A199918 -id:A199918 - cp's OEIS frontend

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

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

Michael Somos, Jul 21 2008

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).

			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 + ...

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, see p. 41, 10th equation numerator.

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973.
I. Pak, On Fine's partition theorems, Dyson, Andrews and missed opportunities, Math. Intelligencer, 25 (No. 1, 2003), 10-16.

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

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 *)

{a(n) = if( issquare( 24*n + 1, &n), (-1)^(n \ 3) )};

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

A026836 Triangular array T read by rows: T(n,k) = number of partitions of n into distinct parts, the greatest being k, for k=1,2,...,n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 1, 0, 0, 0, 1, 2, 2, 1, 1, 1, 0, 0, 0, 1, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 2, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 2, 3, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 1, 3, 4, 3, 2, 2, 1, 1

Offset: 1

Clark Kimberling

Conjecture: A199918(n) = Sum_{k=1..n} (-1)^(n-k) T(n,k). - George Beck, Jan 13 2019

			Triangle begins:
[1]
[0, 1]
[0, 1, 1]
[0, 0, 1, 1]
[0, 0, 1, 1, 1]
[0, 0, 1, 1, 1, 1]
[0, 0, 0, 2, 1, 1, 1]
[0, 0, 0, 1, 2, 1, 1, 1]
[0, 0, 0, 1, 2, 2, 1, 1, 1]
[0, 0, 0, 1, 2, 2, 2, 1, 1, 1]
[0, 0, 0, 0, 2, 3, 2, 2, 1, 1, 1]
[0, 0, 0, 0, 2, 3, 3, 2, 2, 1, 1, 1]
[0, 0, 0, 0, 1, 3, 4, 3, 2, 2, 1, 1, 1]
[0, 0, 0, 0, 1, 3, 4, 4, 3, 2, 2, 1, 1, 1]
... - _N. J. A. Sloane_, Nov 09 2018

Henry Bottomley, Partition calculators using java applets
Index entries for sequences related to partitions

If seen as a square array then transpose of A070936 and visible form of A053632. Central diagonal and those to the right of center are A000009 as are row sums.

with(combinat);
f2:=proc(n) local i,j,p,t0,t1,t2;
t0:=Array(1..n,0);
t1:=partition(n);
p:=numbpart(n);
for i from 1 to p do
t2:=t1[i];
if nops(convert(t2,set))=nops(t2) then
# now have a partition t2 of n into distinct parts
t0[t2[-1]]:=t0[t2[-1]]+1;
od:
[seq(t0[j],j=1..n)];
end proc;
for n from 1 to 12 do lprint(f2(n)); od: # N. J. A. Sloane, Nov 09 2018

T(n, k) = A070936(n-k, k-1) = A053632(k-1, n-k) = T(n-1, k-1)+T(n-2k+1, k-1). - Henry Bottomley, May 12 2002

T(n, k) = coefficient of x^n in x^k*Product_{i=1..k-1} (1+x^i). - Vladeta Jovovic, Aug 07 2003

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula