%I A049218
%S A049218 1,0,1,2,0,1,0,8,0,1,24,0,20,0,1,0,184,0,40,0,1,720,0,784,0,70,0,1,0,8448,
               0,2464,0,112,0,1,
%T A049218 40320,0,52352,0,6384,0,168,0,1,0,648576,0,229760,0,14448,0,240,0,1,3628800,
%U A049218 0,5360256,0,804320,0,29568,0,330,0,1
%V A049218 1,0,1,-2,0,1,0,-8,0,1,24,0,-20,0,1,0,184,0,-40,0,1,-720,0,784,0,-70,0,
               1,0,-8448,0,2464,0,-112,0,1,
%W A049218 40320,0,-52352,0,6384,0,-168,0,1,0,648576,0,-229760,0,14448,0,-240,0,
               1,-3628800,
%X A049218 0,5360256,0,-804320,0,29568,0,-330,0,1
%N A049218 Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.
%C A049218 |T(n,k)| gives the sum of the M_2 multinomial numbers (A036039) for the 
               partitions of n with k odd parts. E.g.: |T(6,2)| = 144 + 40 = 184 
               from the partitions of 6 with two odd parts, namely (1,5) and (3,
               3), with M_2 numbers 144 and 40. Proof via the general Jabotinsky 
               triangle formula for |T(n,k)| using partitions of n into k parts 
               and their M_3 numbers (A036040). Then with the special e.g.f. of 
               the (unsigned) k=1 column, f(x):= artanh(x), only odd parts survive 
               and the M_3 numbers are changed into the M_2 numbers. For the Knuth 
               reference on Jabotinsky triangles see A039692. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_d\
               e), Feb 24 2005.
%C A049218 |T(n,k)| gives the number of permutations of {1,2,...,n} (degree n permutations) 
               with the number of odd cycles equal to k. E.g.: |T(5,3)|= 20 from 
               the 20 degree 5 permutations with cycle structure (.)(.)(...). Proof: 
               Use the cycle index polynomial for the symmetric group S_n (see the 
               M_2 array A036039 or A102189) together with the partition interpretation 
               of |T(n,k)| given above. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Feb 24 2005.
%C A049218 The unsigned triangle with e.g.f. exp(x*artanh(z)) is the associated 
               Jabotinsky type triangle for the Sheffer type triangle A060524. See 
               the comments there. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Feb 24 2005.
%D A049218 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260.
%F A049218 E.g.f.: arctan(x)^k/k!=sum {n=0..inf} T(n, k) x^n/n!
%e A049218 1; 0,1; -2,0,1; 0,-8,0,1; 24,0,-20,0,1; 0,184,0,-40,0,1; ...
%o A049218 (PARI) T(n,k)=polcoeff(serlaplace(atan(x)^k/k!), n)
%Y A049218 Essentially same as A008309, which is the main entry for this sequence.
%Y A049218 Row sums (unsigned) give A000246(n); signed row sums give A002019(n), 
               n>=1.
%Y A049218 Sequence in context: A011328 A048277 A059419 this_sequence A154469 A022902 
               A037273
%Y A049218 Adjacent sequences: A049215 A049216 A049217 this_sequence A049219 A049220 
               A049221
%K A049218 sign,tabl,nice
%O A049218 1,4
%A A049218 N. J. A. Sloane (njas(AT)research.att.com).
%E A049218 Additional comments from Michael Somos.