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 1-7 of 7 results.

A265837 Expansion of Product_{k>=1} 1/(1 - k^3*x^k).

Original entry on oeis.org

1, 1, 9, 36, 164, 505, 2474, 7273, 31008, 103644, 379890, 1226802, 4747529, 14553648, 52167558, 171639695, 583371802, 1851395692, 6427705062, 19983302144, 67235043192, 214615427776, 697704303005, 2194982897304, 7262755260410, 22402942281766, 72461661415093
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 16 2015

Keywords

Links

Crossrefs

Cf. A006906, A077335, A265838, A265839, A265840.
Column k=3 of A292193.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1 - k^3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c * 3^n, where
c = 86.60286320343345379122228784466307940393110978... if n mod 3 = 0
c = 86.27536745612304663727011387030370600864018892... if n mod 3 = 1
c = 86.29819842537784019895326532818285333403267092... if n mod 3 = 2.
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(3*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

A265838 Expansion of Product_{k>=1} 1/(1 - k^4*x^k).

Original entry on oeis.org

1, 1, 17, 98, 610, 2531, 18580, 72453, 449494, 2114440, 10753594, 48572844, 272867295, 1137441506, 5834448870, 27276382027, 129389072144, 576677550870, 2884567552542, 12401875640710, 59474089385344, 270438887909580, 1230979340265033, 5477371267093144
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 16 2015

Keywords

Links

Crossrefs

Cf. A006906, A077335, A265837, A265839, A265841.
Column k=4 of A292193.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1 - k^4*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c * 3^(4*n/3), where
c = 27.2472595510480930563087281042486261391960582835336715327... if n mod 3 = 0
c = 26.8841208067599453033952496040472485838861626762931432887... if n mod 3 = 1
c = 26.9277867007233095885556073185206409643421012262073908850... if n mod 3 = 2.
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(4*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

A265839 Expansion of Product_{k>=1} 1/(1 - k^5*x^k).

Original entry on oeis.org

1, 1, 33, 276, 2324, 13225, 145586, 760057, 6836328, 45996924, 322816122, 2064921330, 16881567137, 96217644312, 708147553326, 4769313137735, 31412238427954, 198869428043476, 1442034056253438, 8596120396405880, 58954590481229064, 387170921610808720
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 16 2015

Keywords

Links

Crossrefs

Cf. A006906, A077335, A265837, A265838, A265842.
Column k=5 of A292193.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1 - k^5*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c * 3^(5*n/3), where
c = 12.8519823810391431573687005461910113782018563173082562291... if n mod 3 = 0
c = 12.4535903496941652158697054030067622653283880393322526099... if n mod 3 = 1
c = 12.5138855694494734654940524026530463555984202132997900068... if n mod 3 = 2.
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(5*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

A292194 Sum of n-th powers of products of terms in all partitions of n.

Original entry on oeis.org

1, 1, 5, 36, 610, 13225, 1173652, 92137513, 27960729094, 14612913824364, 11885159817456154, 23676862215173960082, 144210774157588042096815, 778807208565930895328294712, 15863318347221014170216633451982, 908978343753718115412387406378667615
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2017

Keywords

Examples

			5 = 4 + 1 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1.
So a(5) = 5^5 + (4*1)^5 + (3*2)^5 + (3*1*1)^5 + (2*2*1)^5 + (2*1*1*1)^5 + (1*1*1*1*1)^5 = 13225.

Links

Crossrefs

Main diagonal of A292193.
Cf. A292190.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      `if`(i>n, 0, i^k*b(n-i, i, k))+b(n, i-1, k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 11 2017
  • Mathematica
    nmax = 20; Table[SeriesCoefficient[Product[1/(1 - k^n*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 15 2017 *)
  • PARI
    {a(n) = polcoeff(1/prod(k=1, n, 1-k^n*x^k+x*O(x^n)), n)}

Formula

a(n) = [x^n] Product_{k=1..n} 1/(1 - k^n*x^k).
From Vaclav Kotesovec, Sep 15 2017: (Start)
a(n) ~ 3^(n^2/3) if mod(n,3)=0
a(n) ~ 3^(n*(n-4)/3)*2^(2*n+1) if mod(n,3)=1
a(n) ~ 3^(n*(n-2)/3)*2^n if mod(n,3)=2
(End)

A294579 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(1 + k*n/d).

Original entry on oeis.org

1, 1, 3, 1, 5, 4, 1, 9, 10, 7, 1, 17, 28, 25, 6, 1, 33, 82, 97, 26, 12, 1, 65, 244, 385, 126, 80, 8, 1, 129, 730, 1537, 626, 588, 50, 15, 1, 257, 2188, 6145, 3126, 4508, 344, 161, 13, 1, 513, 6562, 24577, 15626, 35652, 2402, 2049, 163, 18
Offset: 1

Views

Author

Seiichi Manyama, Nov 02 2017

Keywords

Examples

			Square array begins:
  1,  1,   1,   1,    1, ...
  3,  5,   9,  17,   33, ...
  4, 10,  28,  82,  244, ...
  7, 25,  97, 385, 1537, ...
  6, 26, 126, 626, 3126, ...

Links

Crossrefs

Columns k=0..2 give A000203, A078308, A294567.
Rows k=0..1 give A000012, A000051(n+1).
Cf. A292166, A292193.

Formula

L.g.f. of column k: -log(Product_{j>=1} (1 - j^k * x^j)). - Seiichi Manyama, Jun 02 2019
G.f. of column k: Sum_{j>0} j^(k+1) * x^j / (1 - j^k * x^j). - Seiichi Manyama, Jan 14 2023

A294582 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^k*x^j)^j.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 9, 14, 13, 1, 1, 17, 36, 42, 24, 1, 1, 33, 98, 148, 103, 48, 1, 1, 65, 276, 546, 489, 289, 86, 1, 1, 129, 794, 2068, 2467, 1959, 690, 160, 1, 1, 257, 2316, 7962, 12969, 14281, 6326, 1771, 282, 1, 1, 513, 6818, 30988, 70243
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2017

Keywords

Examples

			Square array begins:
    1,  1,   1,   1,    1, ...
    1,  1,   1,   1,    1, ...
    3,  5,   9,  17,   33, ...
    6, 14,  36,  98,  276, ...
   13, 42, 148, 546, 2068, ...

Links

Crossrefs

Columns k=0..2 give A000219, A266941, A285674.
Rows n=0-1 give A000012.
Cf. A292193, A294580.

Formula

A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(2+k*j/d)) * A(n-j,k) for n > 0.

A292068 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^k*x^j).

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, -1, -1, -1, 1, -1, -3, -2, 1, 1, -1, -7, -6, 2, -1, 1, -1, -15, -20, 6, -1, 1, 1, -1, -31, -66, 20, 5, 4, -1, 1, -1, -63, -212, 66, 71, 40, -1, 2, 1, -1, -127, -666, 212, 605, 442, 11, 18, -2, 1, -1, -255, -2060, 666, 4439, 4660, 215, 226, -22, 2
Offset: 0

Views

Author

Seiichi Manyama, Sep 12 2017

Keywords

Examples

			Square array begins:
    1,  1,  1,   1,   1, ...
   -1, -1, -1,  -1,  -1, ...
    0, -1, -3,  -7, -15, ...
   -1, -2, -6, -20, -66, ...
    1,  2,  6,  20,  66, ...

Crossrefs

Columns k=0..2 give A081362, A022693, A292165.
Rows n=0..2 give A000012, (-1)*A000012, (-1)*A000225.
Main diagonal gives A292072.
Cf. A292189, A292193.

Programs

  • Maple
    b:= proc(n, i, k) option remember; (m->
      `if`(mn, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      -add(b(n-i$2, k)*A(i, k), i=0..n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 12 2017
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[# < n, 0, If[n == #, i!^k, b[n, i-1, k] + If[i > n, 0, i^k b[n-i, i-1, k]]]]&[i(i+1)/2];
A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[b[n-i, n-i, k] A[i, k], {i, 0, n-1}]];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Nov 20 2019, after Alois P. Heinz *)
  • Python
    from sympy.core.cache import cacheit
from sympy import factorial as f
@cacheit
def b(n, i, k):
    m=i*(i + 1)/2
    return 0 if mn else i**k*b(n - i, i - 1, k))
@cacheit
def A(n, k): return 1 if n==0 else -sum([b(n - i, n - i, k)*A(i, k) for i in range(n)])
for d in range(13): print([A(n, d - n) for n in range(d + 1)]) # Indranil Ghosh, Sep 14 2017, after Maple program
Showing 1-7 of 7 results.

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