Real-rooted atlas

A075501 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-72 - 12\,x - x^{2}\right)\,P_{n-2}(x) + \left(-36\,x - 6\,x^{2}\right)\,P_{n-2}'(x) + \left(18 + 2\,x\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 18 + 2\,x\)
\(P_{1}(x) = 252 + 72\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
018220
1252723327
23240180018045224
340176388807020360586441
44898887801922268002016063061517676
55925312150232326613488907200478801008728518127
6713836802822688001812585603603398428576809979215128573904016
785828377652236057604773608640132269760014791896076204801890002160912333926385
8103094830089574147123212240201600046045739520697845456049756291217962560332640297010281993025412
9123774262272174396544819230812605908481544574873600308781538560292717877761442692944384912005524203960116833110241783
101485653944320316389265920007656620235264050445593662464130468539801601599952932864103399680384372731673676447800875160514812174890387789688
111783002411417657247198654464018850472463513601615456079677440532743735363840828542479564806829796413440319446167040878009932814270256013333326552134713561486730201
1221397335006412810340155806031872460998418922496005097707871879168021198094572695040411917089660876842449351712768025055574635520886182897600191740348802529727201965600819014133398769940083292
1325677585649336321865507512087019521121906439123038208159077881742008320082685275301551104019844289114480230425181388201505536183813647846400081358495320960225013396608039305474208429287040281736010080153954202583523915615
1430813572964188160336304903888650240027205580665765969920492192397747665469443175517263033350144093268247317003100161440223340007868416128038099210205184694941325752960023890716989568053051415056647636308825670171920039412801224016122476135556272784416
Technical details
Node id
n-a43c161a6b378a05 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-bdc380a726c17e48

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           18,             2;
  1:          252,            72,             3;
  2:         3240,          1800,           180,             4;
  3:        40176,         38880,          7020,           360,            5;
  4:       489888,        780192,        226800,         20160,          630,           6;
  5:      5925312,      15023232,       6613488,        907200,        47880,        1008,          7;
  6:     71383680,     282268800,     181258560,      36033984,      2857680,       99792,       1512,        8;
  7:    858283776,    5223605760,    4773608640,    1322697600,    147918960,     7620480,     189000,     2160,      9;
  8:  10309483008,   95741471232,  122402016000,   46045739520,   6978454560,   497562912,   17962560,   332640,   2970,   10;
  9: 123774262272, 1743965448192, 3081260590848, 1544574873600, 308781538560, 29271787776, 1442692944, 38491200, 552420, 3960, 11;
%F Formula
The row polynomials satisfy P(n,x) = -(72+12*x+x^2)*P(n-2,x) - (36*x+6*x^2)*d/dx P(n-2,x) + (18+2*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 18 + 2*x, P(1,x) = 252 + 72*x + 3*x^2.
%t Mathematica
P[0, x_] := 18 + 2*x;
P[1, x_] := 252 + 72*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(72 + 12*x + x^2)*P[n - 2, x] - (36*x + 6*x^2)*D[P[n - 2, x], x] + (18 + 2*x)*P[n - 1, x] + 6*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] = 18 + 2*x; v[2] = 252 + 72*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(72 + 12*x + x^2)*v[i - 2] - (36*x + 6*x^2)*deriv(v[i - 2], x) + (18 + 2*x)*v[i - 1] + 6*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.