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A219237
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Coefficient of Gauss polynomials [n+4,4]_q (q-binomials).
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4
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1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 1, 1, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1, 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 6, 8, 9, 11, 11, 12, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 6, 9, 10, 13, 14, 16, 16, 18, 16, 16, 14, 13, 10, 9, 6, 5, 3, 2, 1, 1
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OFFSET
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0,9
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COMMENTS
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The length of row n of this table is 4*n + 1 = A016813(n).
The sum of row n is binomial(n+4,4) = A000332(n+4), n>= 0.
The Gauss polynomial [n+4,4]_q := [n+4]_q/([n]_q*[4]_q, with [n]_q = product(1-q^j,j=1..n) = (q;q)_n (in q-shifted factorials notation), n>=0. [n+4,4]_q = product(1-q^j,j=(n+1)..(n+4))/product(1-q^j,j=1..4). This is a polynomial in q (of degree 4*n) because it is the o.g.f. of the numbers p(n,4,k), the number of partitions of k into at most 4 parts, each <= n (see Andrews, p. 33 and 35). p(n,4,k) is also the number of partitions of k into at most n parts, each <= 4, due to the symmetry property [n+4,4]_q = [n+4,n]_q (Andrews, (3,3,2), p.35). With the latter interpretation p(n,4,k) is the number of solutions of the two Diophantine equations sum(j*m(j),j=1..4) = k and sum(m(j),j=0..m) = n, i.e. sum(m(j),j=1..m) = n - m(0), with 0 <= m(j) <= n. Therefore p(n,4,k) = [q^k] [x^n] G(4;x,q) with o.g.f. G(4;x,q) = 1/product(1-x*q^j,j=0..4). Here we will call p(n,4,k) = a(n,k), n >= 0, 0 <= k <= 4*n.
See the comments in A008967 concerning a counting problem of Cayley (there m = 4, Theta = n and q = k), described also in the Hawkins reference (N(p->n,4,w->k) = a(n,k)) given there.
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REFERENCES
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G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 240, 242-3.
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LINKS
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FORMULA
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a(n,k) = [q^k] [x^n](1/product(1-x*q^j,j=0..4)), n >= 0, 0 <= k <= 4*n.
a(n,k) = [q^k]([n+4,4]_q), n >= 0, 0 <= k <= 4*n.
See the comments above.
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EXAMPLE
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The table a(n,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0: 1
1: 1 1 1 1 1
2: 1 1 2 2 3 2 2 1 1
3: 1 1 2 3 4 4 5 4 4 3 2 1 1
4: 1 1 2 3 5 5 7 7 8 7 7 5 5 3 2 1 1
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
...
Row n = 5: [1, 1, 2, 3, 5, 6, 8, 9, 11, 11, 12, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1],
Row n = 6: [1, 1, 2, 3, 5, 6, 9, 10, 13, 14, 16, 16, 18, 16, 16, 14, 13, 10, 9, 6, 5, 3, 2, 1, 1].
Partition interpretation: a(3,5) = 4 because there are 4 partitions of 5 into at most 4 parts, each <= 3, namely 23, 113, 122 and 1112. here are also 4 partitions of 5 into at most 3 parts, each <= 4, namely 14, 23, 113 and 122. Note the conjugacy of the partitions 1112 and 14.
The 4 solutions of the two Diophantine equations given in a comment, with k=5 and n=3, are for (m(0), m(1), m(2), m(3), m(4)): (1,1,0,0,1), (1,0,1,1,0), (0,2,0,1,0) and (0,1,2,0,0).
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MATHEMATICA
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a[0, 0] = 1; a[n_, k_] := SeriesCoefficient[ QBinomial[n+4, 4, q], {q, 0, k}]; Table[a[n, k], {n, 0, 6}, {k, 0, 4*n}] // Flatten (* Jean-François Alcover, Dec 04 2013 *)
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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