A188203
G.f.: exp( Sum_{n>=1} A188202(n)*x^n/n ) where A188202(n) = [x^n] (1 + 2^n*x + x^2)^n.
Original entry on oeis.org
1, 2, 11, 206, 17586, 6878604, 11551087875, 80650796495414, 2307974943300931286, 268728588584911887188180, 126776477973814964972206209838, 241684409250478693507166916367088620
Offset: 0
G.f.: A(x) = 1 + 2*x + 11*x^2 + 206*x^3 + 17586*x^4 + 6878604*x^5 +...
The l.g.f. of A188202 begins:
log(A(x)) = 2*x + 18*x^2/2 + 560*x^3/3 + 68614*x^4/4 + 34210752*x^5/5 +...
The coefficients of x^n in (1 + 2^n*x + x^2)^n begin:
n=1: [1, (2), 1];
n=2: [1, 8, (18), 8, 1];
n=3: [1, 24, 195, (560), 195, 24, 1];
n=4: [1, 64, 1540, 16576, (68614), 16576, 1540, 64, 1];
n=5: [1, 160, 10245, 328320, 5273610, (34210752), 5273610, ...]; ...
where the central coefficients form the logarithmic derivative, A188202.
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{a(n)=polcoeff(exp(sum(k=1,n,polcoeff((1+2^k*x+x^2+x*O(x^k))^k,k)*x^k/k)+x*O(x^n)),n)}
A188140
Central coefficient in (1 + x + 2^n*x^2)^n.
Original entry on oeis.org
1, 1, 9, 49, 1729, 31361, 5613441, 297047297, 310070491137, 43518954344449, 287194749188794369, 100075094464681701377, 4382651775014102039543809, 3633063629528650538489708545, 1089270933915429450356021881831425
Offset: 0
The coefficients of x^n in (1 + x + 2^n*x^2)^n begin:
n=0: [(1)];
n=1: [1, (1), 2];
n=2: [1, 2, (9), 8, 16];
n=3: [1, 3, 27, (49), 216, 192, 512];
n=4: [1, 4, 70, 196, (1729), 3136, 17920, 16384, 65536];
n=5: [1, 5, 170, 650, 11205, (31361), 358560, 665600, ...];
n=6: [1, 6, 399, 1940, 65295, 249606, (5613441), 15974784, ...];
n=7: [1, 7, 917, 5411, 357539, 1738261, 76854407, (297047297), ...]; ...
where the above central coefficients in parenthesis form this sequence.
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a:= n-> coeff((1+x+2^n*x^2)^n, x, n):
seq(a(n), n=0..20);
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Table[Sum[Binomial[n,k] * Binomial[n-k, n-2*k] * 2^(n*k), {k,0,Floor[n/2]}], {n,0,20}] (* Vaclav Kotesovec, Feb 11 2015 *)
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{a(n)=polcoeff((1+x+2^n*x^2)^n,n)}
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{a(n)=sum(k=0,n\2,binomial(n, k)*binomial(n-k, n-2*k)*2^(n*k))}
Showing 1-2 of 2 results.
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