Real-rooted atlas

A075501 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-2}(x) + \left(-6\,x^{2}\right)\,P_{n-2}'(x) + \left(6 + 2\,x\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 36 + 36\,x + 3\,x^{2}\)
\(P_{1}(x) = 216 + 504\,x + 108\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
03636375
12165041084832
2129664802700240510721
37776803525832093604506156264
44665697977611702883024002520075672525083
5279936118506242253484888179841134000574561176844676032
6167961614276736042340320024167808045042480342921611642417289858118113
7100776961716567552783540864063648115201653372000177502752889056021600024301017766849160
8604661762061896601614361220684816320268800057557174400837414547258049006420528640374220330011394027043147
9362797056247548524544261594817228841083474544641930718592000370537846272341504190721648791936433026006138004356129309306518400
10217678233629713078886404745838988800010208826980352063056992078080156562247761921866611755008118171063296419323132884942000962676561613233222423176705
11130606940163566004822835285870797981696025133963284684802019320099596800639292482436608966632892825607805481615360359376937920975566592015697281614545447098146171228061177448
127836416409642794670012825615510233709047808614664558563328006372134839848960025437713487234048480569937937689648513544814592028187521464960984647664000210914383682759702402129400882015171883804881595227
1347018498457651355171298672642798261268130529281495875252164050944198847352177510400099222330361861324823151670633560268828778729373149184206790353827200090398328134400247514736268842878699136465060960303408010800165023990447754998912
142821109907456616271459283763205044573558329753600362741075543546265606152404971845818368038106207156400201728108812955203170283521645969531437563904144042861611480832772157028614400026279788688524857874270970888272667894475569760042228001305617153689866107033861185
Technical details
Node id
n-5562101454b732f4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-867434bb7cffd0f7

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 2)
  • 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:        36,           36,             3;
  1:       216,          504,           108,             4;
  2:      1296,         6480,          2700,           240,             5;
  3:      7776,        80352,         58320,          9360,           450,            6;
  4:     46656,       979776,       1170288,        302400,         25200,          756,           7;
  5:    279936,     11850624,      22534848,       8817984,       1134000,        57456,        1176,          8;
  6:   1679616,    142767360,     423403200,     241678080,      45042480,      3429216,      116424,       1728,        9;
  7:  10077696,   1716567552,    7835408640,    6364811520,    1653372000,    177502752,     8890560,     216000,     2430,     10;
  8:  60466176,  20618966016,  143612206848,  163202688000,   57557174400,   8374145472,   580490064,   20528640,   374220,   3300,   11;
  9: 362797056, 247548524544, 2615948172288, 4108347454464, 1930718592000, 370537846272, 34150419072, 1648791936, 43302600, 613800, 4356, 12;
%F Formula
The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 6*x^2*d/dx P(n-2,x) + (6+2*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 36 + 36*x + 3*x^2, P(1,x) = 216 + 504*x + 108*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 36 + 36*x + 3*x^2;
P[1, x_] := 216 + 504*x + 108*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 6*x^2*D[P[n - 2, x], x] + (6 + 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] = 36 + 36*x + 3*x^2; v[2] = 216 + 504*x + 108*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] - 6*x^2*deriv(v[i - 2], x) + (6 + 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.