Real-rooted atlas

A075500 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-2}(x) + \left(-5\,x^{2}\right)\,P_{n-2}'(x) + \left(5 + 2\,x\right)\,P_{n-1}(x) + \left(5\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 25 + 30\,x + 3\,x^{2}\)
\(P_{1}(x) = 125 + 350\,x + 90\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02530358
1125350904569
26253750187520056455
3312538750337506500375682506
4156253937505643751750001750063071166887
57812539687509056250425250065625039900980818052763
639062539843750141796875971250002172187519845008085014409302944924
71953125399218750218671875021315625006644531258560125051450001500002025105474804535
897656253996093750333996093754554687500019275781250336538125027994312511880000259875275011105885592011
9488281253998046875050698828125095547031250053882812500012409237500013724287500795135000250593754262503630122179953302392
102441406253999023437507664794921875197853906250001466504101562543693650000006251245000004749030000020221987504915625066852546801347559424875093
111220703125399951171875011557177734375040592753906250039135776367187514867899921875026976950000000261404000000014442553125047047000009084075010101005915141095277023806779
12610351562539997558593750173957592773437582726464843750001029135351562500049299992929687501117652082421875135392400000000943994390625039570781250010171411250159705000147875073501526536469349555490
13305175781253999877929687502615363525390625016777236425781250026762464599609375016024960300781250044869488726562500669301266562500057711392578125030274217187500994706212500206783850002691325002107000900016674371172017174141
1415258789062539999389648437503929045104980468753390318798828125000690033160522460937551286376654296875001757389975605468750318999922312500000334997659089843752154948554687500880104805937502325837150000398951962504373250002932500108801717928327660568052147
Technical details
Node id
n-ed64ed06df6a14fd · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3ba766e796b1031e

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:       25,          30,            3;
  1:      125,         350,           90,            4;
  2:      625,        3750,         1875,          200,            5;
  3:     3125,       38750,        33750,         6500,          375,            6;
  4:    15625,      393750,       564375,       175000,        17500,          630,           7;
  5:    78125,     3968750,      9056250,      4252500,       656250,        39900,         980,         8;
  6:   390625,    39843750,    141796875,     97125000,     21721875,      1984500,       80850,      1440,        9;
  7:  1953125,   399218750,   2186718750,   2131562500,    664453125,     85601250,     5145000,    150000,     2025,     10;
  8:  9765625,  3996093750,  33399609375,  45546875000,  19275781250,   3365381250,   279943125,  11880000,   259875,   2750,   11;
  9: 48828125, 39980468750, 506988281250, 955470312500, 538828125000, 124092375000, 13724287500, 795135000, 25059375, 426250, 3630, 12;
%F Formula
The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 5*x^2*d/dx P(n-2,x) + (5+2*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 25 + 30*x + 3*x^2, P(1,x) = 125 + 350*x + 90*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 25 + 30*x + 3*x^2;
P[1, x_] := 125 + 350*x + 90*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 5*x^2*D[P[n - 2, x], x] + (5 + 2*x)*P[n - 1, x] + 5*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] = 25 + 30*x + 3*x^2; v[2] = 125 + 350*x + 90*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] - 5*x^2*deriv(v[i - 2], x) + (5 + 2*x)*v[i - 1] + 5*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.