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 32 results. Next

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.

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.

A299162 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 - k*x^k)).

Original entry on oeis.org

1, 1, 2, 6, 17, 49, 135, 380, 1051, 2925, 8119, 22548, 62574, 173767, 482360, 1339126, 3717700, 10321163, 28653557, 79548612, 220843925, 613110573, 1702128034, 4725475979, 13118945083, 36421037100, 101112695940, 280710759278, 779313926949, 2163544401343, 6006468273440
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A297328.
Cf. A006906, A067687, A299105, A299106, A299108, A299164, A299166, A299167.

Programs

  • Mathematica
    nmax = 30; 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} A006906(k-1)*a(n-k).

A297329 a(n) = [x^n] Product_{k>=1} 1/(1 - k*x^k)^n.

Original entry on oeis.org

1, 1, 7, 37, 219, 1276, 7687, 46551, 285043, 1756243, 10883842, 67751289, 423366831, 2654041235, 16683909711, 105129718102, 663837626163, 4199521413019, 26610335585263, 168864540960165, 1073001606214814, 6826237566223329, 43474472693256491, 277152041235941803
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 28 2017

Keywords

Links

Crossrefs

Main diagonal of A297328.
Cf. A297322, A297324, A297326.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - k x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 23}]

Formula

a(n) = A297328(n,n).
a(n) ~ c * d^n / sqrt(n), where d = 6.51548390811914394587815688142024783108478... and c = 0.2552310487728179222346375591994440863074... - Vaclav Kotesovec, Aug 01 2019

A022726 Expansion of 1/Product_{m>=1} (1 - m*q^m)^2.

Original entry on oeis.org

1, 2, 7, 18, 49, 114, 282, 624, 1422, 3058, 6597, 13700, 28564, 57698, 116479, 230398, 453698, 879080, 1696732, 3230578, 6124326, 11486884, 21439480, 39659598, 73036175, 133445640, 242756058, 438680734, 789328034, 1411926186, 2515574329, 4458203590, 7871211452, 13831782146
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Self-convolution of A006906. - Vaclav Kotesovec, Jan 06 2016

Links

Crossrefs

Cf. A006906, A022662.
Column k=2 of A297328.

Programs

  • Magma
    n:=40; R:=PowerSeriesRing(Integers(), n); Coefficients(R!(&*[(1/(1-m*x^m))^2:m in [1..n]])); // G. C. Greubel, Jul 25 2018
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 06 2016 *)
  • PARI
    N=66; q='q+O('q^N);
gf= 1/prod(n=1,N, (1-n*q^n)^2 );
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

Formula

From Vaclav Kotesovec, Jan 07 2016: (Start)
a(n) ~ c * n * 3^(n/3), where
c = 9588921272.54120308291761424720457... = (c0^2 + 2*c1*c2)/3 if mod(n,3)=0
c = 9588921272.50566179874517327053929... = (c2^2 + 2*c0*c1)/3 if mod(n,3)=1
c = 9588921272.49785814355801212400055... = (c1^2 + 2*c0*c2)/3 if mod(n,3)=2
For the constants c0, c1, c2 see A006906.
(End)
G.f.: exp(2*Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 07 2018

Extensions

Added more terms, Joerg Arndt, Oct 06 2012

A022727 Expansion of Product_{m>=1} (1-m*q^m)^-3.

Original entry on oeis.org

1, 3, 12, 37, 114, 312, 855, 2178, 5496, 13302, 31719, 73482, 168086, 375984, 830976, 1805887, 3880746, 8225460, 17262440, 35809446, 73621776, 149875003, 302635110, 605861124, 1204043358, 2374645746
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 A297328.

Programs

  • Magma
    n:=50; R:=PowerSeriesRing(Integers(), n); Coefficients(R!(&*[(1/(1-m*x^m))^3:m in [1..n]])); // G. C. Greubel, Jul 25 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 - k*q^k)^-3, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Jul 25 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n)^-3)) \\ G. C. Greubel, Jul 25 2018

Formula

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

A022728 Expansion of Product_{m>=1} (1-m*q^m)^-4.

Original entry on oeis.org

1, 4, 18, 64, 219, 676, 2030, 5736, 15793, 41864, 108430, 273240, 675526, 1634780, 3891960, 9108872, 21018870, 47815572, 107446898, 238524144, 523812125, 1138233100, 2449710880, 5223395480, 11042278208
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 A297328.

Programs

  • Magma
    n:=50; R:=PowerSeriesRing(Integers(), n); Coefficients(R!(&*[(1/(1-m*x^m))^4:m in [1..n]])); // G. C. Greubel, Jul 25 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 - k*q^k)^-4, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Jul 25 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n)^-4)) \\ G. C. Greubel, Jul 25 2018

Formula

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

A022729 Expansion of Product_{m>=1} 1/(1 - m*q^m)^5.

Original entry on oeis.org

1, 5, 25, 100, 375, 1276, 4155, 12775, 37935, 108460, 301533, 815075, 2153995, 5567685, 14123030, 35183376, 86259665, 208293520, 496100890, 1166243015, 2708878924, 6220640495, 14134118490, 31792023545
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=5 of A297328.

Programs

  • Magma
    n:=50; R:=PowerSeriesRing(Integers(), n); Coefficients(R!(&*[(1/(1-m*x^m))^5:m in [1..n]])); // G. C. Greubel, Jul 25 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 - k*q^k)^-5, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Jul 25 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n)^-5)) \\ G. C. Greubel, Jul 25 2018

Formula

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

A022730 Expansion of Product_{m>=1} 1/(1 - m*q^m)^6.

Original entry on oeis.org

1, 6, 33, 146, 594, 2196, 7687, 25410, 80664, 246258, 728610, 2093334, 5865853, 16057998, 43063812, 113293158, 292928448, 745216692, 1867840830, 4616732712, 11264133069, 27149243724, 64691795178
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=6 of A297328.

Programs

  • Magma
    n:=50; R:=PowerSeriesRing(Integers(), n); Coefficients(R!(&*[(1/(1-m*x^m))^6:m in [1..n]])); // G. C. Greubel, Jul 25 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 - k*q^k)^-6, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Jul 25 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n)^-6)) \\ G. C. Greubel, Jul 25 2018

Formula

G.f.: exp(6*Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 07 2018
Showing 1-10 of 32 results. Next

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

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