A008309 -id:A008309 - cp's OEIS frontend

A049218 Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.

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

Offset: 1

N. J. A. Sloane

|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

			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 + ....

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

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
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

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.

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

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 *)

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

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

Additional comments from Michael Somos

A002429 Numerators of double sums of reciprocals.

Original entry on oeis.org

1, 1, 14, 818, 141, 13063, 16774564, 1057052, 4651811, 778001383, 1947352646, 1073136102266, 72379420806883, 112229882767, 120372921248744, 13224581478608216, 2077531074698521033, 517938126297258811, 13785854249175914469406, 343586489824688536178, 1958290344469311726833

Offset: 0

N. J. A. Sloane

nonn

Also, numerators of coefficients of expansion of arctan(x)^3. - Ruperto Corso, Dec 09 2011

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 117.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..770
Mohammad K. Azarian, A Double Sum, Problem 440, College Mathematics Journal, Vol. 21, No. 5, Nov. 1990, p. 424. Solution published in Vol. 22. No. 5, Nov. 1991, pp. 448-449.
H. A. Rothe, in C. F. Hindenburg, editor, Sammlung Combinatorisch-Analytischer Abhandlungen, Vol. 2, Chap. XI. Fleischer, Leipzig, 1800, p. 316.

Cf. A008309, A049218.

List([0..25], n-> NumeratorRat( 3*Sum([3..2*n+3], k-> (-1)^(k-1)*2^(k-2)* Binomial(2*(n+1),k-1)*Stirling1(k,3)/Factorial(k)) )) # G. C. Greubel, Jul 03 2019

[Numerator(3*(&+[2^(k-2)*Binomial(2*(n+1), k-1)* StirlingFirst(k,3)/Factorial(k): k in [3..2*n+3]]) ): n in [0..25]]; // G. C. Greubel, Jul 03 2019

p2x:=proc(n) option remember: if(n=1) then RETURN(1) else RETURN(((n-1)*p2x(n-1)+1/(2*n-1))/n) fi: end proc;
p3x:=proc(n) option remember: if(n=1) then RETURN(1) else RETURN(((2*n-1)*p3x(n-1)+3*p2x(n))/(2*n+1)) fi: end proc;
A002429 := proc(n)
    numer(p3x(n)) ;
end proc:
seq(A002429(n),n=1..25) ; # Ruperto Corso, Dec 09 2011

a[n_]:= (-1)^n*SeriesCoefficient[ArcTan[x]^3, {x, 0, 2*n+3}]//Numerator; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 04 2013 *)
a[n_]:= Numerator[3*Sum[2^(k-2)*Binomial[2*(n+1),k-1]*StirlingS1[k,3]/k!, {k,3,2*n+3}]]; Table[a[n], {n,0,25}] (* G. C. Greubel, Jul 03 2019 *)

stirling1(n, k)=if(n<1, 0, n!*polcoeff(binomial(x, n), k))
for(n=0,25,print1(numerator(3/4*sum(i=3,2*n+3,2^i*binomial(2*(n+1),i-1)*stirling1(i,3)/ i!))",")) \\ Ruperto Corso, Dec 09 2011

[numerator( 3*sum((-1)^(k-1)*2^(k-2)*binomial(2*(n+1), k-1)* stirling_number1(k,3)/factorial(k) for k in (3..2*n+3)) ) for n in (0..25)] # G. C. Greubel, Jul 03 2019

a(n) = numerator of 3*Sum_{i=3..2*n+3} 2^(i-2)*binomial(2*(n+1),i-1) *Stirling1(i,3)/ i!. - Ruperto Corso, Dec 09 2011

More terms from Ruperto Corso, Dec 09 2011

cp's OEIS Frontend

A049218 Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.

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