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A262143
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Square array read by antidiagonals upwards: the n-th row o.g.f. is exp( Sum_{i >= 1} c(n,i)*x^i/i ) for n >= 1, where c(n,k) is Shanks' array of generalized Euler and class numbers.
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1
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1, 1, 1, 1, 3, 3, 1, 8, 33, 23, 1, 16, 208, 1011, 371, 1, 30, 768, 14336, 65985, 10515, 1, 46, 2211, 94208, 2091520, 7536099, 461869, 1, 64, 5043, 412860, 24313856, 535261184, 1329205857, 28969177, 1, 96, 9984, 1361948, 164276421, 11025776640, 211966861312, 334169853267, 2454072147
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OFFSET
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1,5
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COMMENTS
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Shanks' array c(n,k) n >= 1, k >= 0, is A235605.
We conjecture that the entries of the present array are all integers. More generally, we conjecture that for r = 0,1,2,... and for each n >= 1, the expansion of exp( Sum_{i >= 1} c(n,i + r)*x^i/i ) has integer coefficients. The case n = 1 was conjectured by Hanna in A255895.
For the similarly defined array associated with Shanks' d(n,k) array see A262144.
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LINKS
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EXAMPLE
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The square array begins (row indexing n starts at 1)
1 1 3 23 371 10515 461869 ..
1 3 33 1011 65985 7536099 1329205857 ..
1 8 208 14336 2091520 535261184 211966861312 ..
1 16 768 94208 24313856 11025776640 7748875976704 ..
1 30 2211 412860 164276421 115699670490 126686112278631 ..
1 46 5043 1361948 778121381 787337024970 1239870854518999 ..
1 64 9984 3716096 2891509760 3978693525504 8522989918683136 ..
...
Array as a triangle
1
1 1
1 3 3
1 8 33 23
1 16 208 1011 371
1 30 768 14336 65985 10515
1 46 2211 94208 2091520 7536099 461869
1 64 5043 412860 24313856 535261184 1329205857 28969177
1 96 9984 1361948 164276421 11025776640 211966861312 ...
...
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CROSSREFS
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Cf. A000233 (column 1), A000364 (c(1,n)), A000281 (c(2,n)), A000436 (c(3,n)), A000490 (c(4,n)), A000187 (c(5,n)), A000192 (c(6,n)), A064068 (c(7,n)), A235605, A235606, A255881, A255895, A262144, A262145.
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KEYWORD
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AUTHOR
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STATUS
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approved
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