Real-rooted atlas

A075497 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
22010131
312088181227
48327922442811897
564967536308054040117693
656128763523865691201040541181351
75299208234884950401464962268018207012019515
8541363294311686541760232288045505649840296888124217393
9593792001143347208963980837051456878035212240489979245841081310514069
1069485568014625059841276070656600640000166896640283120322949072185760678013014232422735
1186294466561967845683218878751232995281689631696403206340428168069846465097123260409680154161030698803
121132316139522777301975042901422694401691211653126066935705613983002880210749017620830656013378992545160134201801927207340633
13156384724582441007509544964628322336768295206363545611768135075843070111616005353542937662851360004955121602590931287516018148208114789151722093
14226583928012806319493166694476552093274112529779748126722321404895846467598200750081339561839616182524266496171281144321100523424477116641356992240242381246897625425367
Technical details
Node id
n-81363024323f6f88 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-13d19411b4b87b8f

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,         1;
  2:       20,        10,        1;
  3:      120,        88,       18,        1;
  4:      832,       792,      244,       28,       1;
  5:     6496,      7536,     3080,      540,      40,       1;
  6:    56128,     76352,    38656,     9120,    1040,      54,     1;
  7:   529920,    823488,   495040,   146496,   22680,    1820,    70,    1;
  8:  5413632,   9431168,  6541760,  2322880,  455056,   49840,  2968,   88,   1;
  9: 59379200, 114334720, 89639808, 37051456, 8780352, 1224048, 99792, 4584, 108, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + (4+2*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4 + x)*P[n - 1, x] + (4 + 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; for(n = 1, N, my(i = n + 1); v[i] = (4 + x)*v[i - 1] + (4 + 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

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.