A384364
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..k*n} 3^i * Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 21, 9, 1, 1, 219, 657, 27, 1, 1, 3045, 119241, 22869, 81, 1, 1, 52923, 40365873, 80850987, 836001, 243, 1, 1, 1103781, 21955523049, 747786838869, 60579666801, 31436181, 729, 1, 1, 26857659, 17512689629457, 14298291269335467, 16117269494868801, 48066954848379, 1204022961, 2187, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, ...
1, 3, 21, 219, 3045, ...
1, 9, 657, 119241, 40365873, ...
1, 27, 22869, 80850987, 747786838869, ...
1, 81, 836001, 60579666801, 16117269494868801, ...
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a(n, k) = sum(i=0, k*n, 3^i*sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));
A384356
Expansion of Product_{k>=1} 1/(1 - k*(k+1)/2 * x)^((1/18) * (2/3)^k).
Original entry on oeis.org
1, 1, 19, 1147, 145606, 31784062, 10617130378, 5033441934298, 3213448742033479, 2657684269018334807, 2763967539211567981613, 3530274805575983022456005, 5432490565296371673408076892, 9912854399723224290769677025316, 21163615551469069985356131546443588
Offset: 0
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a384362(n, k) = sum(i=0, k*n, 2^i*sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));
my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, a384362(2, k)*x^k/k)/4))
A384357
Expansion of Product_{k>=1} 1/(1 - k*(k+1)*(k+2)/6 * x)^((1/54) * (2/3)^k).
Original entry on oeis.org
1, 1, 153, 128793, 319155321, 1744213657689, 17803590830142393, 304609764628470426969, 8095576593110601916260369, 315845539893724747798646514673, 17317064152543324914717101316522961, 1288754843591816442932799782872809777393, 126555732798742295186573610437899751882638209
Offset: 0
-
a384362(n, k) = sum(i=0, k*n, 2^i*sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));
my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, a384362(3, k)*x^k/k)/8))
A384358
Expansion of Product_{k>=1} 1/(1 - k*(k+1)*(k+2)*(k+3)/24 * x)^((1/162) * (2/3)^k).
Original entry on oeis.org
1, 1, 1321, 16210201, 820657237561, 117856012064818489, 38648527065793350391329, 25112088578490906968072202609, 29248901038277816617484354852346429, 56683882435365104654655753669402941927069, 172551008002533192343018045442364399983107657925
Offset: 0
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a384362(n, k) = sum(i=0, k*n, 2^i*sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));
my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, a384362(4, k)*x^k/k)/16))
Showing 1-4 of 4 results.