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.

A022629 Expansion of Product_{m>=1} (1 + m*q^m).

Original entry on oeis.org

1, 1, 2, 5, 7, 15, 25, 43, 64, 120, 186, 288, 463, 695, 1105, 1728, 2525, 3741, 5775, 8244, 12447, 18302, 26424, 37827, 54729, 78330, 111184, 159538, 225624, 315415, 444708, 618666, 858165, 1199701, 1646076, 2288961, 3150951, 4303995, 5870539, 8032571, 10881794, 14749051, 19992626
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Sum of products of terms in all partitions of n into distinct parts. - Vladeta Jovovic, Jan 19 2002
Number of partitions of n into distinct parts, when there are j sorts of part j. a(4) = 7: 4, 4', 4'', 4''', 31, 3'1, 3''1. - Alois P. Heinz, Aug 24 2015

Examples

			The partitions of 6 into distinct parts are 6, 1+5, 2+4, 1+2+3, the corresponding products are 6,5,8,6 and their sum is a(6) = 25.

Links

Crossrefs

Cf. A000009, A006906, A022661, A022693, A265758, A266891, A285222, A304043.
Cf. A092484, A265840, A265841, A265842, A322380, A322381.
Column k=1 of A292189.

Programs

  • Magma
    Coefficients(&*[(1+m*x^m):m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 16 2018
  • Maple
    b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 1] elif i<1 then [0, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i-1));
         [f[1]+g[1], f[2]+g[2]*i]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 02 2012
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, i*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015
  • Mathematica
    nn=20;CoefficientList[Series[Product[1+i x^i,{i,1,nn}],{x,0,nn}],x]  (* Geoffrey Critzer, Nov 02 2012 *)
nmax = 50; CoefficientList[Series[Exp[Sum[(-1)^(j+1)*PolyLog[-j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2015 *)
(* More efficient program: 10000 terms, 4 minutes, 100000 terms, 6 hours *) nmax = 40; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j+1]] += k*poly[[j-k+1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 06 2016 *)
  • PARI
    N=66; q='q+O('q^N); Vec(prod(n=1,N, (1+n*q^n) )) \\ Joerg Arndt, Oct 06 2012

Formula

Conjecture: log(a(n)) ~ sqrt(n/2) * (log(2*n) - 2). - Vaclav Kotesovec, May 08 2018

A092484 Expansion of Product_{m>=1} (1 + m^2*q^m).

Original entry on oeis.org

1, 1, 4, 13, 25, 77, 161, 393, 726, 2010, 3850, 7874, 16791, 31627, 69695, 139560, 255997, 482277, 986021, 1716430, 3544299, 6507128, 11887340, 21137849, 38636535, 70598032, 123697772, 233003286, 412142276, 711896765, 1252360770
Offset: 0

Views

Author

Jon Perry, Apr 04 2004

Keywords

Comments

Sum of squares of products of terms in all partitions of n into distinct parts.

Examples

			The partitions of 6 into distinct parts are 6, 1+5, 2+4, 1+2+3, the corresponding squares of products are 36, 25, 64, 36 and their sum is a(6) = 161.

Links

Crossrefs

Cf. A022629, A077335, A265844, A285737, A292165.
Column k=2 of A292189.

Programs

  • Maple
    b:= proc(n, i) option remember; (m->
      `if`(mn, 0, i^2*b(n-i, i-1)))))(i*(i+1)/2)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 10 2017
  • Mathematica
    Take[ CoefficientList[ Expand[ Product[1 + m^2*q^m, {m, 100}]], q], 31] (* Robert G. Wilson v, Apr 05 2005 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(n=1, N, 1+n^2*x^n)) \\ Seiichi Manyama, Sep 10 2017

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018
Conjecture: log(a(n)) ~ sqrt(2*n) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020

Extensions

More terms from Robert G. Wilson v, Apr 05 2004

A292190 Sum of n-th powers of products of terms in all partitions of n into distinct parts.

Original entry on oeis.org

1, 1, 4, 35, 337, 11925, 371081, 49032439, 3545396034, 3416952655320, 749189363202730, 598250899004413536, 2383502427069445040595, 1729793152213690218766715, 131751643363739706679145099315, 271212858254426215135033141804302
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2017

Keywords

Examples

			5 = 4 + 1 = 3 + 2. So a(5) = 5^5 + (4*1)^5 + (3*2)^5 = 11925.

Links

Crossrefs

Main diagonal of A292189.
Cf. A265949, A292167, A292194, A292305, A292306.

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:= n-> b(n$3):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 11 2017
  • Mathematica
    nmax = 15; Table[SeriesCoefficient[Product[(1 + k^n*x^k), {k, 1, nmax}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 12 2017 *)
  • PARI
    {a(n) = polcoeff(prod(k=1, n, 1+k^n*x^k+x*O(x^n)), n)}

Formula

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

A265840 Expansion of Product_{k>=1} (1 + k^3*x^k).

Original entry on oeis.org

1, 1, 8, 35, 91, 405, 1069, 3799, 8686, 36744, 86310, 235776, 686329, 1605779, 5230579, 13191702, 30608501, 73907925, 206052723, 433747560, 1324608945, 2995740974, 6973434054, 15364943439, 35816669079, 86662644756, 184871083828, 502089539734, 1098571699830
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 16 2015

Keywords

Links

Crossrefs

Cf. A022629, A092484, A265837, A265841, A265842.
Column k=3 of A292189.

Programs

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

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(3*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018
Conjecture: log(a(n)) ~ 3*sqrt(n/2) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020

A265841 Expansion of Product_{k>=1} (1 + k^4*x^k).

Original entry on oeis.org

1, 1, 16, 97, 337, 2177, 7313, 38529, 108594, 717186, 2053522, 7527458, 30757155, 88042387, 448973459, 1390503396, 4087546309, 12699966117, 49599776261, 124699632310, 608410782855, 1651128186296, 4862631132392, 13170300313769, 39285370060347, 130999461143020
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 16 2015

Keywords

Links

Crossrefs

Cf. A022629, A092484, A265838, A265840, A265842.
Column k=4 of A292189.

Programs

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

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(4*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018
Conjecture: log(a(n)) ~ 4*sqrt(n/2) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020

A265842 Expansion of Product_{k>=1} (1 + k^5*x^k).

Original entry on oeis.org

1, 1, 32, 275, 1267, 11925, 51445, 406183, 1406614, 14690040, 51144366, 251885088, 1481359033, 5108404955, 42614629915, 158222158038, 588574803125, 2360755022421, 13255325882835, 39266011999104, 325719196861377, 1031732678138822, 3791401325667894
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 16 2015

Keywords

Links

Crossrefs

Cf. A022629, A092484, A265839, A265840, A265841.
Column k=5 of A292189.

Programs

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

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(5*k)*x^(j*k)/k). - Ilya Gutkovskiy, Oct 18 2018
Conjecture: log(a(n)) ~ 5*sqrt(n/2) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020

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.

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

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