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 1-3 of 3 results.

A143062 Expansion of false theta series variation of Euler's pentagonal number series in powers of x.

Original entry on oeis.org

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

Views

Author

Michael Somos, Jul 21 2008

Keywords

Comments

a(n) = sum over all partitions of n into distinct parts of number of partitions with even largest part minus number with odd largest part.
In the Berndt reference replace {a -> 1, q -> x} in equation (3.1) to get g.f. Replace {a -> x, q -> x} to get f(x). G.f. is 1 - f(x) * x / (1 + x).

Examples

			a(5) = -1 +1 -1 = -1 since 5 = 4 + 1 = 3 + 2. a(7) = -1 +1 -1 +1 +1 = 1 since 7 = 6 + 1 = 5 + 2 = 4 + 3 = 4 + 2 + 1.
G.f. = 1 - x + x^2 - x^5 + x^7 - x^12 + x^15 - x^22 + x^26 - x^35 + x^40 + ...
G.f. = q - q^25 + q^49 - q^121 + q^169 - q^289 + q^361 - q^529 + q^625 - q^841 + ...

References

  • G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See Section 9.4, pp. 232-236.
  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, see p. 41, 10th equation numerator.

Links

Crossrefs

Cf. A010815, A026837, A026838, A080995, A121373, A199918, A203568, A268539.

Programs

  • Mathematica
    a[ n_] := If[ SquaresR[ 1, 24 n + 1] == 2, (-1)^Quotient[ Sqrt[24 n + 1], 3], 0];
a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ@m, (-1)^Quotient[ m, 3], 0]]; (* Michael Somos, Nov 18 2015 *)
  • PARI
    {a(n) = if( issquare( 24*n + 1, &n), (-1)^(n \ 3) )};

Formula

a(n) = b(24*n + 1) where b() is multiplicative with b(p^(2*e)) = (-1)^e if p = 5 (mod 6), b(p^(2*e)) = +1 if p = 1 (mod 6) and b(p^(2*e-1)) = b(2^e) = b(3^e) = 0 if e>0.
G.f.: Sum_{k>=0} x^((3*k^2 + k) / 2) * (1 - x^(2*k + 1)) = 1 - Sum_{k>0} x^((3*k^2 - k) / 2) * (1 - x^k).
G.f.: 1 - x / (1 + x) + x^3 / ((1 + x) * (1 + x^2)) - x^6 / ((1 + x) * (1 + x^2) * (1 + x^3)) + ...
G.f.: 1 - x / (1 + x^2) + x^2 / ((1 + x^2) * (1 + x^4)) - x^3 / ((1 + x^2 ) * (1 + x^4) * (1 + x^6)) + ...
|a(n)| = |A010815(n)| = |A080995(n)| = |A199918(n)| = |A121373(n)|.
From Joerg Arndt, Jun 24 2013: (Start)
a(n) = A026838(n) - A026837(n) (Fine's theorem), see the Pak reference.
a(n)=1 if n = k(3k+1)/2, a(n)=-1 if n = k(3k-1)/2, a(n)=0 otherwise.
G.f.: Sum_{n >= 0} (-q)^n * (Product_{k = 1..n-1} 1 + q^k). (End)
a(n) = - A203568(n) unless n=0. a(0) = 1. - Michael Somos, Jul 12 2015
From Peter Bala, Feb 04 2021: (Start)
A conjectural g.f: 1 + Sum_{n >= 0} (-1)^n*x^(2*n-1)/Product_{k = 1..n} 1 + x^(2*k-1).
G.f.: 1 - Sum_{n >= 1} x^(n*(2*n-1))/Product_{k = 1..2*n} 1 + x^k [added Dec 19 2024: see Berndt et al., Entry 9.44]. (End)
Conjectural g.f.: (1/(1 + x)) * (2 - Sum_{n >= 0} (-1)^n * x^(3*n)/Product_{k = 1..n} 1 + x^(2*k)). - Peter Bala, Jan 19 2025

A269819 Numbers that are congruent to {5, 11, 13, 19} mod 24.

Original entry on oeis.org

5, 11, 13, 19, 29, 35, 37, 43, 53, 59, 61, 67, 77, 83, 85, 91, 101, 107, 109, 115, 125, 131, 133, 139, 149, 155, 157, 163, 173, 179, 181, 187, 197, 203, 205, 211, 221, 227, 229, 235, 245, 251, 253, 259, 269, 275, 277, 283, 293, 299, 301, 307, 317
Offset: 1

Views

Author

Bob Selcoe, Mar 05 2016

Keywords

Comments

No terms are multiples of 3.
Numbers such that (j+5)*(j-5)/48 are positive integers. Equivalent to positive integers (m+3)*(m-2)/12, with m == {2,5,6,9} mod 12 (observation made in A268539 by M. F. Hasler, Mar 02 2016).

Links

Crossrefs

Subsequence of A001651.
Cf. A268539.

Programs

  • Magma
    I:=[5,11,13,19]; [n le 4 select I[n] else Self(n-4) + 24 : n in [1..60]]; // Vincenzo Librandi, Mar 06 2016
  • Magma
    [n : n in [0..400] | n mod 24 in [5, 11, 13, 19]]; // Wesley Ivan Hurt, Jun 04 2016
  • Maple
    A269819:=n->6*n-3-(1-I)*I^(-n)-(1+I)*I^n: seq(A269819(n), n=1..80); # Wesley Ivan Hurt, Jun 04 2016
  • Mathematica
    Table[24 n + {5, 11, 13, 19}, {n, 0, 12}] // Flatten (* Michael De Vlieger, Mar 07 2016 *)
Table[6n-3-(1-I)*I^(-n)-(1+I)*I^n, {n, 80}] (* Wesley Ivan Hurt, Jun 04 2016 *)
LinearRecurrence[{2,-2,2,-1},{5,11,13,19},60] (* Harvey P. Dale, Nov 17 2017 *)
  • PARI
    Vec(x*(1+x)*(5-4*x+5*x^2)/((1-x)^2*(1+x^2)) + O(x^100)) \\ Colin Barker, Mar 06 2016

Formula

a(n) = a(n-4) + 24.
a(n) = sqrt(48*A268539(n) + 25).
G.f.: x*(1+x)*(5-4*x+5*x^2) / ((1-x)^2*(1+x^2)). - Colin Barker, Mar 06 2016
From Wesley Ivan Hurt, Jun 04 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(n) = 6*n-3-(1-i)*i^(-n)-(1+i)*i^n for i=sqrt(-1). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (2-sqrt(2))*Pi/12. - Amiram Eldar, Dec 31 2021

Extensions

Incorrect term 252 replaced by two missing terms 251 and 253 by Colin Barker, Mar 06 2016

A377979 List of exponents in the expansion of (1 - q)*Sum_{n >= 0} q^(2*n*(n+1))*Product_{k >= 2*n+1} 1 - q^k.

Original entry on oeis.org

0, 1, 3, 4, 6, 8, 11, 13, 18, 20, 26, 29, 35, 39, 46, 50, 59, 63, 73, 78, 88, 94, 105, 111, 124, 130, 144, 151, 165, 173, 188, 196, 213, 221, 239, 248, 266, 276, 295, 305, 326, 336, 358, 369, 391, 403, 426, 438, 463, 475, 501, 514, 540, 554, 581, 595, 624, 638, 668, 683, 713, 729, 760, 776, 809, 825
Offset: 1

Views

Author

Peter Bala, Dec 16 2024

Keywords

Comments

Compare with the expansions Sum_{n >= 0} q^(2*n*(n+1))*Product_{k >= 2*n+2} 1 - q^k = 1 - q^2 - q^3 + q^7 + q^17 - q^25 - q^28 + + - - ... (see A268539) and Sum_{n >= 0} q^(n*(n+1))*Product_{k >= 2*n+1} 1 - q^k = 1 - q - q^8 + q^13 + q^17 - q^24 - q^45 + + - - .... (see A204221).
Conjectures:
1) apart from the coefficient of q, the coefficients of the series expansion (see below) belong to {-1, 0, 1}.
2) starting at q^3, the signs of the nonzero coefficients follow the pattern + + - - + + - - ....
It appears that the sequence terms are the exponents in the expansion of Sum_{n >= 0} x^(3*n)/(Product_{k = 1..2*n} 1 + x^k) = 1 + x^3 - x^4 + x^6 - x^8 + x^11 - x^13 + - .... - Peter Bala, Jan 21 2025

Examples

			(1 - q)*Sum_{n >= 0} q^(2*n*(n+1))*Product_{k >= 2*n+1} 1 - q^k = 1 - 2*q + q^3 + q^4 - q^6 - q^8 + q^11 + q^13 - q^18 - q^20 + q^26 + q^29 - q^35 - q^39 + q^46 + q^50 - q^59 - q^63 + + - - ....

Crossrefs

Cf. A033577, A152749, A204221, A244806, A268539, A033578.

Programs

  • Maple
    series(add((1 - q)*q^(2*n*(n+1))*mul(1 - q^k, k = 2*n+1..1000), n = 0..21), q, 1001);

Formula

The following are conjectural:
a(n) is quasi-polynomial in n:
a(8*n+1) = 12*n^2 + 5*n + 1 = A244806(n+1) for n >= 1;
a(8*n+2) = 12*n^2 + 7*n + 1 = A033577(n); a(8*n+3) = 12*n^2 + 11*n + 3;
a(8*n+4) = 12*n^2 + 13*n + 4; a(8*n+5) = 12*n^2 + 17*n + 6 = A033578(n+1);
a(8*n+6) = 12*n^2 + 19*n + 8; a(8*n+7) = 12*n^2 + 23*n + 11;
a(8*n+8) = 12*n^2 + 25*n + 13.
G.f.: x^2*(x^8 - x^7 - 2*x^6 + 3*x^5 + 2*x^4 - 2*x^3 - x^2 + 2*x + 1)/((1 + x)^2*(1 - x)^3*(1 + x^4)) = x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 11*x^7 + ....
Showing 1-3 of 3 results.

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