734, 2585895, -3784566, 5678826, -9514614, 15635424, -25331990, 37563810, -57387042, 100038145, -156346224
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 30 2017

Links

Crossrefs

Column k=3 of A297325.

Programs

  • Magma
    Coefficients(&*[1/(1+m*x^m)^3:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018
  • Mathematica
    With[{nmax=50}, CoefficientList[Series[Product[1/(1+k*q^k)^3, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 25 2018 *)
  • PARI
    first(n) = Vec(prod(m=1, n, (1+m*x^m)^(-3)) + O(x^n)) \\ Iain Fox, Dec 30 2017

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

A022696 Expansion of Product_{m>=1} (1 + m*q^m)^-4.

Original entry on oeis.org

1, -4, 2, 0, 27, -36, 14, -104, 209, -392, 670, -728, 2278, -4444, 4808, -9800, 21750, -35604, 51906, -91120, 176285, -290444, 455168, -741336, 1372544, -2419348, 3490310, -5765744, 10788815, -17086420, 26221946, -44374160
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=4 of A297325.

Programs

  • Maple
    N:= 100: # to get a(0)..a(N)
P:= mul((1+m*q^m)^(-4),m=1..N):
S:=series(P,q,N+1):
[seq(coeff(S,q,j),j=0..N)]; # Robert Israel, Jan 23 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^-4, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Jul 19 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^-4)) \\ G. C. Greubel, Jul 19 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

A022697 Expansion of Product_{m>=1} 1/(1 + m*q^m)^5.

Original entry on oeis.org

1, -5, 5, 0, 35, -76, 35, -155, 455, -720, 1369, -1935, 4415, -10405, 12990, -22512, 54405, -92480, 143150, -253015, 488512, -859795, 1377670, -2332365, 4276230, -7666511, 12092880, -19796225, 36845455, -62053775
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=5 of A297325.

Programs

  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^-5, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Jul 19 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^-5)) \\ G. C. Greubel, Jul 19 2018

Formula

G.f.: exp(-5*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018

A022698 Expansion of Product_{m>=1} 1/(1 + m*q^m)^6.

Original entry on oeis.org

1, -6, 9, -2, 42, -132, 95, -210, 840, -1394, 2442, -4374, 8589, -20862, 31812, -48758, 119856, -222228, 347038, -631992, 1220781, -2228812, 3730962, -6390948, 11861066, -21539358, 35874624, -59882714, 110055054
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=6 of A297325.

Programs

  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^-6, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Jul 19 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^-6)) \\ G. C. Greubel, Jul 19 2018

Formula

G.f.: exp(-6*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018
Showing 1-10 of 14 results. Next

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

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