Real-rooted atlas

A008277 → θ+2 (shift +1) → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
022
1527
21513230
35272272153
4203393247532898
58772202198375210325919
64140128501523484962179201243102
72114778488116189866503361061923952342673
81159755023278956198427334443711276271751178122946946
967857033666487042695804041254071762148120474857496561551227209687
104213597235972975677991276474704628732853221386110031958175150614171530892268080926
1127644437172691956470600342732154948714109267450794792183173774459337646449116407624616322803966185
12190899322131727640040149977167098093416803153424060494152143053000966101114826620813537123803765118176712309230999498754
131382958545104551353503527917368669949892568902458853177920279994647974055002198992313365479135535939018350882957953450912245992360899056943
14104801421478620036309331927414371270241250723410191593360691023386650555725515824622363371967084126390294762293639887804235264050329618948101395234917724410130289758
Technical details
Node id
n-87b0212127393f6f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f58116f4a2b7bfc5

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:      2;
  1:      5,       2;
  2:     15,      13,       2;
  3:     52,      72,      27,       2;
  4:    203,     393,     247,      53,       2;
  5:    877,    2202,    1983,     752,     103,       2;
  6:   4140,   12850,   15234,    8496,    2179,     201,      2;
  7:  21147,   78488,  116189,   86650,   33610,    6192,    395,     2;
  8: 115975,  502327,  895619,  842733,  444371,  127627,  17511,   781,    2;
  9: 678570, 3366648, 7042695, 8040412, 5407176, 2148120, 474857, 49656, 1551, 2;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n+1)*x+(n-1)*x^2)*P(n-2,x) - (x+2*x^2+x^3)*d/dx P(n-2,x) + (n+1+(n-1)*x)*P(n-1,x) - (1+2*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 5 + 2*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[(1 + (n + 1)*x + (n - 1)*x^2)*P[n - 2, x] - (x + 2*x^2 + x^3)*D[P[n - 2, x], x] + (n + 1 + (n - 1)*x)*P[n - 1, x] - (1 + 2*x + x^2)*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] = 2; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + (n + 1)*x + (n - 1)*x^2)*v[i - 2] - (x + 2*x^2 + x^3)*deriv(v[i - 2], x) + (n + 1 + (n - 1)*x)*v[i - 1] - (1 + 2*x + x^2)*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

Same fingerprint ?

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.