Real-rooted atlas

A075501 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-24\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-72\,x + 24\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-36\,x^{2} + 24\,x^{3}\right)\,P_{n-2}''(x) + \left(6 + 4\,x\right)\,P_{n-1}(x) + \left(12\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 36 + 72\,x + 9\,x^{2}\)
\(P_{1}(x) = 216 + 1008\,x + 324\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
036729117
12161008324161564
212961296081009602523341
3777616070417496037440225036383166
4466561959552351086412096001260004536496857257
52799362370124867604544352719365670000344736823264132880696
6167961628553472012702096009667123202252124002057529681496813824812770752825
71007769634331351042350622592025459246080826686000010650165126223392017280002187010061804545202
860466176412379320324308366205446528107520002877858720005024487283240634304481642291203367980330001211467207576253
9362797056495097049088784784451686416433389817856965359296000022232270776322390529335041319033548838972340061380004791614436906153396948
10217678233659426157772801423751696640004083530792140803152849603904009393734865715213066282285056945368506368377390819528494200001058943667392169979945600435621
111306069401671320096456704257612393945088010053585313873920100966004979840003835754894619648676643024977920624438529228803234392441280975566592001726700976174545289227419627375818374328422
1278364164096855893400256512465307011271434242458658234253312003186067419924480001526262809234042883363989565563827238810835851673602536876931846409846476640000232005822048331164288027682200123480225802270267988648625
13470184984576102710342597345288394783804391587845983501008656203776994236760887552000059533398217116794881620616944349218816230229834985193472186111318444480009039832813440002722662098956851454438963260457924804247712016200025624599347965841296496
14282110990745612325429185675264015133720674989260800145096430217418506240307620248592290918400228637242938401210368761690686422191984641316775625150051123212963857545033274887721570286144000028907767557377286944912516505610754468262721057976640063342000208896289787324287199122079729
Technical details
Node id
n-8e1f40621980b6fa · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c952f3936870db53

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,           72,             9;
  1:       216,         1008,           324,             16;
  2:      1296,        12960,          8100,            960,            25;
  3:      7776,       160704,        174960,          37440,          2250,            36;
  4:     46656,      1959552,       3510864,        1209600,        126000,          4536,           49;
  5:    279936,     23701248,      67604544,       35271936,       5670000,        344736,         8232,          64;
  6:   1679616,    285534720,    1270209600,      966712320,     225212400,      20575296,       814968,       13824,        81;
  7:  10077696,   3433135104,   23506225920,    25459246080,    8266860000,    1065016512,     62233920,     1728000,     21870,     100;
  8:  60466176,  41237932032,  430836620544,   652810752000,  287785872000,   50244872832,   4063430448,   164229120,   3367980,   33000,   121;
  9: 362797056, 495097049088, 7847844516864, 16433389817856, 9653592960000, 2223227077632, 239052933504, 13190335488, 389723400, 6138000, 47916, 144;
%F Formula
The row polynomials satisfy P(n,x) = -(24*x-9*x^2)*P(n-2,x) - (72*x-24*x^2-4*x^3)*d/dx P(n-2,x) - (36*x^2-24*x^3)*d^2/dx^2 P(n-2,x) + (6+4*x)*P(n-1,x) + (12*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 36 + 72*x + 9*x^2, P(1,x) = 216 + 1008*x + 324*x^2 + 16*x^3.
%t Mathematica
P[0, x_] := 36 + 72*x + 9*x^2;
P[1, x_] := 216 + 1008*x + 324*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(24*x - 9*x^2)*P[n - 2, x] - (72*x - 24*x^2 - 4*x^3)*D[P[n - 2, x], x] - (36*x^2 - 24*x^3)*D[P[n - 2, x], {x, 2}] + (6 + 4*x)*P[n - 1, x] + (12*x - 4*x^2)*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 + 72*x + 9*x^2; v[2] = 216 + 1008*x + 324*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(24*x - 9*x^2)*v[i - 2] - (72*x - 24*x^2 - 4*x^3)*deriv(v[i - 2], x) - (36*x^2 - 24*x^3)*deriv(deriv(v[i - 2], x), x) + (6 + 4*x)*v[i - 1] + (12*x - 4*x^2)*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.