1,4

|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_de), Feb 24 2005.

|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.

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.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260.

E.g.f.: arctan(x)^k/k!=sum {n=0..inf} T(n, k) x^n/n!

1; 0,1; -2,0,1; 0,-8,0,1; 24,0,-20,0,1; 0,184,0,-40,0,1; ...

(PARI) T(n, k)=polcoeff(serlaplace(atan(x)^k/k!), n)

Essentially same as A008309, which is the main entry for this sequence.

Row sums (unsigned) give A000246(n); signed row sums give A002019(n), n>=1.

Sequence in context: A011328 A048277 A059419 this_sequence A154469 A022902 A037273

Adjacent sequences: A049215 A049216 A049217 this_sequence A049219 A049220 A049221

sign,tabl,nice

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Michael Somos.