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A060690
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a(n) = binomial(2^n + n - 1, n).
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30
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1, 2, 10, 120, 3876, 376992, 119877472, 131254487936, 509850594887712, 7145544812472168960, 364974894538906616240640, 68409601066028072105113098240, 47312269462735023248040155132636160, 121317088003402776955124829814219234385920
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OFFSET
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0,2
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COMMENTS
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Also the number of n X n (0,1) matrices modulo rows permutation (by symmetry this is the same as the number of (0,1) matrices modulo columns permutation), i.e., the number of equivalence classes where two matrices A and B are equivalent if one of them is the result of permuting the rows of the other. The total number of (0,1) matrices is in sequence A002416.
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LINKS
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FORMULA
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a(n) = [x^n] 1/(1-x)^(2^n).
a(n) = (1/n!)*Sum_{k=0..n} ( (-1)^(n-k)*Stirling1(n, k)*2^(k*n) ). - Vladeta Jovovic, May 28 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2^n+n,k) - Vladeta Jovovic, Jan 21 2008
a(n) = Sum_{k=0..n} Stirling1(n,k)*(2^n+n-1)^k/n!. - Vladeta Jovovic, Jan 21 2008
G.f.: A(x) = Sum_{n>=0} [ -log(1 - 2^n*x)]^n / n!. More generally, Sum_{n>=0} [ -log(1 - q^n*x)]^n/n! = Sum_{n>=0} C(q^n+n-1,n)*x^n ; also Sum_{n>=0} log(1 + q^n*x)^n/n! = Sum_{n>=0} C(q^n,n)*x^n. - Paul D. Hanna, Dec 29 2007
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MAPLE
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with(combinat): for n from 0 to 20 do printf(`%d, `, binomial(2^n+n-1, n)) od:
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MATHEMATICA
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Table[Binomial[2^n+n-1, n], {n, 0, 20}] (* Harvey P. Dale, Apr 19 2012 *)
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PROG
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(PARI) a(n)=binomial(2^n+n-1, n)
(PARI) {a(n)=polcoeff(sum(k=0, n, (-log(1-2^k*x +x*O(x^n)))^k/k!), n)} \\ Paul D. Hanna, Dec 29 2007
(PARI) a(n) = sum(k=0, n, stirling(n, k, 1)*(2^n+n-1)^k/n!); \\ Paul D. Hanna, Nov 20 2014
(Sage) [binomial(2^n +n-1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021
(Magma) [Binomial(2^n +n-1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
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CROSSREFS
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Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), this sequence (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
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KEYWORD
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nonn
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AUTHOR
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Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
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EXTENSIONS
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STATUS
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approved
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