

A049597


Triangular array T(n,k) in which kth column gives coefficients of sum of Gaussian polynomials [k,m] for m=0..k.


6



1, 0, 2, 0, 0, 3, 0, 0, 1, 4, 0, 0, 0, 2, 5, 0, 0, 0, 2, 3, 6, 0, 0, 0, 0, 4, 4, 7, 0, 0, 0, 0, 3, 6, 5, 8, 0, 0, 0, 0, 1, 6, 8, 6, 9, 0, 0, 0, 0, 0, 6, 9, 10, 7, 10, 0, 0, 0, 0, 0, 2, 11, 12, 12, 8, 11, 0, 0, 0, 0, 0, 2, 9, 16, 15, 14, 9, 12, 0, 0, 0, 0, 0, 0, 7, 16, 21, 18, 16, 10, 13, 0, 0, 0, 0, 0, 0
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OFFSET

0,3


COMMENTS

It appears that T(n1,k1) is the number of partitions of n with k objects in the first hook; i.e., with (largest part size) + (number of parts)  1 = k. If this is correct, we have T(n1,k1) = sum_{j<=min(k,nk2)} (kj) * T(k1,j1) with T(n1,n1) = n. Equivalently, T(n1,k1) = T(n2,k2) + sum(j<=min(k,nk2)} T(k1,j1) and thus T(n1,k1) = 2*T(n2,k2)  T(n3,k3) + T(k1,nk3).  Franklin T. AdamsWatters, May 27 2008


REFERENCES

Cf. G. E. Andrews, Theory of Partitions, 1976, pages 240243


LINKS

Table of n, a(n) for n=0..96.


FORMULA

The g.f. for the nth row as polynomial in q, sum(k=0..n, T(n,k)*q^k) is sum(k>=0, x^(k*(k+1))*q^(2*k)/(1x^(k+1)*q)/prod(j=1..k, 1x^j*q)^2). For example, the 5th row is the coefficient of x^6 of the g.f., 2*q^4 + 3*q^5 + 6*q^6.  T. Amdeberhan, Jul 31 2012


EXAMPLE

Table begins:
1
0 2
0 0 3
0 0 1 4
0 0 0 2 5
0 0 0 2 3 6
0 0 0 0 4 4 7
0 0 0 0 3 6 5 8
For k=4 the 5 polynomials have coefficients 1; 1 1 1 1; 1 1 2 1 1; 1 1 1 1; 1; which sum to 5 3 4 3 1, giving column 4.


MAPLE

a := n>sort(simplify(sum(product((1q^i), i=nr+1..n)/product((1q^j), j=1..r), r=0..n))):T := proc(n, k) if k=n then n+1 elif k>n then 0 else coeff(a(k), q^(nk)) fi end:seq(seq(T(n, k), k=0..n), n=0..21);


MATHEMATICA

a [n_] := Sum[Product[1q^i, {i, nr+1, n}]/Product[1q^j, {j, 1, r}], {r, 0, n}] // Simplify; T [n_, k_] := Which[k == n, n+1, k>n, 0, True, Coefficient[a[k], q^(n  k)]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 21}] // Flatten (* JeanFrançois Alcover, Feb 19 2015, after Maple *)


CROSSREFS

The nonzero entries of the columns are the rows of A083906.
Sequence in context: A132825 A259480 A280164 * A210951 A233440 A280728
Adjacent sequences: A049594 A049595 A049596 * A049598 A049599 A049600


KEYWORD

nonn,tabl


AUTHOR

Alford Arnold


EXTENSIONS

More terms from Emeric Deutsch, Feb 23 2004


STATUS

approved



