Real-rooted atlas

A075500 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-50 - 10\,x - x^{2}\right)\,P_{n-2}(x) + \left(-25\,x - 5\,x^{2}\right)\,P_{n-2}'(x) + \left(15 + 2\,x\right)\,P_{n-1}(x) + \left(5\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 15 + 2\,x\)
\(P_{1}(x) = 175 + 60\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
015217
1175603238
21875125015043279
319375225004875300547055
4196875376250131250140005256718906
519843756037500318937552500033250840711770347
61992187594531250728437501737750016537506930012608206398693
719960937514578125001598671875531562500713343754410000131250180093863533684

All 15 rows

Technical details
Node id
n-6afa27ca0f35db12 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-76477e71dc51917c

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          15,            2;
  1:         175,           60,            3;
  2:        1875,         1250,          150,            4;
  3:       19375,        22500,         4875,          300,            5;
  4:      196875,       376250,       131250,        14000,          525,           6;
  5:     1984375,      6037500,      3189375,       525000,        33250,         840,         7;
  6:    19921875,     94531250,     72843750,     17377500,      1653750,       69300,      1260,        8;
  7:   199609375,   1457812500,   1598671875,    531562500,     71334375,     4410000,    131250,     1800,      9;
  8:  1998046875,  22266406250,  34160156250,  15420625000,   2804484375,   239951250,  10395000,   231000,   2475,   10;
  9: 19990234375, 337992187500, 716602734375, 431062500000, 103410312500, 11763675000, 695743125, 22275000, 383625, 3300, 11;
%F Formula
The row polynomials satisfy P(n,x) = -(50+10*x+x^2)*P(n-2,x) - (25*x+5*x^2)*d/dx P(n-2,x) + (15+2*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 15 + 2*x, P(1,x) = 175 + 60*x + 3*x^2.
%t Mathematica
P[0, x_] := 15 + 2*x;
P[1, x_] := 175 + 60*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(50 + 10*x + x^2)*P[n - 2, x] - (25*x + 5*x^2)*D[P[n - 2, x], x] + (15 + 2*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 15 + 2*x; v[2] = 175 + 60*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(50 + 10*x + x^2)*v[i - 2] - (25*x + 5*x^2)*deriv(v[i - 2], x) + (15 + 2*x)*v[i - 1] + 5*x*deriv(v[i - 1], 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. [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.