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.

A078616 a(n) = Sum_{k=0..n} A010815(k).

Original entry on oeis.org

1, 0, -1, -1, -1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 0

Views

Author

Benoit Cloitre, Dec 10 2002

Keywords

Comments

To construct the sequence: a(0)=1, a(1)=0, then (2*1+1) (-1)'s followed by 2 0's, followed by (2*2+1) 1's, followed by 3 0's, followed by (2*3+1) (-1)'s, etc.
From George Beck, May 05 2017: (Start)
a(n) = (Number of ones in the distinct partitions of n with an odd number of parts) - (number of ones in the distinct partitions of n with an even number of parts) (conjectured).
The partial sums give A246575. (End) [corrected by Ilya Gutkovskiy, Aug 18 2018]

Links

Crossrefs

Cf. A010815, A000326, A246575.

Programs

  • PARI
    a(n)=polcoeff(eta(x)/(1-x)+O(x^n),n)

Formula

For m > 0, a(k)=0 if A000326(m) <= k < A000326(m) + m; a(k)=(-1)^m if A000326(m) + m <= k < A000326(m+1).
G.f.: eta(x)/(1-x). - Benoit Cloitre, Jan 31 2004
G.f.: exp(-Sum_{k>=1} (sigma_1(k) - 1)*x^k/k). - Ilya Gutkovskiy, Aug 18 2018

A246578 Expansion of g.f. (Product_{r>=1} (1 - x^r))*x^(k^2)/Product_{i=1..k} ((1-x^i)^2) with k=4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 5, 7, 7, 8, 6, 5, 1, -2, -9, -15, -23, -30, -39, -46, -52, -56, -58, -57, -51, -43, -29, -13, 10, 33, 63, 90, 124, 152, 184, 207, 233, 245, 258, 255, 250, 227, 202, 157, 110, 45, -22, -104, -185, -278, -366
Offset: 0

Views

Author

N. J. A. Sloane, Aug 31 2014

Keywords

Links

Crossrefs

k=0 gives A010815. Cf. A246575-A246578.

Programs

  • Maple
    fGL:=proc(k) local a,i,r;
a:=x^(k^2)/mul((1-x^i)^2,i=1..k);
a:=a*mul(1-x^r,r=1..101);
series(a,x,101);
seriestolist(%);
end;fGL(4);
  • Mathematica
    With[{k = 4}, CoefficientList[Product[(1-x^r), {r, 1, nmax}]* x^(k^2)/Product[(1-x^i)^2, {i, 1, k}] + O[x]^nmax, x]] (* Jean-François Alcover, Mar 09 2023 *)

A246576 G.f.: (Product_{r>=1} (1 - x^r))*x^(k^2)/Product_{i=1..k} ((1 - x^i)^2) with k=2.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 1, 1, -1, -2, -4, -5, -6, -6, -5, -4, -1, 1, 5, 7, 11, 12, 15, 14, 15, 12, 11, 6, 3, -3, -7, -13, -17, -22, -25, -28, -29, -29, -28, -25, -22, -16, -11, -3, 3, 12, 18, 27, 32, 40, 43, 49, 49, 52, 49, 49, 43, 40, 31, 25, 14, 6, -6, -15, -27, -36, -47, -55, -64
Offset: 0

Views

Author

N. J. A. Sloane, Aug 31 2014

Keywords

References

  • Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70.

Crossrefs

k=0 gives A010815. Cf. A246575-A246578.

Programs

  • Maple
    fGL:=proc(k) local a,i,r;
a:=x^(k^2)/mul((1-x^i)^2,i=1..k);
a:=a*mul(1-x^r,r=1..101);
series(a,x,101);
seriestolist(%);
end;fGL(2);

A246577 G.f.: (Product_{r>=1} (1 - x^r))*x^(k^2)/Product_{i=1..k} ((1 - x^i)^2) with k=3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 3, 3, 3, 1, -1, -3, -7, -10, -13, -16, -18, -18, -18, -15, -11, -5, 2, 11, 20, 30, 39, 48, 55, 60, 63, 63, 59, 53, 43, 29, 14, -5, -26, -47, -69, -92, -111, -130, -146, -157, -164, -168, -163, -155, -141, -120, -94, -65, -28, 10, 51, 95
Offset: 0

Views

Author

N. J. A. Sloane, Aug 31 2014

Keywords

References

  • Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70.

Crossrefs

k=0 gives A010815. Cf. A246575-A246578.

Programs

  • Maple
    fGL:=proc(k) local a,i,r;
a:=x^(k^2)/mul((1-x^i)^2,i=1..k);
a:=a*mul(1-x^r,r=1..101);
series(a,x,101);
seriestolist(%);
end;fGL(3);

A334895 G.f.: (Sum_{k>=1} prime(k) * x^k) * (Product_{j>=1} (1 - x^j)).

Original entry on oeis.org

0, 2, 1, 0, -1, -1, -3, -4, -4, -3, 3, -1, 5, 3, 2, 6, 8, 11, 3, 3, 1, -5, -5, -3, -4, -8, -12, -16, -19, -13, -5, 9, 3, 1, -7, 3, 7, 0, 20, 18, 18, 18, 23, 19, 15, 9, 5, 5, 15, -9, -25, -27, -25, -20, -6, -12, -20, -10, -20, -17, -27, -9, -1, 5, -5, -13, -23, 3, 1, 15, 19
Offset: 0

Views

Author

Ilya Gutkovskiy, May 14 2020

Keywords

Comments

Convolution of primes with A010815.

Links

Crossrefs

Cf. A000040, A000041, A010815, A086717, A246575 (convolution of nonnegative integers with A010815).

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[Prime[k] x^k, {k, 1, nmax}] Product[(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]
A010815[0] = 1; A010815[n_] := A010815[n] = -(1/n) Sum[DivisorSigma[1, k] A010815[n - k], {k, 1, n}]; a[n_] := Sum[Prime[k] A010815[n - k], {k, 1, n}]; Table[a[n], {n, 0, 70}]

Formula

Sum_{k=1..n} a(k) * A000041(n-k) = prime(n).
Showing 1-5 of 5 results.

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

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