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A001752
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Expansion of 1/((1+x)*(1-x)^5).
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21
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1, 4, 11, 24, 46, 80, 130, 200, 295, 420, 581, 784, 1036, 1344, 1716, 2160, 2685, 3300, 4015, 4840, 5786, 6864, 8086, 9464, 11011, 12740, 14665, 16800, 19160, 21760, 24616, 27744, 31161, 34884, 38931, 43320, 48070, 53200, 58730, 64680
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Define a unit column of a binary matrix to be a column with only one 1. a(n)=number of 3 x n binary matrices with 1 unit column up to row and column permutations(if offset is 1) - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 09 2000
Generally, number of 3 x n binary matrices with k=0,1,2,... unit columns, up to row and column permutations, is coefficient of x^k in 1/6*(Z(S_n; 5 + 3*x,5 + 3*x^2, ...) + 3*Z(S_n; 3 + x,5 + 3*x^2,3 + x^3,5 + 3*x^4, ...) + 2*Z(S_n; 2,2,5 + 3*x^3,2,2,5 + 3*x^6, ...)), where Z(S_n; x_1,x_2,...,x_n) is cycle index of symmetric group S_n of degree n.
First differences of a(n) give number of minimal 3-covers of an unlabeled n-set that cover 4 points of that set uniquely (if offset is 4).
Transform of tetrahedral numbers binomial(n+3,3) under Riordan array (1/(1-x^2),x). - Paul Barry, Apr 16 2005
a(n) = A108561(n+5,n) for n>0. - Reinhard Zumkeller, Jun 10 2005
Equals triangle A152205 as an infinite lower triangular matrix * [1, 2, 3,...]. [From Gary W. Adamson, Feb 14 2010]
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Dragomir Z. Djokovic, Poincare series of some pure and mixed trace algebras of two generic matrices. See Table 4.
Index to sequences with linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).
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FORMULA
| a(n) = [((n+3)^2-1)*((n+3)^2-3)/48].
G.f.: 1/((1+x)*(1-x)^5).
A002620(n)=a(n-2)-2*a(n-3)+a(n-4). A000332(n)=a(n-4)+a(n-5). A000292(n)=a(n)-a(n-2).
a(n) = Sum{k=0..n, (-1)^(n-k)*C(k+4, 4) } - Paul Barry, Jul 01 2003
{a[n] == a[n - 2] + (n*(n + 1)*(n - 1))/6, a[1] == 0, a[2] == 1}; (3*(-1)^n - 3*(-1)^(2*n) + 12*n - 20*(-1)^(2*n)*n + 22*n^2 - 18*(-1)^(2*n)*n^2 + 12*n^3 - 4*(-1)^(2*n)*n^3 + 2*n^4)/96 - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004
a(n)=sum{k=0..floor(n/2), binomial(n-2k+3, 3)}; a(n)=sum{k=0..n, binomial(k+3, 3)*(1-(-1)^(n+k-1))/2}; - Paul Barry, Apr 16 2005
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EXAMPLE
| There are 4 binary 3 X 2 matrices with 1 unit column up to row and column permutations:
[0 0] [0 0] [0 1] [0 1]
[0 0] [0 1] [0 1] [0 1]
[0 1] [1 1] [1 0] [1 1].
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PROG
| (PARI) a(n)=if(n<0, 0, ((n+3)^2-1)*((n+3)^2-3)/48-if(n%2, 1/16))
(MAGMA) [Floor(((n+3)^2-1)*((n+3)^2-3)/48): n in [0..40]]; // Vincenzo Librandi, Aug 15 2011
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CROSSREFS
| Cf. A057524, A056885, A152205.
Sequence in context: A006527 A167875 A057304 * A160860 A192748 A143075
Adjacent sequences: A001749 A001750 A001751 * A001753 A001754 A001755
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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