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 1-5 of 5 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

A292189 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j^k*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 13, 7, 3, 1, 1, 16, 35, 25, 15, 4, 1, 1, 32, 97, 91, 77, 25, 5, 1, 1, 64, 275, 337, 405, 161, 43, 6, 1, 1, 128, 793, 1267, 2177, 1069, 393, 64, 8, 1, 1, 256, 2315, 4825, 11925, 7313, 3799, 726, 120, 10
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2017

Keywords

Examples

			Square array begins:
   1, 1,  1,  1,   1, ...
   1, 1,  1,  1,   1, ...
   1, 2,  4,  8,  16, ...
   2, 5, 13, 35,  97, ...
   2, 7, 25, 91, 337, ...

Links

Crossrefs

Columns k=0..5 give A000009, A022629, A092484, A265840, A265841, A265842.
Rows 0+1, 2, 3 give A000012, A000079, A007689.
Main diagonal gives A292190.
Cf. A292166.

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, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Sep 11 2017
  • Mathematica
    m = 14;
col[k_] := col[k] = Product[1 + j^k*x^j, {j, 1, m}] + O[x]^(m+1) // CoefficientList[#, x]&;
A[n_, k_] := col[k][[n+1]];
Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)

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

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
Showing 1-5 of 5 results.

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

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