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A262143
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.
1
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
OFFSET
1,5
COMMENTS
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.
LINKS
William Y. C. Chen, Neil J. Y. Fan, Jeffrey Y. T. Jia, The generating function for the Dirichlet series Lm(s) Mathematics of Computation, Vol. 81, No. 278, April 2012.
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
EXAMPLE
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 ...
...
CROSSREFS
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.
Sequence in context: A200342 A181330 A350253 * A284554 A078033 A221712
KEYWORD
nonn,tabl
AUTHOR
Peter Bala, Sep 13 2015
STATUS
approved