Real-rooted atlas

A166961 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3\,x^{2} + 4\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-3\,x^{2} + 2\,x^{3} - 6\,n x^{2} + 4\,n x^{3}\right)\,P_{n-2}'(x) + \left(-2\,x^{3} - 4\,n x^{3}\right)\,P_{n-2}''(x) + \left(1 - 2\,x + 4\,n x\right)\,P_{n-1}(x) + \left(5\,x - 2\,x^{2}\right)\,P_{n-1}'(x) + \left(2\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
2118322
311181264249
41722277561654119
51435049206357482535691846
61261227968151460200337500952272630167
71156758123832385044738027964250265345233964893639051

All 15 rows

Technical details
Node id
n-1a6a5162ada875b1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-572d7783088b9b6d

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: 1,       2;
  2: 1,      18,          3;
  3: 1,     118,        126,           4;
  4: 1,     722,       2775,         616,            5;
  5: 1,    4350,      49206,       35748,         2535,           6;
  6: 1,   26122,     796815,     1460200,       337500,        9522,          7;
  7: 1,  156758,   12383238,    50447380,     27964250,     2653452,      33964,         8;
  8: 1,  940578,  188805351,  1594147464,   1825924275,   410027532,   18569194,    117408,      9;
  9: 1, 5643502, 2853243270, 47908755076, 104086120005, 46781799282, 5038354832, 120198640, 397629, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(4*n^2-4*n-3)*x^2*P(n-2,x) - ((6*n+3)*x^2-(4*n+2)*x^3)*d/dx P(n-2,x) - (4*n+2)*x^3*d^2/dx^2 P(n-2,x) + (1+(4*n-2)*x)*P(n-1,x) + (5*x-2*x^2)*d/dx P(n-1,x) + 2*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 4*n - 3)*x^2*P[n - 2, x] - ((6*n + 3)*x^2 - (4*n + 2)*x^3)*D[P[n - 2, x], x] - (4*n + 2)*x^3*D[P[n - 2, x], {x, 2}] + (1 + (4*n - 2)*x)*P[n - 1, x] + (5*x - 2*x^2)*D[P[n - 1, x], x] + 2*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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 4*n - 3)*x^2*v[i - 2] - ((6*n + 3)*x^2 - (4*n + 2)*x^3)*deriv(v[i - 2], x) - (4*n + 2)*x^3*deriv(deriv(v[i - 2], x), x) + (1 + (4*n - 2)*x)*v[i - 1] + (5*x - 2*x^2)*deriv(v[i - 1], x) + 2*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.