Real-rooted atlas

A176230 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 3\,x + 6\,x^{2} + 4\,n x - 10\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x^{2} + 14\,x^{3} - 8\,n x^{3}\right)\,P_{n-1}'(x) + \left(4\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
216310
3115451576
4128210420105764
5145630315047259459496
6166148513860519756237010395140152
71913003450453153159459459459451351352390480
811205460120120135135075675601891890016216200202702546206736
91153918027846045945904135131019297278041351310031013482534459425997313824
1011901453558140013226850174594420130945815052378326009820936125654729075065472907523758664096
1112312194511191953357585061108047067218851704321211895015124241632525207069387515124241632513749310575618884638912
12127631878201894077224455185338692028109701620265034329560149081810377546381007673006957151150950379480971870031623414322517492190577600
1313254485034534501640388755019589575100391791500130509328950010767019638375538350981918751507382749372502055521930962501027760965481257905853580625532985208200576
14137861425565110032635102512401338950316234143225542115674100061665657928875452214824811750203496671165287551799152660255006474894082531875298841265347625021345804667687517411277367391104
Technical details
Node id
n-01922c46a12adf16 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-86230d7253d1861d

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,   6,    3;
  3: 1,  15,   45,     15;
  4: 1,  28,  210,    420,     105;
  5: 1,  45,  630,   3150,    4725,      945;
  6: 1,  66, 1485,  13860,   51975,    62370,     10395;
  7: 1,  91, 3003,  45045,  315315,   945945,    945945,    135135;
  8: 1, 120, 5460, 120120, 1351350,  7567560,  18918900,  16216200,   2027025;
  9: 1, 153, 9180, 278460, 4594590, 41351310, 192972780, 413513100, 310134825, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (1+(4*n-3)*x+(4*n^2-10*n+6)*x^2)*P(n-1,x) - (4*x^2+(8*n-14)*x^3)*d/dx P(n-1,x) + 4*x^4*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 + (4*n - 3)*x + (4*n^2 - 10*n + 6)*x^2)*P[n - 1, x] - (4*x^2 + (8*n - 14)*x^3)*D[P[n - 1, x], x] + 4*x^4*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 + (4*n - 3)*x + (4*n^2 - 10*n + 6)*x^2)*v[i - 1] - (4*x^2 + (8*n - 14)*x^3)*deriv(v[i - 1], x) + 4*x^4*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.