Real-rooted atlas

A371259 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-144 - 60\,x + 9\,x^{2} - 168\,n - 32\,n x - 73\,n^{2} - 4\,n^{2} x - 14\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(-231\,x + 69\,x^{2} + 4\,x^{3} - 182\,n x + 32\,n x^{2} - 47\,n^{2} x + 4\,n^{2} x^{2} - 4\,n^{3} x\right)\,P_{n-2}'(x) + \left(-75\,x^{2} + 41\,x^{3} - 40\,n x^{2} + 8\,n x^{3} - 5\,n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(-6\,x^{3} + 4\,x^{4} - 2\,n x^{3}\right)\,P_{n-2}'''(x) + \left(34 + 4\,x + 16\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(17\,x - 4\,x^{2} + 4\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 36 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
136440
2176440092173
311289639456180016154168
49144576439718430657655042513853865
591445760056863756854040608147520013500361538624512
6110649369600855416401921041229728038521139253425002865649206993889669
715933509222400148991346708482233278957312105183313920196957280015974784548806433173091767008

All 15 rows

Technical details
Node id
n-2e9729dba03bdbf4 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-5a3a97678ce8350c

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:                 36,                  4;
  2:               1764,                400,                  9;
  3:             112896,              39456,               1800,               16;
  4:            9144576,            4397184,             306576,             5504,              25;
  5:          914457600,          568637568,           54040608,          1475200,           13500,            36;
  6:       110649369600,        85541640192,        10412297280,        385211392,         5342500,         28656,          49;
  7:     15933509222400,     14899134670848,      2233278957312,     105183313920,      1969572800,      15974784,       54880,       64;
  8:   2692763058585600,   2983964432375808,    536010818404608,   30897202065408,    733555467200,    8012014848,    41555136,    97280,     81;
  9: 527781559482777600, 682163049518899200, 143932689484425216, 9882199518511104, 285948849600000, 3948657232896, 27527296832, 97176576, 162324, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4+14*n^3+73*n^2+168*n+144+(4*n^2+32*n+60)*x-9*x^2)*P(n-2,x) - ((4*n^3+47*n^2+182*n+231)*x-(4*n^2+32*n+69)*x^2-4*x^3)*d/dx P(n-2,x) - ((5*n^2+40*n+75)*x^2-(8*n+41)*x^3)*d^2/dx^2 P(n-2,x) - ((2*n+6)*x^3-4*x^4)*d^3/dx^3 P(n-2,x) + (2*n^2+16*n+34+4*x)*P(n-1,x) + ((4*n+17)*x-4*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 36 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 36 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 + 14*n^3 + 73*n^2 + 168*n + 144 + (4*n^2 + 32*n + 60)*x - 9*x^2)*P[n - 2, x] - ((4*n^3 + 47*n^2 + 182*n + 231)*x - (4*n^2 + 32*n + 69)*x^2 - 4*x^3)*D[P[n - 2, x], x] - ((5*n^2 + 40*n + 75)*x^2 - (8*n + 41)*x^3)*D[P[n - 2, x], {x, 2}] - ((2*n + 6)*x^3 - 4*x^4)*D[P[n - 2, x], {x, 3}] + (2*n^2 + 16*n + 34 + 4*x)*P[n - 1, x] + ((4*n + 17)*x - 4*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
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] = 36 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 14*n^3 + 73*n^2 + 168*n + 144 + (4*n^2 + 32*n + 60)*x - 9*x^2)*v[i - 2] - ((4*n^3 + 47*n^2 + 182*n + 231)*x - (4*n^2 + 32*n + 69)*x^2 - 4*x^3)*deriv(v[i - 2], x) - ((5*n^2 + 40*n + 75)*x^2 - (8*n + 41)*x^3)*deriv(deriv(v[i - 2], x), x) - ((2*n + 6)*x^3 - 4*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (2*n^2 + 16*n + 34 + 4*x)*v[i - 1] + ((4*n + 17)*x - 4*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.