Real-rooted atlas

A008955 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-4\,n^{2} - 16\,n^{2} x + 36\,n^{2} x^{2} + 8\,n^{3} + 48\,n^{3} x - 144\,n^{3} x^{2} - 8\,n^{4} - 56\,n^{4} x + 216\,n^{4} x^{2} + 4\,n^{5} + 32\,n^{5} x - 144\,n^{5} x^{2} - n^{6} - 8\,n^{6} x + 36\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(16\,n^{2} x^{2} + 16\,n^{2} x^{3} - 48\,n^{3} x^{2} - 64\,n^{3} x^{3} + 56\,n^{4} x^{2} + 96\,n^{4} x^{3} - 32\,n^{5} x^{2} - 64\,n^{5} x^{3} + 8\,n^{6} x^{2} + 16\,n^{6} x^{3}\right)\,P_{n-2}'(x) + \left(1 - 2\,n + 4\,n^{2} + 8\,n^{2} x - 4\,n^{3} - 16\,n^{3} x + 2\,n^{4} + 8\,n^{4} x\right)\,P_{n-1}(x) + \left(-8\,n^{2} x^{2} + 16\,n^{3} x^{2} - 8\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
210104144258
310017605652460812120
4170042720222804399872230400897496
54420014507201059890430201472372304001658880096114496
6163540066406240627176088247411417647751904004473833472162570240014044058176
7817700003961388800460013803202329737205766176082532008975665428486780447959042080899072002684327758848

All 15 rows

Technical details
Node id
n-50073e7cd4eca286 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-cf0961e8793f3f0e

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,              8;
  2:           10,            104,             144;
  3:          100,           1760,            5652,             4608;
  4:         1700,          42720,          222804,           399872,            230400;
  5:        44200,        1450720,        10598904,         30201472,          37230400,          16588800;
  6:      1635400,       66406240,       627176088,       2474114176,        4775190400,        4473833472,        1625702400;
  7:     81770000,     3961388800,     46001380320,     232973720576,      617608253200,      897566542848,      678044795904,      208089907200;
  8:   5315050000,   299356512000,   4130969695200,   25611161492480,    86739280573200,   172179378514944,   200448949421056,   126883863724032,    33710564966400;
  9: 435834100000, 27991386384000, 447854963630400, 3289834514600960, 13595446259786400, 34353248410341504, 54402366815071744, 52817838204911616, 28779423511412736, 6742112993280000;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(8*n^6-32*n^5+56*n^4-48*n^3+16*n^2)*x-(36*n^6-144*n^5+216*n^4-144*n^3+36*n^2)*x^2)*P(n-2,x) + ((8*n^6-32*n^5+56*n^4-48*n^3+16*n^2)*x^2+(16*n^6-64*n^5+96*n^4-64*n^3+16*n^2)*x^3)*d/dx P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(8*n^4-16*n^3+8*n^2)*x)*P(n-1,x) - (8*n^4-16*n^3+8*n^2)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (8*n^6 - 32*n^5 + 56*n^4 - 48*n^3 + 16*n^2)*x - (36*n^6 - 144*n^5 + 216*n^4 - 144*n^3 + 36*n^2)*x^2)*P[n - 2, x] + ((8*n^6 - 32*n^5 + 56*n^4 - 48*n^3 + 16*n^2)*x^2 + (16*n^6 - 64*n^5 + 96*n^4 - 64*n^3 + 16*n^2)*x^3)*D[P[n - 2, x], x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (8*n^4 - 16*n^3 + 8*n^2)*x)*P[n - 1, x] - (8*n^4 - 16*n^3 + 8*n^2)*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] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (8*n^6 - 32*n^5 + 56*n^4 - 48*n^3 + 16*n^2)*x - (36*n^6 - 144*n^5 + 216*n^4 - 144*n^3 + 36*n^2)*x^2)*v[i - 2] + ((8*n^6 - 32*n^5 + 56*n^4 - 48*n^3 + 16*n^2)*x^2 + (16*n^6 - 64*n^5 + 96*n^4 - 64*n^3 + 16*n^2)*x^3)*deriv(v[i - 2], x) + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (8*n^4 - 16*n^3 + 8*n^2)*x)*v[i - 1] - (8*n^4 - 16*n^3 + 8*n^2)*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

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.