Real-rooted atlas

A136124 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
2117119
35044121107
4274296113181702
51764220410452352515274
6130681823610099284543033144712
7109584166984104132345246580721421422568
8102657616817041153032433272977691358011345214407120
91062864018507312137210285700428146561124219325746169863150292720
1012054384022120891217519235679081500225833154270623542283456902445751623471040
111486442880285567782423938081201159137496361109870761417361110463211213737689034108818344624320
12198027590403961769587234899301656179524921846020876026138971236822673225326378814217407312386046311021119938250880
13283465647360588026239488541228897872293526966512104909686024261480384884683717753611318279582028593995563192335614911711842662908800
14433916300160093001465958408902329149232505925321428819117740438165091918431609875154789514187202601151906211612074762770190122894718008133130136443724800
Technical details
Node id
n-6d8f246bcb836a95 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-42f458d827ed1a04

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:        3,        1;
  2:       11,        7,        1;
  3:       50,       44,       12,       1;
  4:      274,      296,      113,      18,       1;
  5:     1764,     2204,     1045,     235,      25,      1;
  6:    13068,    18236,    10099,    2845,     430,     33,     1;
  7:   109584,   166984,   104132,   34524,    6580,    721,    42,    1;
  8:  1026576,  1681704,  1153032,  433272,   97769,  13580,  1134,   52,  1;
  9: 10628640, 18507312, 13721028, 5700428, 1465611, 242193, 25746, 1698, 63, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n*x+x^2)*P(n-2,x) + (2*n+1+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n*x + x^2)*P[n - 2, x] + (2*n + 1 + 2*x)*P[n - 1, 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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 2*n*x + x^2)*v[i - 2] + (2*n + 1 + 2*x)*v[i - 1]); 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.