Real-rooted atlas

A176230 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(6 + 4\,x\right)\,P_{n-1}'(x) + \left(4 + 4\,x\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
2108119
37678181173
47649283003212025
5949613100532082050128787
6140152214176103140204801830721479851
72390480397952821935765273806146035709819156093
846206736827962885101532814287616203812015545663281281196506001
9997313824190619380812908592004102230726883329664668243470881044016214680247715
1023758664096480970144003534815304012507325440240222360026700307217719800704640162902001122398824579
116188846389121319479621856104219294608040477105416087244064160111589595048793412324334748013272602431024213484655325197
12174921905776003909412659148832929508437056138832770841603308170702320477200782080434221085762547781632968754602353120349802881107230543328761
1353298520820057612438457688874881110279042208128503659740748736131133088783840210084641601122162909850816146883381216665890711220111234039685364882833813545227970258067
141741127736739110442289474443657728397947335640400001928377493793587254345355785680969554096450778881095812255297768438554375680443495316264159894454723926762846418048664303921125287685200499035
Technical details
Node id
n-83eeed08410aaf7f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6ec6238b9895326e

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:         2,          1;
  2:        10,          8,          1;
  3:        76,         78,         18,         1;
  4:       764,        928,        300,        32,        1;
  5:      9496,      13100,       5320,       820,       50,       1;
  6:    140152,     214176,     103140,     20480,     1830,      72,      1;
  7:   2390480,    3979528,    2193576,    527380,    61460,    3570,     98,     1;
  8:  46206736,   82796288,   51015328,  14287616,  2038120,  155456,   6328,   128,   1;
  9: 997313824, 1906193808, 1290859200, 410223072, 68833296, 6466824, 347088, 10440, 162, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (6+4*x)*d/dx P(n-1,x) + (4+4*x)*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[(2 + x)*P[n - 1, x] + (6 + 4*x)*D[P[n - 1, x], x] + (4 + 4*x)*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] = (2 + x)*v[i - 1] + (6 + 4*x)*deriv(v[i - 1], x) + (4 + 4*x)*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

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.