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A049218
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Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.
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9
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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, 40320, 0, -52352, 0, 6384, 0, -168, 0, 1, 0, 648576, 0, -229760, 0, 14448, 0, -240, 0, 1, -3628800, 0, 5360256, 0, -804320, 0, 29568, 0, -330, 0, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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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):= 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. 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. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 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*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.
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260.
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LINKS
| P. Bala, Diagonals of triangles with generating function exp(t*F(x)).
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565
Eric Weisstein's World of Mathematics, Mittag-Leffler Polynomial
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FORMULA
| E.g.f.: arctan(x)^k/k!=sum {n=0..inf} 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! . [From 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*ln((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
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EXAMPLE
| 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+....
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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
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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}]] (* From Jean-François Alcover, Dec 06 2011, after Vladimir Kruchinin *)
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PROG
| (PARI) T(n, k)=polcoeff(serlaplace(atan(x)^k/k!), n)
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CROSSREFS
| 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. A137513.
Sequence in context: A048277 A059419 A185415 * A154469 A022902 A037273
Adjacent sequences: A049215 A049216 A049217 * A049219 A049220 A049221
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KEYWORD
| sign,tabl,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional comments from Michael Somos.
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