A006950 G.f.: Product_{k>=1} (1 + x^(2*k - 1)) / (1 - x^(2*k)).
1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 86, 105, 130, 161, 196, 236, 287, 350, 420, 501, 602, 722, 858, 1016, 1206, 1431, 1687, 1981, 2331, 2741, 3206, 3740, 4368, 5096, 5922, 6868, 7967, 9233, 10670, 12306, 14193, 16357, 18803, 21581
Offset: 0
Keywords
Examples
G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 13*x^9 + ... G.f. = q^-1 + q^7 + q^15 + 2*q^23 + 3*q^31 + 4*q^39 + 5*q^47 + 7*q^55 + 10*q^63 + ... From _Seiichi Manyama_, Jul 07 2018: (Start) n | the ways to stack n triangles in a valley --+------------------------------------------------------ 1 | *---* | \ / | * | 2 | * | / \ | *---* | \ / | * | 3 | *---* *---* | / \ / \ / \ | *---* *---* | \ / \ / | * * | 4 | * * | / \ / \ | *---* *---*---* *---* | / \ / \ / \ / \ / \ | *---* *---* *---* | \ / \ / \ / | * * * | 5 | *---* * * *---* | / \ / / \ / \ \ / \ | *---* *---*---* *---*---* *---* | / \ / \ / \ / \ / \ / \ / \ | *---* *---* *---* *---* | \ / \ / \ / \ / | * * * * | 6 | * | / \ | *---* *---* * * *---* | / \ / / \ / / \ / \ \ / \ | *---* *---*---* *---*---* *---*---* | / \ / \ / \ / \ / \ / \ / \ / | *---* *---* *---* *---* | \ / \ / \ / \ / | * * * * | * | / \ | *---* | \ / \ | *---* | \ / \ | *---* | \ / | * (End)
References
- A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 108.
- M. D. Hirschhorn, The Power of q, Springer, 2017. See pod, page 297.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
- N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
- Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
- Luca Ferrari, Schröder partitions, Schröder tableaux and weak poset patterns, arXiv:1606.06624 [math.CO], 2016. Mentions this sequence.
- M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic Properties of Partition k-tuples with Odd Parts Distinct, JIS, Vol. 19 (2016), Article 16.5.7
- Mircea Merca, New relations for the number of partitions with distinct even parts, Journal of Number Theory 176 (July 2017), 1-12.
- Victor S. Miller, Counting Matrices that are Squares, arXiv:1606.09299 [math.GR], 2016.
- Michael Somos, Introduction to Ramanujan theta functions
- Maxie D. Schmidt, Exact Formulas for the Generalized Sum-of-Divisors Functions, arXiv:1705.03488 [math.NT], 2017. See Example 4.2 p. 13.
- Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
- L. Wang, New Congruences for Partitions where the Odd Parts are Distinct, J. Int. Seq. 18 (2015) # 15.4.2.
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
- M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - From _N. J. A. Sloane_, Dec 25 2012
Crossrefs
Programs
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Maple
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i-irem(i, 2))))) end: a:= n-> b(n, n): seq(a(n), n=0..50); # Alois P. Heinz, Jan 06 2013 -
Mathematica
CoefficientList[ Series[ Product[(1 + x^(2k - 1))/(1 - x^(2k)), {k, 25}], {x, 0, 50}], x] (* Robert G. Wilson v, Jun 28 2012 *) CoefficientList[Series[x*QPochhammer[-1/x, x^2] / ((1+x)*QPochhammer[x^2, x^2]), {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 17 2015 *) CoefficientList[Series[2*(-x)^(1/8) / EllipticTheta[2, 0, Sqrt[-x]], {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 17 2015 *) b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i-Mod[i, 2]]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 11 2018, after Alois P. Heinz *) -
PARI
{a(n)=polcoeff(exp(sum(m=1, n+1, sumdiv(m, d, (-1)^(m-d)*d)*x^m/m)+x*O(x^n)), n)} \\ Paul D. Hanna, Jul 22 2009 (GW-BASIC) ' A program with two A-numbers (Note that here A000217 are the generalized hexagonal numbers): 10 Dim A000217(100), A057077(100), a(100): a(0)=1 20 For n = 1 to 51: For j = 1 to n 30 If A000217(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A000217(j)) 40 Next j: Print a(n-1);: Next n ' Omar E. Pol, Jun 10 2012
Formula
a(n) = (1/n)*Sum_{k=1..n} (-1)^(k+1)*A002129(k)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic, Feb 05 2002
G.f.: 1/Sum_{k>=0} (-x)^(k*(k+1)/2). - Vladeta Jovovic, Sep 22 2002 [corrected by Vaclav Kotesovec, Aug 17 2015]
G.f.: Product_{m>=1} (1+x^m)^(if A001511(m) > 1, A001511(m)-1 else A001511(m)). - Jon Perry, Apr 15 2005
Expansion of 1 / psi(-x) in powers of x where psi() is a Ramanujan theta function.
Expansion of q^(1/8) * eta(q^2) / (eta(q) * eta(q^4)) in powers of q.
Convolution inverse of A106459. - Michael Somos, Nov 02 2005
G.f.: exp( Sum_{n>=1} [Sum_{d|n} (-1)^(n-d)*d] * x^n/n ). - Paul D. Hanna, Jul 22 2009
a(n) ~ (8*n+1) * cosh(sqrt(8*n-1)*Pi/4) / (16*sqrt(2)*n^2) - sinh(sqrt(8*n-1)*Pi/4) / (2*Pi*n^(3/2)) ~ exp(Pi*sqrt(n/2))/(4*sqrt(2)*n) * (1 - (2/Pi + Pi/16)/sqrt(2*n) + (3/16 + Pi^2/1024)/n). - Vaclav Kotesovec, Aug 17 2015, extended Jan 09 2017
Can be computed recursively by Sum_{j>=0} (-1)^(ceiling(j/2)) a(n - j(j+1)/2) = 0, for n > 0. [Merca, Theorem 4.3] - Eric M. Schmidt, Sep 21 2017
Extensions
G.f. and more terms from Vladeta Jovovic, Feb 05 2002
Comments