A262144 Square array read by antidiagonals upwards: the n-th row o.g.f. is exp( Sum_{i >= 1} d(n,i+1)*x^i/i ) for n >= 1, where d(n,k) is Shanks's array of generalized Euler and class numbers.

1, 1, 2, 1, 11, 10, 1, 46, 241, 108, 1, 128, 2739, 10411, 2214, 1, 272, 16384, 265244, 836321, 75708, 1, 522, 64964, 2883584, 45094565, 112567243, 3895236, 1, 904, 212325, 18852096, 822083584, 12975204810, 22949214033

Offset: 1

Author

Peter Bala, Sep 18 2015

Keywords

• nonn
• tabl
• easy

Comments

Shanks's array d(n,k) n >= 1, k >= 1, is A235606.

We conjecture that the entries of the present array are all integers. More generally, we conjecture that for r = 1, 2, ... and for each n >= 1, the expansion of exp( Sum_{i >= 1} d(n,i + r)*x^i/i ) has integer coefficients. This is the case r = 1.

For the similarly defined array associated with Shanks' c(n,k) array see A262143.

Examples

			The triangular array begins
1
1   2
1  11     10
1  46    241      108
1 128   2739    10411      2214
1 272  16384   265244    836321       75708
1 522  64964  2883584  45094565   112567243     3895236
1 904 212325 18852096 822083584 12975204810 22949214033 ...
The square array begins (row indexing n starts at 1)
1, 2, 10, 108, 2214, 75708, 3895236, 280356120, 26824493574, ...
1, 11, 241, 10411, 836321, 112567243, 22949214033, 6571897714923, 2507281057330113, ...
1, 46, 2739, 265244, 45094565, 12975204810, 5772785327575, 3656385436507960, 3107332328608143945, ...
1, 128, 16384, 2883584, 822083584, 395136991232, 300338473074688, 330739694704787456, 493338658405976375296, ...
1, 272, 64864, 18852096, 8133183744, 5766226378752, 6562478680375296, 11019751545852395520, 25333348417380699340800, ...
1, 522, 212325, 94501768, 57064909374, 54459242196516, 84430282319806062, 197625548666434041000, 642556291067409622713543, ...
1, 904, 586452, 382674008, 311514279098, 379982635729752, 753288329161251844, 2308779464340711480136, 10003494921382094286802995, ...
		

Links

• P. Bala, Notes on logarithmic differentiation, the binomial transform and series reversion
• 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]

Crossrefs

Cf. A000182 (d(1,n)), A000464 (d(2,n)), A000191 (d(3,n)), A000318 (d(4,n)), A000320 (d(5,n)), A000411 (d(6,n)), A064072 (d(7,n)), A235605, A235606, A262143, A262145 (row 1 of square array).