Real-rooted atlas

A130534 → basis A013610 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(81\,x^{2} - 12\,n x - n^{2}\right)\,P_{n-2}(x) + \left(36\,x^{3} + 12\,n x^{2}\right)\,P_{n-2}'(x) + \left(1 + 12\,x + 2\,n\right)\,P_{n-1}(x) + \left(-12\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
121214
266081147
3243127294321497
4120184857516048202515792
5720125284698066960405008748176436
6504096336413424719280597375236196357212103372
7403208311683957660795916881506254470228125023513996826799372
83628807964352412293249274003211066422575442752295055466158592531441364599144
93628800839980804660526161147290048154136115012324969726031133641771994882869781419683005285806632
10399168009675244805693565816151058044802233289475020216146860116669429734312414080988480260127939500714492981458774928
11479001600120892953607485358003221163533811233880319550033907186764022255258202197464337344282354603305196312000550159533255091681330956638640
1262270208001629088588801054699284096315047848896053964821889005871564083880427772702580321390791558407371208901431721278350002601468648922958251208981352922996815157440
1387178291200235544827392015865424774784497317616939529031861856160010551470023036883863731703752467088140787841844150863183251398670609009890296518961250204323689340607016312487308419150756479680
1413076743680003637786360320025388064747072083059202420812815878969113644001972896935641632168880766816361610292411190821764539701579328631453998897351003349316220950754063875567525802852108693718598965210761680258039888988704640
Technical details
Node id
n-bb38c1d358a3999c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-78e837b0826284b4

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       2,       12;
  2:       6,       60,        81;
  3:      24,      312,       729,        432;
  4:     120,     1848,      5751,       6048,       2025;
  5:     720,    12528,     46980,      66960,      40500,       8748;
  6:    5040,    96336,    413424,     719280,     597375,     236196,     35721;
  7:   40320,   831168,   3957660,    7959168,    8150625,    4470228,   1250235,    139968;
  8:  362880,  7964352,  41229324,   92740032,  110664225,   75442752,  29505546,   6158592,   531441;
  9: 3628800, 83998080, 466052616, 1147290048, 1541361150, 1232496972, 603113364, 177199488, 28697814, 1968300;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+12*n*x-81*x^2)*P(n-2,x) + (12*n*x^2+36*x^3)*d/dx P(n-2,x) + (2*n+1+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 12*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 12*n*x - 81*x^2)*P[n - 2, x] + (12*n*x^2 + 36*x^3)*D[P[n - 2, x], x] + (2*n + 1 + 12*x)*P[n - 1, x] - 12*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 12*n*x - 81*x^2)*v[i - 2] + (12*n*x^2 + 36*x^3)*deriv(v[i - 2], x) + (2*n + 1 + 12*x)*v[i - 1] - 12*x^2*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.