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 A218449 Gaussian binomial coefficient [2*n-1,n] for q=2, n>=0. 3
 1, 1, 7, 155, 11811, 3309747, 3548836819, 14877590196755, 246614610741341843, 16256896431763117598611, 4274137206973266943778085267, 4488323837657412597958687922896275, 18839183877670041942218307147122500601235 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Compare to: [x^n] Product_{k=0..n-1} 1+2^k*x  =  2^(n*(n-1)/2). LINKS G. C. Greubel, Table of n, a(n) for n = 0..50 Eric Weisstein's World of Mathematics, q-Binomial Coefficient. 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 EXAMPLE 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. MATHEMATICA Table[QBinomial[2n-1, n, 2], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 12 2016 *) PROG (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), ", ")) CROSSREFS Cf. A022166, A006098, A006125. Sequence in context: A139226 A220367 A220311 * A197595 A197979 A203584 Adjacent sequences:  A218446 A218447 A218448 * A218450 A218451 A218452 KEYWORD nonn AUTHOR Paul D. Hanna, Oct 28 2012 STATUS approved

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Last modified December 2 05:01 EST 2021. Contains 349437 sequences. (Running on oeis4.)