cp's OEIS Frontend

Number of Joyce trees with 2n-1 nodes. Number of tremolo permutations of {0,1,...,2n}. - Ralf Stephan, Mar 28 2003

The Hankel transform of this sequence is A000178(n) for n odd = 1, 12, 34560, ...; example: det([1, 2, 16; 2, 16, 272, 16, 272, 7936]) = 34560. - Philippe Deléham, Mar 07 2004

a(n) is the number of increasing labeled full binary trees with 2n-1 vertices. Full binary means every non-leaf vertex has two children, distinguished as left and right; labeled means the vertices are labeled 1,2,...,2n-1; increasing means every child has a label greater than its parent. - David Callan, Nov 29 2007

From Micha Hofri (hofri(AT)wpi.edu), May 27 2009: (Start)

a(n) was found to be the number of permutations of [2n] which when inserted in order, to form a binary search tree, yield the maximally full possible tree (with only one single-child node).

The e.g.f. is sec^2(x)=1+tan^2(x), and the same coefficients can be manufactured from the tan(x) itself, which is the e.g.f. for the number of trees as above for odd number of nodes. (End)

a(n) is the number of increasing strict binary trees with 2n-1 nodes. For more information about increasing strict binary trees with an associated permutation, see A245894. - Manda Riehl, Aug 07 2014

For relations to alternating permutations, Euler and Bernoulli polynomials, zigzag numbers, trigonometric functions, Fourier transform of a square wave, quantum algebras, and integrals over and in n-dimensional hypercubes and over Green functions, see Hodges and Sukumar. For further discussion on the quantum algebra, see the later Hodges and Sukumar reference and the paper by Hetyei presenting connections to the general combinatorial theory of Viennot on orthogonal polynomials, inverse polynomials, tridiagonal matrices, and lattice paths (thereby related to continued fractions and cumulants). - Tom Copeland, Nov 30 2014

The Zigzag Hankel transform is A000178. That is, A000178(2*n - k) = det( [a(i+j - k)]{i,j = 1..n} ) for n>0 and k=0,1. - _Michael Somos, Mar 12 2015

a(n) is the number of standard Young tableaux of skew shape (n,n,n-1,n-2,...,3,2)/(n-1,n-2,n-3,...,2,1). - Ran Pan, Apr 10 2015

For relations to the Sheffer Appell operator calculus and a Riccati differential equation for generating the Meixner-Pollaczek and Krawtchouk orthogonal polynomials, see page 45 of the Feinsilver link and Rzadkowski. - Tom Copeland, Sep 28 2015

For relations to an elliptic curve, a Weierstrass elliptic function, the Lorentz formal group law, a Lie infinitesimal generator, and the Eulerian numbers A008292, see A155585. - Tom Copeland, Sep 30 2015

Absolute values of the alternating sums of the odd-numbered rows (where the single 1 at the apex of the triangle is counted as row #1) of the Eulerian triangle, A008292. The actual alternating sums alternate in sign, e.g., 1, -2, 16, -272, etc. (Even-numbered rows have alternating sums always 0.) - Gregory Gerard Wojnar, Sep 28 2018

The sequence is periodic modulo any odd prime p. The minimal period is (p-1)/2 if p == 1 mod 4 and p-1 if p == 3 mod 4 [Knuth & Buckholtz, 1967, Theorem 1]. - Allen Stenger, Aug 03 2020

From Peter Bala, Dec 24 2021: (Start)

Conjectures:

1) The sequence taken modulo any integer k eventually becomes periodic with period dividing phi(k).

2) The Gauss congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k, except when p = 2, n = 1 and k = 1 or 2.

3) For i >= 1 define a_i(n) = a(n+i). The Gauss congruences a_i(n*p^k) == a_i(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k. If true, then for each i >= 1 the expansion of exp(Sum_{n >= 1} a_i(n)*x^n/n) has integer coefficients. For an example, see A262145.(End)

Differs from the absolute values of A275994 the first time at index 60.

Consider the C(k)-summation process for divergent series: the series Sum((-1)^n*(n+1)^k) == 1 - 2^k + 3^k - 4^k + ..., summable C(1) to the value 1/2 for k = 0, is for each k >= 1 exactly summable C(k+1) to the sum s(k+1) = (2^(k+1)-1)*B(k+1)/ (k+1) and so a(n) = abs(numerator(s(2n))). - Benoit Cloitre, Apr 27 2002

Odd part of tangent numbers A000182 (even part is 2^A101921(n)). - Ralf Stephan, Dec 21 2004

(-1)^n*a(n+1) is the numerator of Euler(2n+1,1). - N. J. A. Sloane, Nov 10 2009 (a misprint corrected by Vladimir Shevelev, Sep 18 2017)

a(n) is the absolute value of the constant term of the Euler polynomial E_{2n-1} times the even part of 2n. - Peter Luschny, Nov 26 2010

From Vladimir Shevelev, Aug 31 2017: (Start)

Let E_m(x) = x^m + Sum_{odd k=1..m} e_k(m)*x^(m-k) be the Euler polynomial, let 2*n-1 <= m. Show that the expression c(m,n) = |e_(2*n-1)(m)|/binomial(m,2*n-1) does not depend on m and c(m,n) = a(n)/A006519(2*n). Indeed, by the formula in the Shevelev link |e_(2*n-1)(m)| = binomial(m,2*n-1)*(4^n-1)*B_(2*n)/n. On the other hand, by Cloitre's formula, we have a(n) = (4^n-1)*|B_(2*n)|*2^A001511(n) /n. Taking into account that 2^A001511 = A006519(2*n) we obtain the claimed equality. Since sign(e_k(n)) = (-1)^((k+1)/2), we have the following application of the sequence: e_k(n) = (-1)^((k+1)/2))*a((k+1)/2)*binomial(n,k)/A006519(k+1). (End)

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

Equals central terms of A174449. - G. C. Greubel, Nov 29 2021