%I #59 Nov 30 2021 11:37:58

%S 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,

%T 0,1,0,-8448,0,2464,0,-112,0,1,40320,0,-52352,0,6384,0,-168,0,1,0,

%U 648576,0,-229760,0,14448,0,-240,0,1,-3628800,0,5360256,0,-804320,0,29568,0,-330,0,1

%N Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.

%C |T(n,k)| gives the sum of the M_2 multinomial numbers (A036039) for those partitions of n with exactly k odd parts. E.g.: |T(6,2)| = 144 + 40 = 184 from the partitions of 6 with exactly 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):= arctanh(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. - _Wolfdieter Lang_, Feb 24 2005 [The first two sentences have been corrected thanks to the comment by _José H. Nieto S._ given below. - _Wolfdieter Lang_, Jan 16 2012]

%C |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. - _Wolfdieter Lang_, Feb 24 2005 [See the following _José H. Nieto S._ correction. - _Wolfdieter Lang_, Jan 16 2012]

%C The first sentence of the above comment is inexact, it should be "|T(n,k)| gives the number of degree n permutations which decompose into exactly k odd cycles". The number of degree n permutations with k odd cycles (and, possibly, other cycles of even length) is given by A060524. - _José H. Nieto S._, Jan 15 2012

%C The unsigned triangle with e.g.f. exp(x*arctanh(z)) is the associated Jabotinsky type triangle for the Sheffer type triangle A060524. See the comments there. - _Wolfdieter Lang_, Feb 24 2005

%C Also the Bell transform of the sequence (-1)^(n/2)*A005359(n) without column 0. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 28 2016

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

%H Peter Bala, <a href="/A112007/a112007_Bala.txt">Diagonals of triangles with generating function exp(t*F(x)).</a>

%H Steven Finch, <a href="https://arxiv.org/abs/2111.14487">Rounds, Color, Parity, Squares</a>, arXiv:2111.14487 [math.CO], 2021.

%H Kruchinin Vladimir Victorovich, <a href="https://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Mittag-LefflerPolynomial.html">Mittag-Leffler Polynomial</a>

%F E.g.f.: arctan(x)^k/k! = Sum_{n>=0} T(n, k) x^n/n!.

%F T(n,k) = ((-1)^((3*n+k)/2)*n!/2^k)*Sum_{i=k..n} 2^i*binomial(n-1,i-1)*Stirling1(i,k)/i!. - _Vladimir Kruchinin_, Feb 11 2011

%F E.g.f.: exp(t*arctan(x)) = 1 + t*x + t^2*x^2/2! + t*(t^2-2)*x^3/3! + .... The unsigned row polynomials are the Mittag-Leffler polynomials M(n,t/2). See A137513. The compositional inverse (with respect to x) (x-t/2*log((1+x)/(1-x)))^(-1) = x/(1-t) + 2*t/(1-t)^4*x^3/3!+ (24*t+16*t^2)/(1-t)^7*x^5/5! + .... The rational functions in t generate the (unsigned) diagonals of the table. See the Bala link. - _Peter Bala_, Dec 04 2011

%e Triangle begins:

%e 1;

%e 0, 1;

%e -2, 0, 1;

%e 0, -8, 0, 1;

%e 24, 0, -20, 0, 1;

%e 0, 184, 0, -40, 0, 1;

%e ...

%e O.g.f. for fifth subdiagonal: (24*t+16*t^2)/(1-t)^7 = 24*t + 184*t^2 + 784*t^3 + 2404*t^4 + ....

%p A049218 := proc(n,k)(-1)^((3*n+k)/2) *add(2^(j-k)*n!/j! *stirling1(j,k) *binomial(n-1,j-1),j=k..n) ; end proc: # _R. J. Mathar_, Feb 14 2011

%p # The function BellMatrix is defined in A264428.

%p # Adds (1,0,0,0, ..) as column 0.

%p BellMatrix(n -> `if`(n::odd, 0, (-1)^(n/2)*n!), 10); # _Peter Luschny_, Jan 28 2016

%t t[n_, k_] := (-1)^((3n+k)/2)*Sum[ 2^(j-k)*n!/j!*StirlingS1[j, k]*Binomial[n-1, j-1], {j, k, n}]; Flatten[ Table[ t[n, k], {n, 1, 11}, {k, 1, n}]] (* _Jean-François Alcover_, Dec 06 2011, after _Vladimir Kruchinin_ *)

%t BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];

%t rows = 12;

%t M = BellMatrix[If[OddQ[#], 0, (-1)^(#/2)*#!]&, rows];

%t Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)

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

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

%Y Row sums (unsigned) give A000246(n); signed row sums give A002019(n), n>=1. A137513.

%K sign,tabl,nice

%O 1,4

%A _N. J. A. Sloane_

%E Additional comments from _Michael Somos_