Real-rooted atlas

A163936 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-16\,x + 4\,x^{2} + 2\,n + 24\,n x - 28\,n x^{2} - 5\,n^{2} - 24\,n^{2} x + 53\,n^{2} x^{2} + 4\,n^{3} + 16\,n^{3} x - 38\,n^{3} x^{2} - n^{4} - 4\,n^{4} x + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-11\,x^{2} + 5\,x^{3} - 2\,n x^{2} - 32\,n x^{3} + 3\,n^{2} x + 16\,n^{2} x^{2} + 53\,n^{2} x^{3} - 2\,n^{3} x - 12\,n^{3} x^{2} - 30\,n^{3} x^{3} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(2\,x^{2} - 3\,x^{3} + x^{4} + 6\,n x^{3} - 6\,n x^{4} - n^{2} x^{2} - 8\,n^{2} x^{3} + 9\,n^{2} x^{4} + 4\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-4\,x + 3\,n + 12\,n x - 5\,n^{2} - 12\,n^{2} x + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(-x - 2\,n x - 4\,n x^{2} + 2\,n^{2} x + 8\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
2246
3632947
42423219816470
512017762952832255705
672014832396002323228503681270
750401359365232605139201402508640491327095
84032013645447067736103043204895000718200242066424414390
936288014935680992366641990410241441375003825000032957406425681499323825
10362880017735616014596822083822478464388393460016115400002609818401400627216410610011233772550
1139916800227342592022567330944742746232321000673772005934183494415404116280161560985656298078407200121275640940575
124790016003131087616036709471641614751966435842529510825600201749035622476189907841613129489766493049864562170626009880861447323799432950
13622702080046134434304062813649312003012927458918463746636835600654547911229443369828227281685966988943361026255239856507259276008106632162355264169209452414196025
14871782912007243970227200112958136161280635033248892928161820937849680020697974447253121386695593239488488629257561088878587975056967501605222800264969406768295315027255327221966414282538413750
Technical details
Node id
n-03fa9c0ed408d1dc · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3d72d99fe3345b32

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:      1;
  2:      2,        4;
  3:      6,       32,        9;
  4:     24,      232,      198,        16;
  5:    120,     1776,     2952,       832,        25;
  6:    720,    14832,    39600,     23232,      2850,       36;
  7:   5040,   135936,   523260,    513920,    140250,     8640,      49;
  8:  40320,  1364544,  7067736,  10304320,   4895000,   718200,   24206,    64;
  9: 362880, 14935680, 99236664, 199041024, 144137500, 38250000, 3295740, 64256, 81;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^4-4*n^3+5*n^2-2*n+(4*n^4-16*n^3+24*n^2-24*n+16)*x-(9*n^4-38*n^3+53*n^2-28*n+4)*x^2)*P(n-2,x) - ((2*n^3-3*n^2)*x-(4*n^4-12*n^3+16*n^2-2*n-11)*x^2-(4*n^4-30*n^3+53*n^2-32*n+5)*x^3)*d/dx P(n-2,x) - ((n^2-2)*x^2-(4*n^3-8*n^2+6*n-3)*x^3+(4*n^3-9*n^2+6*n-1)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3-5*n^2+3*n+(4*n^3-12*n^2+12*n-4)*x)*P(n-1,x) + ((2*n^2-2*n-1)*x-(4*n^3-8*n^2+4*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - 4*n^3 + 5*n^2 - 2*n + (4*n^4 - 16*n^3 + 24*n^2 - 24*n + 16)*x - (9*n^4 - 38*n^3 + 53*n^2 - 28*n + 4)*x^2)*P[n - 2, x] - ((2*n^3 - 3*n^2)*x - (4*n^4 - 12*n^3 + 16*n^2 - 2*n - 11)*x^2 - (4*n^4 - 30*n^3 + 53*n^2 - 32*n + 5)*x^3)*D[P[n - 2, x], x] - ((n^2 - 2)*x^2 - (4*n^3 - 8*n^2 + 6*n - 3)*x^3 + (4*n^3 - 9*n^2 + 6*n - 1)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 - 5*n^2 + 3*n + (4*n^3 - 12*n^2 + 12*n - 4)*x)*P[n - 1, x] + ((2*n^2 - 2*n - 1)*x - (4*n^3 - 8*n^2 + 4*n)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 - 4*n^3 + 5*n^2 - 2*n + (4*n^4 - 16*n^3 + 24*n^2 - 24*n + 16)*x - (9*n^4 - 38*n^3 + 53*n^2 - 28*n + 4)*x^2)*v[i - 2] - ((2*n^3 - 3*n^2)*x - (4*n^4 - 12*n^3 + 16*n^2 - 2*n - 11)*x^2 - (4*n^4 - 30*n^3 + 53*n^2 - 32*n + 5)*x^3)*deriv(v[i - 2], x) - ((n^2 - 2)*x^2 - (4*n^3 - 8*n^2 + 6*n - 3)*x^3 + (4*n^3 - 9*n^2 + 6*n - 1)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^3 - 5*n^2 + 3*n + (4*n^3 - 12*n^2 + 12*n - 4)*x)*v[i - 1] + ((2*n^2 - 2*n - 1)*x - (4*n^3 - 8*n^2 + 4*n)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.