Real-rooted atlas

A286785 → reverse

nonnegative coefficients

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

Recurrence

\(\left(56 + 30\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(56 + 182\,x + 2\,x^{2} + 30\,n + 129\,n x + 3\,n x^{2} + 4\,n^{2} + 22\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(28\,x - 26\,x^{2} - 2\,x^{3} + 8\,n x - 7\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2 + 14\,x + 14\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02141430
12277242143
2244220330132728
3265520143014304293876
42901050455081906006143021318
521191904119003332043316247524862120175
6215231922713210852821705621705610077616796690690
7218950405586030164485465813023361046520406980587864032015

All 15 rows

Technical details
Node id
n-23bacd54faa778ef · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-ca8fcf8c70f1b4a8

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: 2,  14,    14;
  1: 2,  27,    72,     42;
  2: 2,  44,   220,    330,     132;
  3: 2,  65,   520,   1430,    1430,     429;
  4: 2,  90,  1050,   4550,    8190,    6006,     1430;
  5: 2, 119,  1904,  11900,   33320,   43316,    24752,     4862;
  6: 2, 152,  3192,  27132,  108528,  217056,   217056,   100776,    16796;
  7: 2, 189,  5040,  55860,  301644,  854658,  1302336,  1046520,   406980,    58786;
  8: 2, 230,  7590, 106260,  743820, 2826516,  6056820,  7354710,  4903140,  1634380,  208012;
  9: 2, 275, 11000, 189750, 1669800, 8182020, 23377200, 39657750, 39657750, 22472725, 6537520, 742900;
%F Formula
The row polynomials satisfy (4*n^2+30*n+56)*P(n,x) = (4*n^2+30*n+56+(22*n^2+129*n+182)*x+(n^2+3*n+2)*x^2)*P(n-1,x) + ((8*n+28)*x-(7*n+26)*x^2-(n+2)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + 14*x + 14*x^2.
%t Mathematica
P[0, x_] := 2 + 14*x + 14*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 30*n + 56 + (22*n^2 + 129*n + 182)*x + (n^2 + 3*n + 2)*x^2)*P[n - 1, x] + ((8*n + 28)*x - (7*n + 26)*x^2 - (n + 2)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 30*n + 56)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2 + 14*x + 14*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 30*n + 56 + (22*n^2 + 129*n + 182)*x + (n^2 + 3*n + 2)*x^2)*v[i - 1] + ((8*n + 28)*x - (7*n + 26)*x^2 - (n + 2)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 30*n + 56)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.