A384362
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..k*n} 2^i * Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 10, 4, 1, 1, 74, 148, 8, 1, 1, 730, 13540, 2440, 16, 1, 1, 9002, 2308756, 3087368, 42256, 32, 1, 1, 133210, 632363044, 10208479240, 778026256, 752800, 64, 1, 1, 2299754, 253970683348, 69754997963528, 52520969994256, 207633589664, 13660480, 128, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, ...
1, 2, 10, 74, 730, ...
1, 4, 148, 13540, 2308756, ...
1, 8, 2440, 3087368, 10208479240, ...
1, 16, 42256, 778026256, 52520969994256, ...
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a(n, k) = sum(i=0, k*n, 2^i*sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));
A384359
Expansion of Product_{k>=1} 1/(1 - k*(k+1)/2 * x)^((1/48) * (3/4)^k).
Original entry on oeis.org
1, 1, 37, 4453, 1126375, 489185863, 324848377243, 306044183298331, 388203452145317314, 637855747987693348770, 1317841032827800659419754, 3343784211346797764798294634, 10221662989279986155378379955158, 37051850653048390530321630384383382, 157140052593846256021318451838028238910
Offset: 0
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a384364(n, k) = sum(i=0, k*n, 3^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, a384364(2, k)*x^k/k)/9))
A384360
Expansion of Product_{k>=1} 1/(1 - k*(k+1)*(k+2)/6 * x)^((1/192) * (3/4)^k).
Original entry on oeis.org
1, 1, 424, 998584, 6925040260, 105920615923684, 3026129933925315784, 144928319460945421096936, 10782220800085014574469693026, 1177609713750570874317795178806210, 180749886489278186545417627942230436008, 37658177020555445685152123914054243838809128
Offset: 0
-
a384364(n, k) = sum(i=0, k*n, 3^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, a384364(3, k)*x^k/k)/27))
Showing 1-3 of 3 results.