cp's OEIS Frontend

T(n,k) is simply constructed from Pascal's triangle PT and A084358 through multiplication along the diagonals. Taking the matrix inverse gives TI = 2I-A129652 = PT times diagonal multiplication by -A000262 with the sign of the first term flipped to positive.

T and TI are also reciprocals under the list partition transform described in A133314.

Number of partitions of {1,..,n} into any number of lists of size >1, where a list means an ordered subset, cf. A000262. - Vladeta Jovovic, Vladimir Baltic, Oct 29 2002

Coefficients may be generated from a modified Riordan array (MRA) formed from Rgf(z,t) = (t/(1+z))/(exp(-z/(1+z))-t) with each row of the array acting to generate the succeeding polynomial P(n,t) from the preceding n polynomials.

The MRA is constructed by appending an n! to the left of the n-th row of the Riordan array A129652 and removing the unit diagonal.

The MRA is partially

1;

1, 1;

2, 3, 2;

6, 13, 9, 3;

24, 73, 52, 18, 4;

120, 501, 365, 130, 30, 5;

720, 4051, 3006, 1095, 260, 45, 6;

For the MRA:

1) First column is the n!'s.

2) Second column is A000262.

Then, e.g., from the terms in the MRA,

P(0,t) = 0!*(t-1)^0 = 1 from the n=0 row,

P(1,t) = 1!*(t-1)^1 + 1*P(0,t) = t from the n=1 row,

P(2,t) = 2!*(t-1)^2 + 3*P(0,t)*(t-1)^1 + 2*P(1,t)

P(3,t) = 3!*(t-1)^3 + 13*P(0,t)*(t-1)^2 + 9*P(1,t)*(t-1)^1 + 3*P(2,t)

generating

P(0,t) = (1)

P(1,t) = (0, 1)

P(2,t) = (-1, 1, 2)

P(3,t) = (4, -14, 10, 6) = 4 + -14 t + 10 t^2 + 6 t^3

P(4,t) = (-15, 83, -157, 89, 24)

P(5,t) = (56, -424, 1266, -1724, 826, 120)

P(6,t) = (-185, 1887, -8038, 17642, -19593, 8287, 720)

P(7,t) = (204, -4976, 36226, -126944, 239576, -234688, 90602, 5040)

For the polynomial array:

1) The first column is A009940 = (-1)^n * n!*Lag(n,1) =(-1)^n* n!* Lag(n,-1,-1).

2) Row sums are n!.

3) Highest order coefficient is n!.

4) Alternating row sum is below.

Then, with Rf(n,t) = [ t/(1-t)^(n+1) ] * P(n,t)/n!, the polylogs are given umbrally by

Li(-n,t)/n! = [ 1 + Rf(.,t) ]^n for n = 0,1,2,... so conversely

Rf(n,t) = {[ Li(-(.),t))/(.)! ]-1}^n.

Note umbrally [ Rf(.,t) ]^n = Rf(n,t) and

(1+Rf)^0 = 1^0 * [ Rf(.,t) ]^0 = Rf(0,t) = t/(1-t) = Li(0,t).

More generally, Newton interpolation holds and for Re(s) >= 0,

Li(-s,t)/(s)! = [ 1 + Rf(.,t) ]^s, when convergent in t.

Alternatively, the Rf's may be formed through differentiation of their o.g.f., the Rgf(z,t) above, which may also be written as

Rgf(z,t) = Sum_{k>=1} [ t^k ] * exp[ k * z/(z+1) ]/(z+1)

= Sum_{n>=0} [ (-z)^n ] * Sum_{k>=1} [ (t^k * Lag(n,k) ]

= Sum_{k>=0} [ (-z)^k ] * Lag(k,Li(-(.),t))

= Sum_{k>=0} [ z^k ] * {[ Li(-(.),t))/(.)! ]-1}^k

= exp[ Li(-(.),t)*z/(1+z) ]/(1+z),

and operationally as

Rgf(z,t) = {Sum_{k>=0} (-z)^k * Lag(k,tD)} [ t/(1-t) ]

= {Sum_{k>=0} (-z)^k * Lag(k,T(.,:tD:))} [ t/(1-t) ]

= {Sum_{k>=0} (-z)^k * Sum_{j>=0} Lag(k,j) (tD)^j /j!} [ x/(1-x) ]

where D is w.r.t. x at 0

= {Sum_{k>=0} (-z)^k*Sum_{j=0..k} (-1)^j*[ 1-Lag(k,.) ]^j*(:tD:)^j/ j!} [ t/(1-t) ]

= {Sum_{k>=0} (-z)^k * exp[ -[ 1-Lag(k,.) ]* :tD: ]} [ t/(1-t) ]

where (:tD:)^n = t^n * D^n, D is the derivative w.r.t. t unless otherwise stated, Lag(n,x) is a Laguerre polynomial and T(n,t) is a Touchard / Bell / exponential polynomial.

Hence [ t/(1-t)^(n+1) ] * P(n,t)/n! = Rf(n,t)

= {Sum_{k=0..n} (-1)^n-k)*[ C(m,k)/k! ]*(tD)^k} [ t/(1-t) ]

= {Sum_{k=0..n} (-1)^(n-k)*[ C(m,k)/k! ]*Sum_{j=0..k} S2(k,j)*(:tD:)^j} [ t/(1-t) ]

= {Sum_{k>=0} (-1)^(n-k) * Lag(n,k) * (tD)^k/k!} [ x/(1-x) ] where D is w.r.t. x at 0

= {Sum_{k=0..n} (-1)^(n-k)* [ 1-Lag(n,.) ]^k *(:tD:)^k/k!}[ t/(1-t) ],

where S2(k,j) are the Stirling numbers of the second kind and C(m,k), binomial coefficients.

The P(n,t) are related to the Laguerre polynomials through

P(n,t) = (-1)^n n! [ (1-t)^(n+1)} ] Sum_{k>=0} [ (t^k*Lag(n,k+1) ] = Sum_{m=0..n} a(n,m) * t^m

where a(n,m) = (-1)^n n! [ Sum_{k=0..m} (-1)^k * C(n+1,k) *Lag(n,m-k+1) ] .

Conjecture for the polynomial array:

The greatest common divisor of the coefficients of each polynomial is given by A060872(n)/n or, equivalently, by A038548(n).

Some e.g.f.'s for the Rf's are

exp[ -Rf(.,t)*z ] = exp{[ 1-Li(-(.),t)/(.)! ]*z}

= Sum_{n>=0} { (z^n/n!) * Sum_{k>=1} [ t^k * Lag(n,k) ] }

= Sum_{k>=1} { t^k * (e^z) * J_0[ 2*sqrt(k*z)}

= Sum_{n>=0} {(-1)^n*(z^n/n!)*(z^/j!)*Lag(n,-1)*Sum_{k>=1} [ t^k*k^n*(k+1)^j ]}

where J_0(x) is the zeroth Bessel function of the first kind.

The expressions (:tD:)^j}[ t/(1-t) ] and the Laguerre polynomials are intimately connected to Lah numbers and rook polynomials.

Some interesting relations to physics, probability and number theory are, for abs(t) < 1 and abs(z) < 1 at least,

BE(t,z) = Sum_{k>=0} [ (-z)^k ] *[ 1 + Rf(.,t) ]^k

= Rgf(-z/(1+z),t)/(1+z) = t/{exp(z)-t}, a Bose-Einstein distribution,

FD(t,z) = Sum_{k>=0} [ (-z)^k+1 ] *[ 1 + Rf(.,-t) ]^k

= -Rgf(-z/(1+z),-t)/(1+z) = t/{exp(z)+t}, a Fermi-Dirac distribution

and as t tends to 1 from below, z*BE(t,z) tends to the Bernoulli e.g.f., which is related by the Mellin transform to (s-1)!*Zeta(s). Taking Mellin transforms of BE and FD w.r.t. z gives the polylogarithm over different domains.

Since BE(2,z) is essentially the e.g.f. for the ordered Bell numbers, it follows that umbrally

n! * Lag(n,OB(.)) = P(n,2) and

n! * Lag(n,P(.,2)) = OB(n)

where OB(n) are the signed ordered Bell/Fubini numbers A000670.

I.e., P(n,2) and the ordered Bell numbers form a reciprocal Laguerre combinatorial transform pair,

or, equivalently, P(n,2)/n! and OB(n)/n! form a reciprocal finite difference pair, so

P(n,2)/n! = (-1)^(n+1) * Rf(n,2) = -{1-[ Li(-(.),2))/(.)! ]}^n and

OB(n) = -Li(-n,2).

Note that n!*Lag(n,(.)!*Lag(.,x)) = x^n is a true identity for general Laguerre polynomials Lag(n,x,a) with a = -1,0,1,..., so one could look at analogous higher-order reciprocal pairs containing OB(n).

In addition, a mixed-order iterated Laguerre transform gives

n!*Lag{n,(.)!*Lag[ .,P(.,2),0 ],-1}

= P(n,2) - n*P(n-1,2)

= n!*Lag[ n,OB(.),-1 ] = A084358(n), lists of sets of lists.

For Eulerian polynomials, E(n,t), given by A173018 (A008292),

E(n,t)/n! = [ 1-t+P(.,t)/(.)! ]^n

P(n,t)/n! = [ E(.,t)/(.)!-(1-t) ]^n, or equivalently

[ E(.,t)/(1-t) ]^n = n!*Lag[ n,-P(.,t)/(1-t) ]

[ -P(.,t)/(1-t) ]^n = n!*Lag[ n,E(.,t)/(1-t) ], a Laguerre transform pair.

Then from known relations for the Eulerian polynomials, the alternating row sum of the polynomial array is

P(n,-1) = (-2)^(n+1) * n! * Lag[ n,c(.)*Zeta(-(.)) ]

where c(n) = [ 2^(n+1) - 1 ] and Zeta is the Riemann zeta function. And so

Zeta(-n) = n! * Lag[ n,-P(.,-1)/2 ] / [ 2 - 2^(n+2) ],

which also holds, with the summation limit of Lag extended to infinity, for n = s, any complex number with Re(s) > 0.

Then from standard formulas for the signed Euler numbers EN(n), the Bernoulli numbers Ber(n), the Genocchi numbers GN(n), the Euler polynomials EP(n,t), the Eulerian polynomials E(n,t), the Touchard / Bell polynomials T(n,t) and the binomial C(x,y) = x!/[ (x-y)!*y! ]

2^(n+1) * (1-2^(n+1)) * (-1)^n * Zeta(-n)

= 2^(n+1) * (1-2^(n+1)) * Ber(n+1)/(n+1)

= [ -(1+EN(.)) ]^n

= 2^n * GN(n+1)/(n+1)

= 2^n * EP(n,0)

= (-1)^n * E(n,-1)

= (-2)^n * n! * Lag[ n,-P(.,-1)/2 ]

= (-2)^n * n! * C{T[ .,P(.,-1)/2 ] + n, n}

= an integer = Q(n)

These are related to the zag numbers A000182 by Zag(n) = abs[ Q(2*n-1) ]. And, abs[ Q(2*n-1) ] / 2^q(n) = Zag(n) / 2^q(n) = A002425(n) with q(n) = A101921.

These may be generalized by letting n = s, a complex number, (or interpolating) to obtain generalized Laguerre functions or confluent hypergeometric functions of the first kind, M(a,b,x), or second kind, U(a,b,x), whose arguments are P(.,-1)/2, such as

E(s,-1)/[ 2^s*s! ] = -2*Li(-s,-1)/s! = (2-2^(s+2)) * Zeta(-s)/s!

= C{T[ .,P(.,-1)/2 ] + s, s} = Lag[ s,-P(.,-1)/2 ] = M[ -s,1,-P(.,-1)/2 ] or,

GN(s+1)/(s+1)! = EP(s,0)/s! = C{-T[ .,P(.,-1)/2 ]-1, n} = U[ -s,1,-P(.,-1)/2 ]/(s)!

And even more generally

E(s,t)/(1-t)^s = [ (1-t)/t ] Li(-s,t) = s!*Lag[ s,-P(.,t)/(1-t) ]

= s! * C{T[ .,P(.,t)/(1-t) ] + s, s} = s! * M[ -s,1,-P(.,t)/(1-t) ] .

The Laguerre polynomial expressions are fundamental as they can be interpolated to form general M[ a,b,-P(.,t)/(1-t) ] or U[ a,b,-P(.,t)/(1-t) ] which can then be related either directly or by binomial transforms to many important Sheffer sequences, not to mention group theory and Riemann surfaces.

Note for frequently occurring expressions above: The Laguerre polynomials of order -1 and 0 are intimately connected to Lah numbers and rook polynomials and (tD)^n [t/(1-t)] = T(n,:tD:) [t/(1-t)] generates an Eulerian polynomial in the numerator of a rational function. - Tom Copeland, Sep 09 2008

The deformed Todd operator on p. 12 of Gunnells and Villegas is Td(a,D) = -D / (a*exp(-D) - 1) = [-D/(1-D)] * Rgf(D/(1-D), 1/a) = -D * BE(1/a,-D) = D * FD(-1/a,-D), where BE and FD are the Bose-Einstein and Fermi-Dirac distributions given above. See also connections among the Eulerian polynomials, Ehrhart polynomials, and the Todd operator in Beck and Robins, especially pages 31 and 37. - Tom Copeland, Jun 20 2017

An unspecified number of sign-in sheets are available at a meeting of n people. The attendees sign in on one of the sheets in the order that they arrive at the meeting. But some, none, or all of the attendees forget to sign in. a(n) is the number of ways this can happen.

Previous name was: A simple grammar.