Real-rooted atlas

A143495 → D+1 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 - 6\,x - x^{2}\right)\,P_{n-2}(x) + \left(-3\,x - x^{2}\right)\,P_{n-2}'(x) + \left(7 + 2\,x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2426
31618337
464122454235
52567384538851540
61024420238251220150610427
740962305829253136402725234773013
8163841234822101051344843885053763438529032
9655366501781445853122152847890595004970248093967195
10262144338196296503851047850053581501429722207858163686481030785747
111048576174340986297525386216680559633251929727837465894176002619085011247126450
12419430489267642404002665687483764555477450240831492599793818875416784080401501089122050937445
131677721645472681825579174535351056568530317430528340563688807638841661774401939380313931505941113681317585497797
146710886423071885221602959494540867952580491142151503187031647812114037936283453913641967525639659620236421985332169014155666739742
1526843545611670160338996383524533074484613204439986619253457699064821583224152514502454528082249596009818809007672464838591281194832058151421428484337
Technical details
Node id
n-3853df329c867d5a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-36cd35885abbd71d

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      4,       2;
   3:     16,      18,       3;
   4:     64,     122,      45,        4;
   5:    256,     738,     453,       88,       5;
   6:   1024,    4202,    3825,     1220,     150,       6;
   7:   4096,   23058,   29253,    13640,    2725,     234,      7;
   8:  16384,  123482,  210105,   134484,   38850,    5376,    343,     8;
   9:  65536,  650178, 1445853,  1221528,  478905,   95004,   9702,   480,   9;
  10: 262144, 3381962, 9650385, 10478500, 5358150, 1429722, 207858, 16368, 648, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(12+6*x+x^2)*P(n-2,x) - (3*x+x^2)*d/dx P(n-2,x) + (7+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(12 + 6*x + x^2)*P[n - 2, x] - (3*x + x^2)*D[P[n - 2, x], x] + (7 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = -(12 + 6*x + x^2)*v[i - 2] - (3*x + x^2)*deriv(v[i - 2], x) + (7 + 2*x)*v[i - 1] + 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, v[i])))};
T(10)

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.