Real-rooted atlas

A390433 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-144 - 32\,x - x^{2} + 168\,n + 16\,n x - 73\,n^{2} - 2\,n^{2} x + 14\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(80\,x + 6\,x^{2} - 72\,n x - 2\,n x^{2} + 21\,n^{2} x - 2\,n^{3} x\right)\,P_{n-2}'(x) + \left(-16\,x^{2} - x^{3} + 8\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(20 + 2\,x - 12\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(-3\,x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2022
308311
4072604136
501152160822452989
602880059616131206006102142
701036800296438491916865000132074986679
805080320019127577678405120764896023772025488328373332

All 15 rows

Technical details
Node id
n-34db167902c8336b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c387372c4d7fcb32

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0,            2;
   3: 0,            8,             3;
   4: 0,           72,            60,             4;
   5: 0,         1152,          1608,           224,            5;
   6: 0,        28800,         59616,         13120,          600,          6;
   7: 0,      1036800,       2964384,        919168,        65000,       1320,         7;
   8: 0,     50803200,     191275776,      78405120,      7648960,     237720,      2548,       8;
   9: 0,   3251404800,   15569542656,    8095546368,   1023530560,   43410432,    707952,    4480,    9;
  10: 0, 263363788800, 1561831372800, 1000320500736, 157507833600, 8564599680, 189404096, 1817088, 7344, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-14*n^3+73*n^2-168*n+144+(2*n^2-16*n+32)*x+x^2)*P(n-2,x) - ((2*n^3-21*n^2+72*n-80)*x+(2*n-6)*x^2)*d/dx P(n-2,x) - ((n^2-8*n+16)*x^2+x^3)*d^2/dx^2 P(n-2,x) + (2*n^2-12*n+20+2*x)*P(n-1,x) + (2*n-3)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 14*n^3 + 73*n^2 - 168*n + 144 + (2*n^2 - 16*n + 32)*x + x^2)*P[n - 2, x] - ((2*n^3 - 21*n^2 + 72*n - 80)*x + (2*n - 6)*x^2)*D[P[n - 2, x], x] - ((n^2 - 8*n + 16)*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (2*n^2 - 12*n + 20 + 2*x)*P[n - 1, x] + (2*n - 3)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = -(n^4 - 14*n^3 + 73*n^2 - 168*n + 144 + (2*n^2 - 16*n + 32)*x + x^2)*v[i - 2] - ((2*n^3 - 21*n^2 + 72*n - 80)*x + (2*n - 6)*x^2)*deriv(v[i - 2], x) - ((n^2 - 8*n + 16)*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 12*n + 20 + 2*x)*v[i - 1] + (2*n - 3)*x*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, v[i])))};
T(10)

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.