Real-rooted atlas

A166972 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 2\,n x + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(x + 5\,x^{2} - 6\,n x^{2}\right)\,P_{n-1}'(x) + \left(3\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
2110112
3141831126
41122111066811902
513098210259825349139852
61714447154323285723674279811092924
71156120268946357872015495512276495342391137613880
81329881568038001292402100930879857170260203132273913611583720640
916821302748826049060856586286823141570347037015708724548617886021913097179861657752
10113922106171891574800320629383942747317225742922301130686618152451777984011540697393317530478614737477145176
11128185357497278688068805591856827690127611418841948635328193835816176963686889062019373958997242529950045314024382991325930717263600
12156778116860266447621987804914099619575180733652978604230170484416630895840721610334881107320076210882325050279533644105094671633080211219506404125700656288224240
13111403737374622221889814582637103913756275219007433227599947921056833398460382799321562386868101972973431819425574480832907071759510076153201421522210700156770236908975605124512299317476750025696
14122863411760904631028081041880260163836508041235295418454129948282141225958889951149650057517722390460089877376661201739105342137785253824430495209182417530111174043301468692893160224714960461636757180484099741231367212519167899008
Technical details
Node id
n-c78c031797242175 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e7b08fefd490fef1

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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: 1;
  1: 1,    1;
  2: 1,   10,       1;
  3: 1,   41,      83,         1;
  4: 1,  122,    1110,       668,          1;
  5: 1,  309,    8210,     25982,       5349,           1;
  6: 1,  714,   44715,    432328,     572367,       42798,           1;
  7: 1, 1561,  202689,   4635787,   20154955,    12276495,      342391,          1;
  8: 1, 3298,  815680,  38001292,  402100930,   879857170,   260203132,    2739136,        1;
  9: 1, 6821, 3027488, 260490608, 5658628682, 31415703470, 37015708724, 5486178860, 21913097, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(3*n^2-2*n)*x)*P(n-1,x) + (x-(6*n-5)*x^2)*d/dx P(n-1,x) + 3*x^3*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (3*n^2 - 2*n)*x)*P[n - 1, x] + (x - (6*n - 5)*x^2)*D[P[n - 1, x], x] + 3*x^3*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + (3*n^2 - 2*n)*x)*v[i - 1] + (x - (6*n - 5)*x^2)*deriv(v[i - 1], x) + 3*x^3*deriv(deriv(v[i - 1], x), 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

Same fingerprint ?

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.