Real-rooted atlas

A166962 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
2112114
31691031174
41316217083413322
51132927830539106685189756
615412281195165230012190155349613211420
712178124879173480067581939195264552514279871146132808
81873002031229258684678233272534103700851816563692024342391818202467544
9134942515755084485700026569959158540227548510070615874422109611912858700273913691554489060200
10113979801181476533113428975104243450096347413968553809888209226717467146611851735240250854816925219130980144303264052840
11155922618661770131140012197096551612012143466547285489100250173851192893681415054344017116902707454398327255274086822865175304787114120273900929040
1212236945262539351918164265153463309859836013321831784028196285401610582069670013271620001670201559132410462187210707797510977492024349220110810167764166140243830021440390341680552304
13189478289446902199842185591072467338174172111142907795072908217054054315150482862764828420184745201054606603562181715019617371647971077914636418465354425347961746579462327490352069554112195064053153519701097646685024
1413579137163171354455903203824682220891628971030019714238512993472259287905016211999609276247961955131292442616674432975230025399289306107393232293464579814648484964475725023818134421101177805265189566826764888144861083059589756051246417327768200849533588768
Technical details
Node id
n-9bc39e02344a1087 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-53c493e1859e92ba

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,     12,         1;
  3: 1,     69,       103,          1;
  4: 1,    316,      2170,        834,           1;
  5: 1,   1329,     27830,      53910,        6685,            1;
  6: 1,   5412,    281195,    1652300,     1219015,        53496,            1;
  7: 1,  21781,   2487917,   34800675,    81939195,     26455251,       427987,           1;
  8: 1,  87300,  20312292,  586846782,  3327253410,   3700851816,    563692024,     3423918,        1;
  9: 1, 349425, 157550844, 8570002656, 99591585402, 275485100706, 158744221096, 11912858700, 27391369, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(3*n^2-2*n)*x)*P(n-1,x) + (3*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] + (3*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] + (3*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.