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A125806 Triangle of the sum of squared coefficients of q in the q-binomial coefficients, read by rows. 4
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 16, 16, 5, 1, 1, 6, 29, 48, 29, 6, 1, 1, 7, 47, 119, 119, 47, 7, 1, 1, 8, 72, 256, 390, 256, 72, 8, 1, 1, 9, 104, 500, 1070, 1070, 500, 104, 9, 1, 1, 10, 145, 900, 2592, 3656, 2592, 900, 145, 10, 1, 1, 11, 195, 1525, 5674, 10762 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Central terms equal A063075 (number of partitions of 2n^2 whose Ferrers-plot fits within a 2n X 2n box and cover an n X n box).

The matrix inverse starts

1;

-1,1;

1,-2,1;

-1,3,-3,1;

-1,0,4,-4,1;

9,-21,12,4,-5,1;

-1,34,-73,44,1,-6,1;

-219,479,-219,-139,109,-5,-7,1;

101,-1536,3072,-1776,-54,216,-16,-8,1; - R. J. Mathar, Mar 22 2013

LINKS

Paul D. Hanna, Rows n = 0..45, listed in flattened form as n, a(n), for n = 0..1080.

EXAMPLE

The triangle of q-binomial coefficients:

C_q(n,k) = [Product_{i=n-k+1..n}(1-q^i)]/[Product_{j=1..k}(1-q^j)]

begins:

1;

1, 1;

1, 1+q, 1;

1, 1+q+q^2, 1+q+q^2, 1;

1, 1+q+q^2+q^3, 1+q+2*q^2+q^3+q^4, 1+q+q^2+q^3, 1; ...

recurrence: C_q(n+1,k) = C_q(n,k-1) + q^k * C_q(n,k).

Element T(n,k) of this triangle equals the sum of the squares

of the coefficients of q in q-binomial coefficient C_q(n,k).

This triangle begins:

1;

1, 1;

1, 2, 1;

1, 3, 3, 1;

1, 4, 8, 4, 1;

1, 5, 16, 16, 5, 1;

1, 6, 29, 48, 29, 6, 1;

1, 7, 47, 119, 119, 47, 7, 1;

1, 8, 72, 256, 390, 256, 72, 8, 1;

1, 9, 104, 500, 1070, 1070, 500, 104, 9, 1;

1, 10, 145, 900, 2592, 3656, 2592, 900, 145, 10, 1;

1, 11, 195, 1525, 5674, 10762, 10762, 5674, 1525, 195, 11, 1;

1, 12, 256, 2456, 11483, 28160, 37834, 28160, 11483, 2456, 256, 12, 1;

The central terms equal A063075:

1, 2, 8, 48, 390, 3656, 37834, 417540, 4836452, 58130756, ...

MAPLE

C := proc(q, n, k)

    local i, j ;

    mul(1-q^i, i=n-k+1..n)/mul(1-q^j, j=1..k) ;

    expand(factor(%)) ;

end proc:

A125806 := proc(n, k)

    local qbin , q;

    qbin := [coeffs(C(q, n, k), q)] ;

    add( e^2, e=qbin) ;

end proc: # R. J. Mathar, Mar 22 2013

PROG

(PARI) T(n, k)=local(C_q=if(n==0 || k==0, 1, prod(j=n-k+1, n, 1-q^j)/prod(j=1, k, 1-q^j))); sum(i=0, (n-k)*k, polcoeff(C_q, i)^2)

for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A063075 (central terms); A125807, A125808, A125809 (row sums).

Sequence in context: A300260 A026692 A114202 * A202756 A156354 A295205

Adjacent sequences:  A125803 A125804 A125805 * A125807 A125808 A125809

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 11 2006

STATUS

approved

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Last modified April 25 13:31 EDT 2019. Contains 322461 sequences. (Running on oeis4.)