Real-rooted atlas

A075497 → θ+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
21918340
3104140484296
4641110057010052416
54380906262401700180621568
63280379030674732506041652947208832
726676873064874121634882480080890444882176896
82337505715164083774284768344140595021722417220648924275968
9219252367394292297979520653326002359812047176925208003084090010288048640
10218946003805483162118876662390741772038842199095897340137477341133880519751210113619867648
11231693925692203743081497409742412859157564636588744018777446163339705603581925622849208338015841247986360320
12258785933131106321397481958041888141866249808281048761549553605739165677017300361027166712852109834321460128414202813668692008960
13304020964876138813464062226563274162882779706318844174749284710068701823790617197324144027627187016285235750818799781077517441911002548149765349154816
1437452102679391817402479801437354731788013425390695347802956723659571513091907406578376790051738171184074676088659338625727227501038929190313295100276185315015149048256135168
Technical details
Node id
n-df0240fc2993b1a0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-2fc873d32a0822ba

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:        4,        2;
  2:       19,       18,        3;
  3:      104,      140,       48,        4;
  4:      641,     1100,      570,      100,        5;
  5:     4380,     9062,     6240,     1700,      180,       6;
  6:    32803,    79030,    67473,    25060,     4165,     294,      7;
  7:   266768,   730648,   741216,   348824,    80080,    8904,    448,     8;
  8:  2337505,  7151640,  8377428,  4768344,  1405950,  217224,  17220,   648,   9;
  9: 21925236, 73942922, 97979520, 65332600, 23598120, 4717692, 520800, 30840, 900, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(1+2*x+x^2)*P(n-2,x) - (2+4*x+2*x^2)*d/dx P(n-2,x) + (4+2*x)*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(1 + 2*x + x^2)*P[n - 2, x] - (2 + 4*x + 2*x^2)*D[P[n - 2, x], x] + (4 + 2*x)*P[n - 1, x] + (2 + 2*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] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + 2*x + x^2)*v[i - 2] - (2 + 4*x + 2*x^2)*deriv(v[i - 2], x) + (4 + 2*x)*v[i - 1] + (2 + 2*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.