OFFSET
1,4
COMMENTS
|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]
|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]
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
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
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
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260.
LINKS
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Eric Weisstein's World of Mathematics, Mittag-Leffler Polynomial
FORMULA
E.g.f.: arctan(x)^k/k! = Sum_{n>=0} T(n, k) x^n/n!.
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
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
EXAMPLE
Triangle begins:
1;
0, 1;
-2, 0, 1;
0, -8, 0, 1;
24, 0, -20, 0, 1;
0, 184, 0, -40, 0, 1;
...
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 + ....
MAPLE
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
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> `if`(n::odd, 0, (-1)^(n/2)*n!), 10); # Peter Luschny, Jan 28 2016
MATHEMATICA
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 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 12;
M = BellMatrix[If[OddQ[#], 0, (-1)^(#/2)*#!]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
PROG
(PARI) T(n, k)=polcoeff(serlaplace(atan(x)^k/k!), n)
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
Additional comments from Michael Somos
STATUS
approved