Real-rooted atlas

A166979 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-10 + 7\,x + 8\,x^{2}\right)\,P_{n-2}(x) + \left(-9\,x + 72\,x^{2}\right)\,P_{n-2}'(x) + \left(11 - 7\,x\right)\,P_{n-1}(x) + \left(9\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 12 + 13\,x + x^{2}\)
\(P_{2}(x) = 112 + 171\,x + 33\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
11213126
2112171331317
3111224738556214503
41111239211209652685100174074
5111112667233513831101870652514711400719
611111121189965112830013365430135546013482203129864223
71111111221831559332798585512821050217654021998522249062681704300780
811111111240702184918573101605447341812583346354065692683240781842390342118129456107
91111111112765563735532282806339111565067777903826160332540232469692027551475184480677704251504947754483
101111111111214467933197316166633073293551087673830117305656436202355165987241563917840154384458710229340103125517115117689840752
111111111111122741129529711316836979053745519553896047314277727670847541133929186431041137578539812515872688761307876427188291951507776181483742239575173
121111111111112520036832867371463256469360184569907254655251653483080364143380747560213925525794703840490564564439571120149622066594288222849232754007213291728116459266042057687
13111111111111129872922046702273128156145366693991251627802030769710156146372506886125412391734892647515396870947961493384250297413840158220313765803912872063105913611704543650382934758471592850867083657250
141111111111111121875077411095654113558797748571769373910978418077477750170132554932185909802258687807231217726359255534037318919311912015931417331579399206133320487271019449509398251322461143161455886733920394380975122520219895299447105
151111111111111112356186930330396503399084258181125290655331380547281045090982315810034796328769061123201776516286059046235793429717529227182464449641989258924150981185946358053568973847208481013611063471748621022692458042101196351813377266051930011121899240198961411130275
Technical details
Node id
n-2c4a421bbeed7ecb · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-87f160794a2bbf3d

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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:          12,            13,             1;
   2:         112,           171,            33,             1;
   3:        1112,          2473,           855,            62,             1;
   4:       11112,         39211,         20965,          2685,           100,            1;
   5:      111112,        667233,        513831,        101870,          6525,          147,           1;
   6:     1111112,      11899651,      12830013,       3654301,        355460,        13482,         203,         1;
   7:    11111112,     218315593,     327985855,     128210502,      17654021,       998522,       24906,       268,        1;
   8:   111111112,    4070218491,    8573101605,    4473418125,     833463540,     65692683,     2407818,     42390,      342,      1;
   9:  1111111112,   76556373553,  228280633911,  156506777790,   38261603325,   4023246969,   202755147,   5184480,    67770,    425,   1;
  10: 11111111112, 1446793319731, 6166633073293, 5510876738301, 1730565643620, 235516598724, 15639178401, 543844587, 10229340, 103125, 517, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(10-7*x-8*x^2)*P(n-2,x) - (9*x-72*x^2)*d/dx P(n-2,x) + (11-7*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 12 + 13*x + x^2, P(2,x) = 112 + 171*x + 33*x^2 + x^3.
%t Mathematica
P[1, x_] := 12 + 13*x + x^2;
P[2, x_] := 112 + 171*x + 33*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(10 - 7*x - 8*x^2)*P[n - 2, x] - (9*x - 72*x^2)*D[P[n - 2, x], x] + (11 - 7*x)*P[n - 1, x] + 9*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] = 12 + 13*x + x^2; v[2] = 112 + 171*x + 33*x^2 + x^3; for(n = 3, N, my(i = n); v[i] = -(10 - 7*x - 8*x^2)*v[i - 2] - (9*x - 72*x^2)*deriv(v[i - 2], x) + (11 - 7*x)*v[i - 1] + 9*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

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.