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 1-9 of 9 results.

A261053 Expansion of Product_{k>=1} (1+x^k)^(k^k).

Original entry on oeis.org

1, 1, 4, 31, 289, 3495, 51268, 891152, 17926913, 409907600, 10499834497, 297793199060, 9262502810645, 313457634240463, 11464902463397642, 450646709610954343, 18943070964019019671, 847932498252050293971, 40266255926484893366914, 2021845081107882645459639
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 08 2015

Keywords

Links

Crossrefs

Cf. A023880, A261052, A026007, A027998, A248882, A102866, A256142.

Programs

  • Magma
    m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1+x^k)^(k^k): k in [1..(m+2)]]))); // G. C. Greubel, Nov 08 2018
  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(i^i, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 08 2015
  • Mathematica
    nmax=20; CoefficientList[Series[Product[(1+x^k)^(k^k),{k,1,nmax}],{x,0,nmax}],x]
  • PARI
    m=20; x='x+O('x^m); Vec(prod(k=1,m, (1+x^k)^(k^k))) \\ G. C. Greubel, Nov 08 2018

Formula

a(n) ~ n^n * (1 + exp(-1)/n + (exp(-1)/2 + 4*exp(-2))/n^2).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d^(d+1) ) * x^k/k). - Ilya Gutkovskiy, Nov 08 2018

A283499 Expansion of exp( Sum_{n>=1} -A283498(n)/n*x^n ) in powers of x.

Original entry on oeis.org

1, -1, -4, -23, -223, -2767, -42268, -759008, -15672223, -365639304, -9512549191, -273072804420, -8575012101043, -292422232720311, -10762617713743350, -425245537127322111, -17953822507629389009, -806668679245000383731
Offset: 0

Views

Author

Seiichi Manyama, Mar 09 2017

Keywords

Links

Crossrefs

Cf. A023880, A283498.

Programs

  • Mathematica
    A[n_] :=  Sum[d^(d+ 1), {d, Divisors[n]}]; a[n_] := If[n==0, 1, -(1/n)*Sum[A[k]*a[n - k], {k, n}]]; Table[a[n], {n, 0, 17}] (* Indranil Ghosh, Mar 11 2017 *)
  • PARI
    a(n) = if(n==0, 1, -(1/n)*sum(k=1, n, sumdiv(k, d, d^(d + 1))*a(n - k)));
for(n=0, 20, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 11 2017

Formula

G.f.: Product_{k>=1} (1 - x^k)^(k^k).
a(n) = -(1/n)*Sum_{k=1..n} A283498(k)*a(n-k) for n > 0.

A283579 Expansion of exp( Sum_{n>=1} A283533(n)/n*x^n ) in powers of x.

Original entry on oeis.org

1, 1, 17, 746, 66418, 9843707, 2187941520, 680615139257, 282199700198462, 150389915598653924, 100155578743010743914, 81505577512720707466924, 79580089689432499741178617, 91814299713761739807846854872
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2017

Keywords

Links

Crossrefs

Cf. Product_{k>=1} 1/(1 - x^k)^(k^(m*k)): A000041 (m=0), A023880 (m=1), this sequence (m=2), A283580 (m=3).
Cf. A283534 (Product_{k>=1} (1 - x^k)^(k^(2*k))).

Programs

  • Mathematica
    A[n_] :=  Sum[d^(2*d + 1), {d, Divisors[n]}]; a[n_] := If[n==0, 1, (1/n)*Sum[A[k]*a[n - k], {k, n}]]; Table[a[n], {n, 0, 13}] (* Indranil Ghosh, Mar 11 2017 *)
  • PARI
    A(n) = sumdiv(n, d, d^(2*d + 1));
a(n) = if(n==0, 1, (1/n)*sum(k=1, n, A(k)*a(n - k)));
for(n=0, 11, print1(a(n),", ")) \\ Indranil Ghosh, Mar 11 2017

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^(k^(2*k)).
a(n) = (1/n)*Sum_{k=1..n} A283533(k)*a(n-k) for n > 0.
a(n) ~ n^(2*n) * (1 + exp(-2)/n^2). - Vaclav Kotesovec, Mar 17 2017

A283580 Expansion of exp( Sum_{n>=1} A283535(n)/n*x^n ) in powers of x.

Original entry on oeis.org

1, 1, 65, 19748, 16799044, 30535636881, 101591759812967, 558649739234980142, 4722932373908389412037, 58154498193439779564557624, 1000058469893323150011227885608, 23226158305362748824532880463596385, 708825166389400019044145225134521489486
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2017

Keywords

Links

Crossrefs

Cf. Product_{k>=1} 1/(1 - x^k)^(k^(m*k)): A000041 (m=0), A023880 (m=1), A283579 (m=2), this sequence (m=3).
Cf. A283536 (Product_{k>=1} (1 - x^k)^(k^(3*k))).

Programs

  • Mathematica
    A[n_] :=  Sum[d^(3*d + 1), {d, Divisors[n]}]; a[n_] := If[n==0, 1, (1/n)*Sum[A[k]*a[n - k], {k, n}]];Table[a[n], {n, 0, 12}] (* Indranil Ghosh, Mar 11 2017 *)
  • PARI
    A(n) = sumdiv(n, d, d^(3*d + 1));
a(n) = if(n==0, 1, (1/n)*sum(k=1, n, A(k)*a(n - k)));
for(n=0, 12, print1(a(n),", ")) \\ Indranil Ghosh, Mar 11 2017

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^(k^(3*k)).
a(n) = (1/n)*Sum_{k=1..n} A283535(k)*a(n-k) for n > 0.
a(n) ~ n^(3*n) * (1 + exp(-3)/n^3). - Vaclav Kotesovec, Mar 17 2017

A283674 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-x^j)^(j^(k*j)) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 32, 5, 1, 1, 65, 746, 298, 7, 1, 1, 257, 19748, 66418, 3531, 11, 1, 1, 1025, 531698, 16799044, 9843707, 51609, 15, 1, 1, 4097, 14349932, 4295531890, 30535636881, 2187941520, 894834, 22, 1, 1, 16385, 387424586, 1099526502508, 95371863221411, 101591759812967, 680615139257, 17980052, 30
Offset: 0

Views

Author

Seiichi Manyama, Mar 14 2017

Keywords

Examples

			Square array begins:
   1,   1,     1,        1, ...
   1,   1,     1,        1, ...
   2,   5,    17,       65, ...
   3,  32,   746,    19748, ...
   5, 298, 66418, 16799044, ...

Links

Crossrefs

Columns k=0-4 give A000041, A023880, A283579, A283580, A283510.
Rows give: 0-1: A000012, 2: A052539, 3: A283716.
Main diagonal gives A283719.
Cf. A283675.

Programs

  • Maple
    with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
      d*d^(k*d), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Mar 15 2017
  • Mathematica
    A[n_, k_] := If[n==0, 1, Sum[Sum[d*d^(k*d), {d, Divisors[j]}] *A[n - j, k], {j, n}] / n]; Flatten[Table[A[d - n,  n],{d, 0, 10},{n, d, 0, -1}]] (* Indranil Ghosh, Mar 17 2017 *)
  • PARI
    A(n, k) = if(n==0, 1, sum(j=1, n, sumdiv(j, d, d*d^(k*d)) * A(n - j, k))/n);
{for(d=0, 10, for(n=0, d, print1(A(n, d - n),", ");); print(););} \\ Indranil Ghosh, Mar 17 2017

Formula

G.f. of column k: Product_{j>=1} 1/(1-x^j)^(j^(k*j)).

A283510 Expansion of exp( Sum_{n>=1} A283369(n)/n*x^n ) in powers of x.

Original entry on oeis.org

1, 1, 257, 531698, 4295531890, 95371863221411, 4738477950914329100, 459991301719292572342573, 79228623778497392212453912974, 22528478894247280128054776211273960, 10000022549030658394108744658459680045250
Offset: 0

Views

Author

Seiichi Manyama, Mar 17 2017

Keywords

Links

Crossrefs

Cf. Product_{k>=1} 1/(1 - x^k)^(k^(m*k)): A000041 (m=0), A023880 (m=1), A283579 (m=2), A283580 (m=3), this sequence (m=4).
Cf. A283803 (Product_{k>=1} (1 - x^k)^(k^(4*k))).

Programs

  • Mathematica
    CoefficientList[Series[Product[1/(1 - x^k)^(k^(4k)), {k, 1, 10}], {x, 0, 10}], x] (* Indranil Ghosh, Mar 17 2017 *)
  • PARI
    A(n) = sumdiv(n, d, d^(4*d + 1));
a(n) = if(n<1, 1, (1/n) * sum(k=1, n, A(k) * a(n - k)));
for(n=0, 10, print1(a(n),", ")) \\ Indranil Ghosh, Mar 17 2017

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^(k^(4*k)).
a(n) = (1/n)*Sum_{k=1..n} A283369(k)*a(n-k) for n > 0.
a(n) ~ n^(4*n) * (1 + exp(-4)/n^4). - Vaclav Kotesovec, Mar 17 2017

A307504 Expansion of Product_{k>=1} 1/(1-x^k)^((-1)^k*k^k).

Original entry on oeis.org

1, -1, 4, -31, 293, -3499, 51284, -891276, 17928335, -409921846, 10500040633, -297796771914, 9262574642871, -313459274848233, 11464944476563718, -450647901022275715, 18943108018829605740, -847933752191806388254, 40266301788890216414608, -2021846883773977115156632
Offset: 0

Views

Author

Seiichi Manyama, Apr 11 2019

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = (-1)^n * n^n, g(n) = 1.

Links

Crossrefs

Cf. A023880, A266964, A307497.

Programs

  • Mathematica
    nmax=20; CoefficientList[Series[Product[1/(1-x^k)^((-1)^k*k^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 12 2019 *)
  • PARI
    N=20; x='x+O('x^N); Vec(1/prod(k=1, N, (1-x^k)^((-1)^k*k^k)))

Formula

a(n) ~ (-1)^n * n^n * (1 + exp(-1)/n + (exp(-1)/2 + 4*exp(-2))/n^2). - Vaclav Kotesovec, Apr 12 2019

A364041 Expansion of 1/Product_{k>0} (1 - x^(2*k-1))^((2*k-1)^(2*k-1)).

Original entry on oeis.org

1, 1, 1, 28, 28, 3153, 3531, 827074, 911449, 388335592, 415455628, 285728307489, 298762259972, 303174312029604, 312427539531172, 438206538943092800, 447594828079035405, 827688010429432132457, 840767646450714838158, 1979260573433349667269165
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2023

Keywords

Crossrefs

Cf. A023880, A363991.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[DivisorSum[k, #^(# + 1) &, OddQ[#] &]*a[n - k], {k, 1, n}]/n; Array[a, 20, 0] (* Amiram Eldar, Jul 09 2023 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/prod(k=1, N, (1-x^(2*k-1))^((2*k-1)^(2*k-1))))

Formula

G.f.: exp( Sum_{k>0} A363991(k) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} A363991(k) * a(n-k).

A321389 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k^k).

Original entry on oeis.org

1, 2, 10, 72, 670, 7896, 113532, 1938948, 38463150, 869969602, 22098936536, 622728174288, 19271479902324, 649553475002720, 23680210649058960, 928276725059295192, 38931910620358040382, 1739307894106738293052, 82457731356894087128054, 4134332188240252347401752, 218571692793801915329820184
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 08 2018

Keywords

Comments

Convolution of A023880 and A261053.

Links

Crossrefs

Cf. A023880, A156616, A206622, A206623, A206624, A261053.

Programs

  • Maple
    a:=series(mul(((1+x^k)/(1-x^k))^(k^k),k=1..100),x=0,21): seq(coeff(a,x,n),n=0..20); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[((-1)^(k/d + 1) + 1) d^(d + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 20}]
  • PARI
    seq(n)={Vec(exp(sum(k=1, n, sumdiv(k,d, ((-1)^(k/d+1) + 1)*d^(d+1) ) * x^k/k) + O(x*x^n)))} \\ Andrew Howroyd, Nov 09 2018

Formula

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} ((-1)^(k/d+1) + 1)*d^(d+1) ) * x^k/k).
a(n) ~ 2 * n^n * (1 + 2*exp(-1)/n + (exp(-1) + 10*exp(-2))/n^2). - Vaclav Kotesovec, Nov 09 2018
Showing 1-9 of 9 results.

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