|
%I #15 May 07 2018 22:04:23
%S 1,1,2,1,11,10,1,46,241,108,1,128,2739,10411,2214,1,272,16384,265244,
%T 836321,75708,1,522,64964,2883584,45094565,112567243,3895236,1,904,
%U 212325,18852096,822083584,12975204810,22949214033
%N 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.
%C Shanks's array d(n,k) n >= 1, k >= 1, is A235606.
%C 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.
%C For the similarly defined array associated with Shanks' c(n,k) array see A262143.
%H P. Bala, <a href="/A100100/a100100.pdf">Notes on logarithmic differentiation, the binomial transform and series reversion</a>
%H William Y. C. Chen, Neil J. Y. Fan, Jeffrey Y. T. Jia , <a href="http://dx.doi.org/10.1090/S0025-5718-2011-02520-2">The generating function for the Dirichlet series Lm(s)</a> Mathematics of Computation, Vol. 81, No. 278, April 2012.
%H D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0223295-5">Generalized Euler and class numbers</a>. Math. Comp. 21 (1967) 689-694.
%H D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1968-0227093-9">Corrigenda to: "Generalized Euler and class numbers"</a>, Math. Comp. 22 (1968), 699.
%H D. Shanks, <a href="/A000003/a000003.pdf">Generalized Euler and class numbers</a>, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
%e The triangular array begins
%e 1
%e 1 2
%e 1 11 10
%e 1 46 241 108
%e 1 128 2739 10411 2214
%e 1 272 16384 265244 836321 75708
%e 1 522 64964 2883584 45094565 112567243 3895236
%e 1 904 212325 18852096 822083584 12975204810 22949214033 ...
%e The square array begins (row indexing n starts at 1)
%e 1, 2, 10, 108, 2214, 75708, 3895236, 280356120, 26824493574, ...
%e 1, 11, 241, 10411, 836321, 112567243, 22949214033, 6571897714923, 2507281057330113, ...
%e 1, 46, 2739, 265244, 45094565, 12975204810, 5772785327575, 3656385436507960, 3107332328608143945, ...
%e 1, 128, 16384, 2883584, 822083584, 395136991232, 300338473074688, 330739694704787456, 493338658405976375296, ...
%e 1, 272, 64864, 18852096, 8133183744, 5766226378752, 6562478680375296, 11019751545852395520, 25333348417380699340800, ...
%e 1, 522, 212325, 94501768, 57064909374, 54459242196516, 84430282319806062, 197625548666434041000, 642556291067409622713543, ...
%e 1, 904, 586452, 382674008, 311514279098, 379982635729752, 753288329161251844, 2308779464340711480136, 10003494921382094286802995, ...
%Y 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).
%K nonn,tabl,easy
%O 1,3
%A _Peter Bala_, Sep 18 2015
|
