A218449 Gaussian binomial coefficient [2*n-1,n] for q=2, n>=0.

1, 1, 7, 155, 11811, 3309747, 3548836819, 14877590196755, 246614610741341843, 16256896431763117598611, 4274137206973266943778085267, 4488323837657412597958687922896275, 18839183877670041942218307147122500601235

Offset: 0

Author

Paul D. Hanna, Oct 28 2012

Keywords

• nonn

Comments

Compare to: [x^n] Product_{k=0..n-1} 1+2^k*x = 2^(n*(n-1)/2).

Examples

			The coefficients in Product_{k=0..n-1} 1/(1 - 2^k*x) begin:
n=0: [(1)];
n=1: [1,(1), 1, 1, 1, 1, 1, 1, 1, 1, ...];
n=2: [1, 3,(7), 15, 31, 63, 127, 255, 511, 1023, ...];
n=3: [1, 7, 35,(155), 651, 2667, 10795, 43435, 174251, ...];
n=4: [1, 15, 155, 1395,(11811), 97155, 788035, 6347715, ...];
n=5: [1, 31, 651, 11811, 200787,(3309747), 53743987, ...];
n=6: [1, 63, 2667, 97155, 3309747, 109221651,(3548836819), ...];
n=7: [1, 127, 10795, 788035, 53743987, 3548836819, 230674393235,(14877590196755), ...]; ...
the coefficients in parenthesis give the initial terms of this sequence;
an adjacent diagonal forms the Gaussian binomial coefficients [2*n,n] for q=2:
[1, 3, 35, 1395, 200787, 109221651, 230674393235, ...] = A006098.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..50
• Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

Crossrefs

Cf. A022166, A006098, A006125.

Programs

Mathematica

Table[QBinomial[2n-1, n, 2], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 12 2016 *)

PARI

{a(n)=polcoeff(prod(k=0,n-1,1/(1-2^k*x +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = [x^n] Product_{k=0..n-1} 1/(1 - 2^k*x).

a(n) ~ c * 2^(n*(n-1)), where c = A065446. - Vaclav Kotesovec, Sep 22 2016