Real-rooted atlas

A160562 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + x - 4\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(8\,x^{2} - 8\,n x^{2}\right)\,P_{n-1}'(x) + \left(4\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
2110112
3135911128
418496682011872
511655082249707381137600
6128618447273988631631664301990784
714555305317681951405704315857205597871132333824
816801299488187608157280838704652312397027996538084011272660224
919692829483014882011479814061344440019034924991284993108074048427561159527313920
10113305629899381212062431319861523501503161123921406674172125565365624832544606143584805013252626013184
11117711043119256515941273508589861207439455950195582895949109275888959425084589852475205620857199957539226324511204354574172160
12123001824130633081900101483965935736138272780022361555765046024593119305783607598059909609455415111134328462015521822262182635303692060114594815769038848
13129253040830143752323033002653183536690248883015187992667140546040250341173506100305174803509591015619593964021650475203559674043568206388049086923771031773322854111174376539738169344
141365448689553046122300963974276265155829826875450124834085985967954632338422397346007107482461832121915375457451886821632905039067075979725668199783045190040798049701258946417129128595990568701105595092426069327872
Technical details
Node id
n-d8236c7dd97f9760 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-af987ff96e7c5701

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,   1;
  2: 1,  10,      1;
  3: 1,  35,     91,        1;
  4: 1,  84,    966,      820,          1;
  5: 1, 165,   5082,    24970,       7381,           1;
  6: 1, 286,  18447,   273988,     631631,       66430,           1;
  7: 1, 455,  53053,  1768195,   14057043,    15857205,      597871,          1;
  8: 1, 680, 129948,  8187608,  157280838,   704652312,   397027996,    5380840,        1;
  9: 1, 969, 282948, 30148820, 1147981406, 13444400190, 34924991284, 9931080740, 48427561, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(4*n^2-4*n+1)*x)*P(n-1,x) - (8*n-8)*x^2*d/dx P(n-1,x) + 4*x^3*d^2/dx^2 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[(1 + (4*n^2 - 4*n + 1)*x)*P[n - 1, x] - (8*n - 8)*x^2*D[P[n - 1, x], x] + 4*x^3*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; for(n = 1, N, my(i = n + 1); v[i] = (1 + (4*n^2 - 4*n + 1)*x)*v[i - 1] - (8*n - 8)*x^2*deriv(v[i - 1], x) + 4*x^3*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

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