Real-rooted atlas

A136124 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21214329
36094364194
43606843577251478
5252055083525980125612664
62016049104364621254022251987120696
718144048225640181416147635700445229481267440
81814400518544047415242150512558845873608134416914546640
919958400606686405993492429977664883543516315741913941387256710181212480
1023950080076794048081022204843964520014349730030181410420023138520022410750112435595840
113113510400104622278401168479734467956836642415021400564555090898146489807864724680346509681235136178560
12435891456001526982105601792805045761107193010884230490418010801796940191605267723995540821179367129030051667122413541572393600
13653837184000237765144960029182548844801899830189088772972689060212792839116413428865855789105608587640339428871302194335747241521148883104025600
1410461394944000393500975616005025855532608034288289538048147423507613204332252652728909744497662139874702968159149892331339127790822852033589404105287186215154499900083200
Technical details
Node id
n-9d3dc6e51ece4f4c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c70223fe9dad004e

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,        2;
  2:       12,       14,        3;
  3:       60,       94,       36,        4;
  4:      360,      684,      357,       72,       5;
  5:     2520,     5508,     3525,      980,     125,       6;
  6:    20160,    49104,    36462,    12540,    2225,     198,      7;
  7:   181440,   482256,   401814,   161476,   35700,    4452,    294,     8;
  8:  1814400,  5185440,  4741524,  2150512,  558845,   87360,   8134,   416,   9;
  9: 19958400, 60668640, 59934924, 29977664, 8835435, 1631574, 191394, 13872, 567, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n+1+(2*n+2)*x+x^2)*P(n-2,x) + (2*n+3+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n + 1 + (2*n + 2)*x + x^2)*P[n - 2, x] + (2*n + 3 + 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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 2*n + 1 + (2*n + 2)*x + x^2)*v[i - 2] + (2*n + 3 + 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.