Search: a059777 -id:a059777
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A006950
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G.f.: Product_{k>=1} (1 + x^(2*k - 1)) / (1 - x^(2*k)).
(Formerly M0524)
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+10
67
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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
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OFFSET
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0,4
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COMMENTS
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Also the number of partitions of n in which all odd parts are distinct. There is no restriction on the even parts. E.g., a(9)=13 because "9 = 8+1 = 7+2 = 6+3 = 6+2+1 = 5+4 = 5+3+1 = 5+2+2 = 4+4+1 = 4+3+2 = 4+2+2+1 = 3+2+2+2 = 2+2+2+2+1". - Noureddine Chair, Feb 03 2005
Number of partitions of n in which each even part occurs with even multiplicity. There is no restriction on the odd parts.
Also the number of partitions of n into parts not congruent to 2 mod 4. - James A. Sellers, Feb 08 2002
Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras o(n) of skew-symmetric n X n matrices, n=0,1,2,3,... (the cases n=0,1 being degenerate). This sequence, A015128 and A000041 together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003
Poincaré series [or Poincare series] (or Molien series) for symmetric invariants in F_2(b_1, b_2, ... b_n) ⊗ E(e_1, e_2, ... e_n) with b_i 2-dimensional, e_i one-dimensional and the permutation action of S_n, in the case n=2.
It appears that this sequence is related to the generalized hexagonal numbers (A000217) in the same way as the partition numbers A000041 are related to the generalized pentagonal numbers A001318. (See the table in comments section of A195825.) Conjecture: this is 1 together with the row sums of triangle A195836, also column 1 of A195836, also column 2 of the square array A195825. - Omar E. Pol, Oct 09 2011
Since this is also column 2 of A195825 so the sequence contains only one plateau [1, 1, 1] of level 1 and length 3. For more information see A210843. - Omar E. Pol, Jun 27 2012
Also the number of ways to stack n triangles in a valley (pointing upwards or downwards depending on row parity). - Seiichi Manyama, Jul 07 2018
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REFERENCES
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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).
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LINKS
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FORMULA
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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.
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
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EXAMPLE
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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 + ...
n | the ways to stack n triangles in a valley
--+------------------------------------------------------
1 | *---*
| \ /
| *
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2 | *
| / \
| *---*
| \ /
| *
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3 | *---* *---*
| / \ / \ / \
| *---* *---*
| \ / \ /
| * *
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4 | * *
| / \ / \
| *---* *---*---* *---*
| / \ / \ / \ / \ / \
| *---* *---* *---*
| \ / \ / \ /
| * * *
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5 | *---* * * *---*
| / \ / / \ / \ \ / \
| *---* *---*---* *---*---* *---*
| / \ / \ / \ / \ / \ / \ / \
| *---* *---* *---* *---*
| \ / \ / \ / \ /
| * * * *
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6 | *
| / \
| *---* *---* * * *---*
| / \ / / \ / / \ / \ \ / \
| *---* *---*---* *---*---* *---*---*
| / \ / \ / \ / \ / \ / \ / \ /
| *---* *---* *---* *---*
| \ / \ / \ / \ /
| * * * *
| *
| / \
| *---*
| \ / \
| *---*
| \ / \
| *---*
| \ /
| *
(End)
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MAPLE
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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):
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MATHEMATICA
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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];
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PROG
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(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):
20 For n = 1 to 51: For j = 1 to n
40 Next j: Print a(n-1); : Next n ' Omar E. Pol, Jun 10 2012
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A316675
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Triangle read by rows: T(n,k) gives the number of ways to stack n triangles in a valley so that the right wall has k triangles for n >= 0 and 0 <= k <= n.
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+10
7
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1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 1, 1, 3, 2, 1, 1, 1, 0, 0, 1, 1, 3, 3, 2, 1, 1, 1, 0, 0, 1, 1, 3, 3, 3, 2, 1, 1, 1, 0, 0, 1, 1, 4, 3, 4, 3, 2, 1, 1, 1, 0, 0, 1, 1, 5, 4, 5, 4, 3, 2, 1, 1, 1
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OFFSET
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0,33
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LINKS
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FORMULA
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For m >= 0,
Sum_{n>=2m} T(n,2m) *x^n = x^(2m) * Product_{j=1..m} (1+x^(2j-1))/(1-x^(2j)).
Sum_{n>=2m+1} T(n,2m+1)*x^n = x^(2m+1) * Product_{j=1..m} (1+x^(2j-1))/(1-x^(2j)).
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EXAMPLE
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T(8,4) = 3.
* *
/ \ / \
*---* * *---*---* *---*
\ / \ / \ \ / \ / \ / \ / \
*---*---* *---*---* *---*---*
\ / \ / \ / \ / \ / \ /
*---* *---* *---*
\ / \ / \ /
* * *
Triangle begins:
1;
0, 1;
0, 0, 1;
0, 0, 1, 1;
0, 0, 1, 1, 1;
0, 0, 1, 1, 1, 1;
0, 0, 1, 1, 1, 1, 1;
0, 0, 1, 1, 2, 1, 1, 1;
0, 0, 1, 1, 3, 2, 1, 1, 1;
0, 0, 1, 1, 3, 3, 2, 1, 1, 1;
0, 0, 1, 1, 3, 3, 3, 2, 1, 1, 1;
0, 0, 1, 1, 4, 3, 4, 3, 2, 1, 1, 1;
0, 0, 1, 1, 5, 4, 5, 4, 3, 2, 1, 1, 1;
0, 0, 1, 1, 5, 5, 6, 5, 4, 3, 2, 1, 1, 1;
0, 0, 1, 1, 5, 5, 8, 6, 5, 4, 3, 2, 1, 1, 1;
0, 0, 1, 1, 6, 5, 10, 8, 7, 5, 4, 3, 2, 1, 1, 1;
0, 0, 1, 1, 7, 6, 11, 10, 10, 7, 5, 4, 3, 2, 1, 1, 1;
0, 0, 1, 1, 7, 7, 13, 11, 12, 10, 7, 5, 4, 3, 2, 1, 1, 1;
0, 0, 1, 1, 7, 7, 16, 13, 14, 12, 10, 7, 5, 4, 3, 2, 1, 1, 1;
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A207641
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G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1+x^k)/(1-x^k).
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+10
4
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1, 1, 3, 5, 9, 15, 25, 39, 61, 93, 139, 205, 299, 429, 611, 861, 1201, 1663, 2285, 3115, 4221, 5683, 7605, 10123, 13405, 17661, 23163, 30245, 39323, 50925, 65699, 84445, 108167, 138089, 175719, 222921, 281965, 355627, 447309, 561139, 702133, 876395, 1091301
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OFFSET
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0,3
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COMMENTS
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In Ramanujan's equation let a = x and b = 1. - Michael Somos, Nov 20 2015
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REFERENCES
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 370, 9th equation.
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LINKS
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FORMULA
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Expansion of 1 / ((1 + x) * phi(-x)) in powers of x where phi() is a Ramanujan theta function. - Michael Somos, Nov 20 2015
G.f.: 1 + x*(1+x) * (1 / (1-x)^2 + 2*x^3 / ((1-x)*(1-x^2))^2 + 2*x^7*(1+x) / ((1-x)*(1-x^2)*(1-x^3))^2 + 2*x^12*(1+x)*(1+x^2) / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4))^2 + ...). [Ramanujan] - Michael Somos, Nov 20 2015
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 5*x^3 + 9*x^4 + 15*x^5 + 25*x^6 + 39*x^7 +...
such that, by definition,
A(x) = 1 + x*(1+x)/(1-x) + x^2*(1+x)*(1+x^2)/((1-x)*(1-x^2)) + x^3*(1+x)*(1+x^2)*(1+x^3)/((1-x)*(1-x^2)*(1-x^3)) +...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {-x}, {}, x, x], {x, 0, n}]; (* Michael Somos, Mar 11 2014 *)
a[ n_] := SeriesCoefficient[ 1 / ((1 + x) EllipticTheta[ 4, 0, x]), {x, 0, n}]; (* Michael Somos, Nov 20 2015 *)
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PROG
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(PARI) {a(n)=polcoeff(sum(m=0, n, x^m*prod(k=1, m, (1+x^k)/(1-x^k +x*O(x^n))) ), n)}
for(n=0, 50, print1(a(n), ", "))
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / ((1 + x) * eta(x + A)^2), n))}; /* Michael Somos, Nov 20 2015 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A059776
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Three-quadrant Ferrers graphs that partition n.
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+10
3
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1, 2, 5, 11, 24, 48, 95, 178, 328, 585, 1025, 1754, 2958, 4897, 8002, 12889, 20523, 32289, 50296, 77550, 118521, 179553, 269881, 402532, 596178, 876942, 1281777, 1862015, 2689405, 3862891, 5519403, 7846393, 11100970, 15632733, 21917280
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OFFSET
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0,2
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REFERENCES
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G. E. Andrews, Three-quadrant Ferrers graphs, Indian J. Math., 42 (No. 1, 2000), 1-7.
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LINKS
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FORMULA
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a(n) ~ exp(Pi*sqrt(2*n)) / (2^(11/2) * n^(3/2)). - Vaclav Kotesovec, Jul 12 2018
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MAPLE
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t1 := add( (-1)^(j)*q^(j*(j+1)/2)*(1-q^(j+1))/(1-q), j=0..101); t3 := mul((1-q^n)^3, n=1..101); series(t1/t3, q, 101);
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MATHEMATICA
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nmax = 50; CoefficientList[Series[Sum[(-1)^k*x^(k*(k+1)/2)*(1 - x^(k + 1))/(1 - x), {k, 0, nmax}]/Product[(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 11 2018 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A059778
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Expansion of 1 / product((1+q^(2*n+3))/(1-q^(2*n+2)), n=0..inf).
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+10
1
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1, 0, -1, -1, -1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1
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OFFSET
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0,1
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COMMENTS
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a(n) = (-1)^n*(t(n)-t(n-1)), n>0, where t(n) = A010054(n) is characteristic function of triangular numbers. - Vladeta Jovovic, Sep 22 2002
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REFERENCES
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G. E. Andrews, Three-quadrant Ferrers graphs, Indian J. Math., 42 (No. 1, 2000), 1-7.
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LINKS
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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