Real-rooted atlas

A225470 → basis A008277 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 + 8\,x - x^{2} + 21\,n - 6\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(4\,x - x^{2} - 3\,n x\right)\,P_{n-2}'(x) + \left(-4 + 2\,x + 6\,n\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21820341
3162236634465
419443392123614456721
529160581602610042402756117941
652488011633606092401238401135046872433145
7110224802664368015757560378448043855025788735857673281
826453952068813696044966208012319104017010000127411252136108891543866945
97142567040197969139201407240576042952806406812988006118358432069529657615391046052954821
1021427701120062798946560048001235520016050746880028533260000295898400018683112072384001671302100111514472783561
117071141369600217801958528001774241663040064182853504001256364956000146681304000107385616805030572801499553027390027831254426342354385
12254561089305600820009529292800706882125734400273969684172800582774549280007521571411200622147778560339041049601225742760289872004293083600132121878761891201
13992788248291840033309503350323200302121814490496001244923623623680028483926671200004008820880064003677181668160022706148608009575202780027533220005292201264833645631489187219264121525
144169710642825728001452164409029734400137930023878366720060049853922938880014658760798281600022259529336844800223274297005760015328506152960073420431084002468105640005780598824920949129487665684154020175011403931801
Technical details
Node id
n-5ec41c80b7b980a7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5e6379f04e944553

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:         18,          20,           3;
  3:        162,         236,          63,          4;
  4:       1944,        3392,        1236,        144,         5;
  5:      29160,       58160,       26100,       4240,       275,        6;
  6:     524880,     1163360,      609240,     123840,     11350,      468,       7;
  7:   11022480,    26643680,    15757560,    3784480,    438550,    25788,     735,     8;
  8:  264539520,   688136960,   449662080,  123191040,  17010000,  1274112,   52136,  1088,    9;
  9: 7142567040, 19796913920, 14072405760, 4295280640, 681298800, 61183584, 3206952, 96576, 1539, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-21*n+12+(6*n-8)*x+x^2)*P(n-2,x) - ((3*n-4)*x+x^2)*d/dx P(n-2,x) + (6*n-4+2*x)*P(n-1,x) + x*d/dx 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[-(9*n^2 - 21*n + 12 + (6*n - 8)*x + x^2)*P[n - 2, x] - ((3*n - 4)*x + x^2)*D[P[n - 2, x], x] + (6*n - 4 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], 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] = -(9*n^2 - 21*n + 12 + (6*n - 8)*x + x^2)*v[i - 2] - ((3*n - 4)*x + x^2)*deriv(v[i - 2], x) + (6*n - 4 + 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 - 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. [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.