cp's OEIS Frontend

2*T(2*n,x) = Sum_{m=0..n} a(n,m)*(2*x)^(2*m).

Closely related to A284982, which has opposite signs and rows begin with 0 of alternating signs instead of +/2. - Eric W. Weisstein, Apr 07 2017

Bisection triangle of A127672 (without zero entries, even part). The odd part is ((-1)^(n-m))*A111125(n,m).

If the leading 2 is replaced by a 1 we get the essentially identical sequence A110162. - N. J. A. Sloane, Jun 09 2007

Also row n gives coefficients of characteristic polynomial of the Cartan matrix for the root system B_n (or, equally, C_n). - Roger L. Bagula, May 23 2007

From Wolfdieter Lang, Oct 04 2013: (Start)

This triangle a(n,m) is used to express the length ratio side/R given by s(4*n+2) = 2*sin(Pi/(4*n+2)) = 2*cos(2*n*Pi/(4*n+2)) in a regular (4*n+2)-gon, inscribed in a circle with radius R, in terms of rho(4*n+2) = 2*cos(Pi/4*n+2), the length ratio of (the smallest diagonal)/side (for n=2 there is no such diagonal).

s(4*n+2) = Sum_{m=0..n}a(n,m)*rho(4*n+2)^(2*m). This formula is needed to show that the total sum of all length ratios in a (4*n+2)-gon is an integer in the algebraic number field Q(rho(4*n+2)). Note that rho(4*n+2) has degree delta(4*n+2) = A055034(4*n+2). Therefore one has to take s(4*n+2) modulo C(4*n+2, x=rho(4*n+2)), the minimal polynomial of rho(4*n+2) (see A187360). Thanks go to Seppo Mustonen for asking me to look into this problem. See ((-1)^(n-m))*A111125(n,m) for the (4*n)-gon situation. (End)

a(1) = 1 by convention.

A prime index of n is a number m such that prime(m) divides n.

"Standard order" here means as produced by Maple's "sort" command.

From Petros Hadjicostas, May 27 2020: (Start)

According to the Maple help files for the "sort" command, polynomials in multiple variables are "sorted in total degree with ties broken by lexicographic order (this is called graded lexicographic order)."

Thus for example, x_1^2*x_3 = x_1*x_1*x_3 > x_1*x_2*x_2 = x_1*x_2^2, while x_1^2*x_4 = x_1*x_1*x_4 > x_1*x_2*x_3. (End)

Row sums are 0 (for n > 1). Numbers of terms in rows are partition numbers A000041.

From Tom Copeland, Nov 06 2015: (Start)

With the formal Taylor series f(x) = 1 + x[1] x + x[2] x^2/2! + ... , the partition polynomials of this entry give d[log(f(x))]/dx = L_1(x[1]) + L_2(x[1], x[2]) x + L_3(...) x^2/2! + ..., and the coefficients of the reduced polynomials with x[n] = t are signed A028246.

The raising operator R = x + d[log(f(D)]/dD = x + L_1(x[1]) + L_2[x[1], x[2]) D + L_3(x[1], x[2], x[3]) D^2/2! + ... with D = d/dx generates an Appell sequence of polynomials, given umbrally by P_n(x[1], ..., x[n]; x) = (x[.] + x)^n = Sum_{k=0..n} binomial(n,k) x[k] * x^(n-k) = R^n 1 with the e.g.f. f(t)*e^(x*t) = exp[t P.(x[1], ..., x[.]; x)]. P_0 = x[0] = 1.

The umbral compositional inverse Appell sequence is generated by R = x - d[log(f(D))]/dD with e.g.f. e^(x*t)/f(t) = exp[t IP.(x[1], ..., x[.]; x)], so umbrally IP_n(x[1], ..., x[n]; P.(x[1], ..., x[n]; x)) = x^n = P_n(x[1], ..., x[n]; IP.(x[1], ..., x[n]; x)). An unsigned array for the reduced IP_n(x[1], ..., x[n]; x) polynomials with IP_0 = x[0] = 1 and x[n] = -1 for n > 0 is A154921, for which f(t) = 2 - e^t. (End)

From Tom Copeland, Sep 08 2016: (Start)

The Appell formalism allows a matrix representation in the power basis x^n of the raising operator R that incorporates this array's partition polynomials L_n(x[1], ..., x[n]):

VP_(n+1) = VP_n * R = VP_n * XPS^(-1) * MX * XPS, where XPS is the matrix formed from multiplying the n-th diagonal of the Pascal matrix PS of A007318 by the indeterminate x[n], with x[0] = 1 for the main diagonal of ones, i.e., XPS[n,k] = PS[n,k] * x[n-k]; the matrix MX is A129185; the matrix XPS^(-1) is the inverse of XPS, which can be formed by multiplying the diagonals of the Pascal matrix by the partition polynomials IPT(n, x[1], ..., x[n]) of A133314, i.e., XPS^(-1)[n,k] = PS[n,k] * IPT(n-k, x[1], ...); and VP_n is the row vector in the power basis representing the Appell polynomial P_n(x) formed from the basic sequence of moments 1, x[1], x[2], ..., i.e., umbrally P_n(x) = (x[.] + x)^n = Sum_{k=0..n} binomial(n,k) * x[k] * x^(n-k).

Then R = XPS^(-1) * MX * XPS is the Pascal matrix PS with an additional first superdiagonal of ones and the other lower diagonals multiplied by the partition polynomials of this array, i.e., R[n,k] = PS[n,k] * L_{n+1-k}(x[1], ..., x[n+1-k]) except for the first superdiagonal of ones.

Consistently, VP_n = (1, 0, 0, ...) * R^n = (1, 0, 0, ...) * XPS^(-1) * MX^n * XPS = (1, 0, 0, ...) * MX^n * XPS = the n-th row vector of XPS, which is the vector representation of P_n(x) = (x[.] + x)^n with x[0] = 1.

See the Copeland link for the umbral representation R = exp[g.*D] * x * exp[h.*D] that reflects the matrix representations.

The Stirling partition polynomials of the first kind St1_n(a[1], a[2], ..., a[n]) of A036039, the Stirling partition polynomials of the second kind St2_n(b[1], b[2], ..., b[n]) of A036040, and the refined Lah polynomials Lah_n[c[1], c[2], ..., c[n]) of A130561 are Appell sequences in the respective distinguished indeterminates a[1], b[1], and c[1]. Comparing the formulas for their raising operators with that in this entry, L_n(x[1], x[2], ..., x[n]) evaluates to

A) (n-1)! * a[n] for x[n] = St1_n(a[1], a[2], ..., a[n]);

B) b[n] for x[n] = St2_n(b[1], b[2], ..., b[n]);

C) n! * c[n] for x[n] = Lah_n(c[1], c[2], ..., c[n]).

Conversely, from the respective e.g.f.s (added Sep 12 2016)

D) x[n] = St1_n(L_1(x[1])/0!, ..., L_n(x[1], ..., x[n])/(n-1)!);

E) x[n] = St2_n(L_1(x[1]), ..., L_n(x[1], ..., x[n]));

F) x[n] = Lah_n(L_1(x[1])/1!, ..., L_n(x[1], ..., x[n])/n!).

Given only the Appell sequence with no closed form for the e.g.f., the raising operator can be generated using this formalism, as has been partially done for A134264. (End)

For the Appell sequences above, the raising operator is related to the recursion P_(n+1)(x) = x * P_n(x) + Sum_{k=0..n} binomial(n,k) * L_(n-k+1)(x[1], ..., x[n+k-1]) * P_k(x). For a derivation and connections to formal cumulants (c_n = L_n(x[1], ...)) and moments (m_n = x[n]), see the Copeland link on noncrossing partitions. With x = 0, the recursion reduces to x[n+1] = Sum_{k = 0..n} binomial(n,k) * L_(n-k+1)(x[1], ..., x[n+k-1]) * x[k] with x[0] = 1. This array is a differently ordered version of A127671. - Tom Copeland, Sep 13 2016

With x[n] = x^(n-1), a signed version of A130850 is obtained. - Tom Copeland, Nov 14 2016

See p. 2 of Getzler for a relation to stable graphs called necklaces used in computations for Deligne-Mumford-Knudsen moduli spaces of stable curves of genus 1. - Tom Copeland, Nov 15 2019

For a relation to a combinatorial Faa di Bruno Hopf algebra related to functional composition, as presented by Connes and Moscovici, see Figueroa et al. - Tom Copeland, Jan 17 2020

From Tom Copeland, May 17 2020: (Start)

The e.g.f. of an Appell sequence is f(t) e^(x*t) with f(0) = 1. Given the Laguerre-Polya class function f(t) = e^(-a*t^2 + b*t) Product_m (1 - t/z_m) e^(t/z_m) with a = 0 for simplicity (more generally a >= 0) and b real and where the product runs over all the zeros z_m of f(t) with all zeros real and Sum_m 1/(z_m)^2 convergent, the raising operator of the Appell polynomials is R = x + b - Sum_{k > 0} c_(k+1) D^k with c_k = Sum_m (1/(z_m)^k), i.e., traces of powers of the reciprocals of the zeros. From R in earlier comments, b = L_1(x_1) and otherwise c_k = -L_k(x_1, ..., x_k).

The Laguerre / Turan / de Gua inequalities (Csordas and Williamson and Skovgaard) imply that all the zeros of each Appell polynomial are real and simple and its extrema are local maxima above the x-axis and local minima below and are located above or below the zeros of the next lower degree Appell polynomial. (End)

From Tom Copeland, Oct 15 2020: (Start)

With a_n = n! * b_n = (n-1)! * c_n for n > 0, represent a function with f(0) = a_0 = b_0 = 1 as an

A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!

B) ordinary generating function (o.g.f.), or formal power series: f(x) = 1/(1-b.x) = 1 + Sum_{n > 0} b_n * x^n

C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0} c_n * x^n /n.

Expansions of log(f(x)) are given in

I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] = e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials

II) A263916 for the o.g.f.: log[ 1/(1-b.x) ] = log[ 1 - F.(b_1,b_2,...)x ] = -Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.

Expansions of exp(f(x)-1) are given in

III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind

IV) A130561 for an o.g.f.: exp[ b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials

V) A036039 for an l.g.f.: exp[ -log(1-c.x) ] = e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.

Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)

Ignoring signs, these polynomials appear in Schröder in the set of equations (II) on p. 343 and in Stewart's translation on p. 31. - Tom Copeland, Aug 25 2021

A refinement of A135278, up the sign these are the coefficients appearing in the expansion of power-sum symmetric functions in terms of elementary or homogeneous symmetric functions.

Defined by the recurrence formula in Theorem 1, page 2 of Lasalle.

From Tom Copeland, Jan 26 2016: (Start)

Let G(t) = Sum_{n>=0} t^(2n)/(n!(n+1)!) = exp(c.t) be the e.g.f. of the aerated Catalan numbers c_n of A126120.

R = x + H(D) = x + d/dD log[G(D)] = x + D - D^3/3! + 5 D^5/5! - 56 D^7/7! + ... = x + e^(r. D) generates a signed, aerated version of this entry's sequence a(n), (r.)^(2n+1) = r(2n+1) = (-1)^n a(n+1) for n>=0 and r(0) = a(0) = 0, and is, with D = d/dx, the raising operator for the Appell polynomials P(n,x) of A097610, where P(n,x) = (c. + x)^n = Sum{k=0 to n} binomial(n,k) c_k x^(n-k) with c_k = A126120(k), i.e., R P(n,x) = P(n+1,x).

d/dt log[G(t)] = e^(r.t) = e^(q.t) / e^(c.t) = Ev[c. e^(c.t)] / Ev[e^(c.t)] = e^(q.t) e^(d.t) = [Sum_{n>=0} 2n t^(2n-1)/(n!(n+1)!)] / [Sum_{n>=0} t^(2n)/(n!(n+1)!)] with Ev[..] denoting umbral evaluation, so q(n) = c(n+1) = A126120(n+1) and d(2n) = (-1)^n A238390(n) and vanishes otherwise. Then (r. + c.)^n = q(n) = Sum_{k=0..n} binomial(n,k) r(k) c(n-k) and (q. + d.)^n = r(n), relating A180874, A126120 (A000108), and A238390 through binomial convolutions.

The sequence can also be represented in terms of the Faber polynomials of A263916 as a(n) = |(2n-1)! F(2n,0,b(2),0,b(4),0,..)| = |h(2n)| where b(2n) = 1/(n!(n + 1)!) = A126120(2n)/(2n)! = A000108(n)/(2n)!, giving h(0) = 1, h(1) = 0, h(2) = 1, h(3) = 0, h(4) = -1, h(5) = 0, h(6) = 5, h(7) = 0, h(8) = -56, ..., implying, among other relations, that A000108(n/2)= A126120(n) = Bell(n,0,h(2),0,h(4),...), the Bell polynomials of A036040 which reduce to A257490 in this case.

(End)

From Colin Defant, Sep 06 2018: (Start)

a(n) is the number of pairs (rho,r), where rho is a matching on [2n] and r is an acyclic orientation of the crossing graph of rho in which the block containing 1 is the only source (see the Josuat-Verges paper or the Defant-Engen-Miller paper for definitions).

a(n) is the number of permutations of [2n-1] that have exactly 1 preimage under West's stack-sorting map.

a(n) is the number of valid hook configurations of permutations of [2n-1] that have n-1 hooks (see the paper by Defant, Engen, and Miller for definitions).

Say a binary tree is full if every vertex has either 0 or 2 children. If u is a left child in such a tree, then we can start at the sibling of u and travel down left edges until reaching a leaf v. Call v the leftmost nephew of u. A decreasing binary plane tree on [m] is a binary plane tree labeled with the elements of [m] in which every nonroot vertex has a label that is smaller than the label of its parent. a(n) is the number of full decreasing binary plane trees on [2n-1] in which every left child has a label that is larger than the label of its leftmost nephew.

(End)

The length of row n is A008619(n).

Essentially equals a signed version of A034807, the triangle of Lucas polynomials. The initial n coefficients of 1/C(x)^n consist of row n followed by floor((n-1)/2) zeros for n > 0.

For the following formula for 1/C(x)^n see the W. Lang reference, proposition 1 on p. 411:

1/C(x)^n = (sqrt(x))^n*(S(n,1/sqrt(x)) - sqrt(x)*S(n-1,1/sqrt(x))*C(x)), n >= 0, with the Chebyshev polynomials S(n,x) with coefficients given in A049310. See also the coefficient array A115139 for P(n,x) = (sqrt(x)^(n-1))*S(n-1, 1/sqrt(x)). - Wolfdieter Lang, Sep 14 2013

This triangular array is composed of interleaved rows of reversed, A127677 (cf. A156308, A217476, A263916) and reversed, signed A111125. - Tom Copeland, Nov 07 2015

It seems that the n-th row lists the coefficients of the HOMFLYPT (HOMFLY) polynomial reduced to one variable for link family n, see Jablan's slide 38. - Andrey Zabolotskiy, Jan 16 2018

For n >= 1 row n gives the coefficients of the Girard-Waring formula for the sum of x1^n + x2^n in terms of the elementary symmetric functions e_1(x1,x2) = x1 + x2 and e_2(x1,x2) = x1*x2. This is an array using the partitions of n, in the reverse Abramowitz-Stegun order, with all partitions with parts larger than 2 eliminated. E.g., n = 4: x1^4 + x2^4 = 1*e1^4 - 4*e1^3*e2 + 2*e1*e2^2. See also A115131, row n = 4, with the mentioned partitions omitted. - Wolfdieter Lang, May 03 2019

Row n lists the coefficients of the n-th Faber polynomial for the replicable function given in A154272 with offset -1. - Ben Toomey, May 12 2020

From Wolfdieter Lang, Jun 26 2011: (Start)

This triangle S(n,m) appears as U_m(n) in the Knuth reference on p. 285. It is related to the Riordan triangle T_m(n) = A111125(n,m) by S(n,m) = A111125(n,m) - A111125(n-1,m), n >= m >= 1 (identity on p. 286).

Also, S(n,m)-S(n-1,m) = A111125(n-1,m-1), n >= 2, m >= 1 (identity on p. 286). (End)

These polynomials may be expressed in terms of the Faber polynomials of A263916 and are embedded in A127677 and A208513. - Tom Copeland, Nov 06 2015

The columns of A208513 are identical to those of A208509. Here, however, the alternating row sums are periodic (with period 1,0,-2,-3,-2,0).

From Tom Copeland, Nov 07 2015: (Start)

These polynomials may be expressed in terms of the Faber polynomials of A263916, similar to A127677.

Rephrasing notes in A111125: Append an initial column of zeros except for a 1 at the top to A111125. Then the rows of this entry contain the partial sums of the column sequences of modified A111125; therefore, the difference of consecutive pairs of rows of this entry, modified by appending an initial row of zeros to it, generates the modified A111125. (End)

Sum of row #n = A000204(2n). (But sum of row #0 = 1.)

Row #n has the unsigned coefficients of the monic polynomial whose roots are 2 cos(Pi*(2k-1)/(4n)) for k=1..2n. [Comment corrected by Barry Brent, Jan 03 2006]

The positive roots are some diagonal lengths of a regular (4n)-gon, inscribed in the unit circle.

Polynomial of row #n = Sum_{m=0..n} (-1)^m * T(n,m) x^(2*n-2*m).

This is the unsigned version of the coefficient table for scaled Chebyshev T(2*n,x) polynomials. - Wolfdieter Lang, Mar 07 2007

Reversed A127677 (cf. A156308, A217476, A263916). - Tom Copeland, Nov 07 2015