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A219237 Coefficient of Gauss polynomials [n+4,4]_q (q-binomials). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

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.

REFERENCES

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 240, 242-3.

LINKS

Table of n, a(n) for n=0..90.

Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

FORMULA

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.

EXAMPLE

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).

MATHEMATICA

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-Fran├žois Alcover, Dec 04 2013 *)

CROSSREFS

Cf. A000012 (as triangle for m=1), A008967 (m=2), A047971 (m=3).

Sequence in context: A194827 A335359 A332205 * A138774 A156988 A304199

Adjacent sequences:  A219234 A219235 A219236 * A219238 A219239 A219240

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang, Dec 04 2012

STATUS

approved

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Last modified May 21 07:22 EDT 2022. Contains 353889 sequences. (Running on oeis4.)