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A111596 Associated Sheffer triangle to Sheffer triangle A111595. 33
1, 0, 1, 0, -2, 1, 0, 6, -6, 1, 0, -24, 36, -12, 1, 0, 120, -240, 120, -20, 1, 0, -720, 1800, -1200, 300, -30, 1, 0, 5040, -15120, 12600, -4200, 630, -42, 1, 0, -40320, 141120, -141120, 58800, -11760, 1176, -56, 1, 0, 362880, -1451520, 1693440, -846720, 211680, -28224, 2016, -72, 1, 0, -3628800 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Coefficients of Laguerre polynomials (-1)^n * n! * L(n,-1,x), which equals (-1)^n * Lag(n,x,-1) below. Lag(n,Lag(.,x,-1),-1) = x^n evaluated umbrally, i.e., with (Lag(.,x,-1))^k = Lag(k,x,-1). - Tom Copeland, Apr 26 2014

Without row n=0 and column m=0 this is, up to signs, the Lah triangle A008297.

The unsigned column sequences are (with leading zeros): A000142, A001286, A001754, A001755, A001777, A001778, A111597-A111600 for m=1..10.

The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A111595 satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0.

Exponential Riordan array [1,x/(1+x)]. Inverse of the exponential Riordan array [1,x/(1-x)], which is the unsigned version of A111596. - Paul Barry, Apr 12 2007

For the unsigned subtriangle without column nr. m=0 and row nr. n=0 see A105278.

Unsigned triangle also matrix product |S1|*S2 of Stirling number matrices.

The unsigned row polynomials are Lag(n,-x,-1), the associated Laguerre polynomials of order -1 with negated argument. See Gradshteyn and Ryzhik, Abramowitz and Stegun and Rota (Finite Operator Calculus) for extensive formulae. - Tom Copeland, Nov 17 2007, Sep 09 2008

An infinitesimal matrix generator for unsigned A111596 is given by A132792. - Tom Copeland, Nov 22 2007

From the formalism of A132792 and A133314 for n > k, unsigned A111596(n,k) = a(k) * a(k+1)...a(n-1) / (n-k)! = a generalized factorial, where a(n) = A002378(n) = n-th term of first subdiagonal of unsigned A111596. Hence Deutsch's remark in A002378 provides an interpretation of A111596(n,k) in terms of combinations of certain circular binary words. - Tom Copeland, Nov 22 2007

Given T(n,k)= A111596(n,k) and matrices A and B with A(n,k) = T(n,k)*a(n-k) and B(n,k) = T(n,k)*b(n-k), then A*B = C where C(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. - Tom Copeland, Aug 27 2008

LINKS

Table of n, a(n) for n=0..56.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, arXiv preprint arXiv:1105.3044, 2011

FORMULA

E.g.f. m-th column: ((x/(1+x))^m)/m!, m>=0.

E.g.f. for row polynomials p(n, x) is exp(x*y/(1+y)).

a(n, m)= ((-1)^(n-m))*|A008297(n, m)| = ((-1)^(n-m))*(n!/m!)*binomial(n-1, m-1), n>=m>=1; a(0, 0)=1; else 0.

a(n, m) = -(n-1+m)*a(n-1, m) + a(n-1, m-1), n>=m>=0, a(n, -1):=0, a(0, 0)=1; a(n, m)=0 if n<m.

|a(n,m)|=sum(|S1(n,k)|*S2(k,m),k=m..n), n>=0. S2(n,m):=A048993. S1(n,m):=A048994. - Wolfdieter Lang, May 04, 2007.

From Tom Copeland, Nov 21 2011: (Start)

For this Lah triangle, the n-th row polynomial is given umbrally by

  (-1)^n n! C(-Bell.(-x),n), where C(x,n)=x!/(n!(x-n)!), the binomial coefficient, and Bell_n(-x)= exp(x)(xd/dx)^n exp(-x), the n-th Bell / Touchard / exponential polynomial with neg. arg., (cf. A008277). E.g., 2! C(-Bell.(-x),2) = -Bell.(-x)*[-Bell.(-x)-1] = Bell_2(-x)+Bell_1(-x) = -2x+x^2.

A Dobinski relation is (-1)^n n! C(-Bell.(-x),n)= (-1)^n n! e^x Sum(j=0 to infin) (-1)^j C(-j,n)x^j/j!= n! e^x Sum(j=0 to infin) (-1)^j C(j-1+n,n)x^j/j!. See the Copeland link for the relation to inverse Mellin transform. (End)

The n-th row polynomial is (-1/x)^n e^x (x^2*D_x)^n e^(-x). - Tom Copeland, Oct 29 2012

Let f(.,x)^n = f(n,x) = x!/(x-n)!, the falling factorial,and r(.,x)^n = r(n,x) = (x-1+n)!/(x-1)!, the rising factorial, then the Lah polynomials, Lah(n,t)= n!*sum(k=1,..,n) binomial(n-1,k)(-t)^k/k! (extra sign factor on odd rows), give the transform Lah(n,-f(.,x))= r(n,x), and Lah(n,r(.,x))= (-1)^n * f(n,x). - Tom Copeland, Oct 04 2014

EXAMPLE

Binomial convolution of row polynomials: p(3,x) = 6*x-6*x^2+x^3; p(2,x) = -2*x+x^2, p(1,x) = x, p(0,x) = 1,

together with those from A111595: s(3,x) = 9*x-6*x^2+x^3; s(2,x) = 1-2*x+x^2, s(1,x) = x, s(0,x) = 1; therefore

9*(x+y)-6*(x+y)^2+(x+y)^3 = s(3,x+y) = 1*s(0,x)*p(3,y) + 3*s(1,x)*p(2,y) + 3*s(2,x)*p(1,y) +1*s(3,x)*p(0,y) = (6*y-6*y^2+y^3) + 3*x*(-2*y+y^2) + 3*(1-2*x+x^2)*y + 9*x-6*x^2+x^3.

From Wolfdieter Lang, Apr 28 2014: (Start)

The triangle a(n,m) begins:

n\m  0        1         2          3         4         5       6       7     8    9  10 ...

0:   1

1:   0        1

2:   0       -2         1

3:   0        6        -6          1

4:   0      -24        36        -12         1

5:   0      120      -240        120       -20         1

6:   0     -720      1800      -1200       300       -30       1

7:   0     5040    -15120      12600     -4200       630     -42       1

8:   0   -40320    141120    -141120     58800    -11760    1176     -56     1

9:   0   362880  -1451520    1693440   -846720    211680  -28224    2016   -72    1

10:  0 -3628800  16329600  -21772800  12700800  -3810240  635040  -60480  3240  -90   1

-------------------------------------------------------------------------------------------

(End)

MATHEMATICA

a[0, 0] = 1; a[n_, m_] := ((-1)^(n-m))*(n!/m!)*Binomial[n-1, m-1]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jul 05 2013 *)

PROG

(Sage)

lah_number = lambda n, k: factorial(n-k)*binomial(n, n-k)*binomial(n-1, n-k)

A111596_row = lambda n: [(-1)^(n-k)*lah_number(n, k) for k in (0..n)]

for n in range(10): print A111596_row(n) # Peter Luschny, Oct 05 2014

CROSSREFS

Row sums: A111884. Unsigned row sums: A000262.

Cf. A130561 for a natural refinement.

Sequence in context: A021830 A247686 A111184 * A129062 A163936 A187555

Adjacent sequences:  A111593 A111594 A111595 * A111597 A111598 A111599

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang, Aug 23 2005

EXTENSIONS

Name changed. Link with first rows erased, - Wolfdieter Lang, Apr 28 2014

STATUS

approved

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Last modified November 22 06:18 EST 2014. Contains 249804 sequences.