Real-rooted atlas

A086915 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
022
1448
28241244
3169614448304
4323209609602402512
564960480096007200144024064
6128268820160672001008006048010080261536
72567168752643763209408001128960564480806403173888
851218432258048180633667737601354752013547520580608072576042483968
9102446080829440774144040642560121927680203212800174182400653184007257600621159424
102048112640253440030412800212889600894136320223534080031933440002395008000798336000798336009841950208
1140962703367434240111513600100362240056202854401967099904042152140800526901760003512678400010538035200958003200167879268352
12819263897621086208386580480434903040031313018880146127421440438382264320821966745600913296384000547977830400149448499200124540416003065723549696
13163841490944581468161279229952175894118401583047065609498282393603799312957440997319651328016621994188800166219941888009066542284800226663557120017435658240059651093528576
14327683440640156549120407027712067159572480738755297280554066472960028494847180800997319651328002327079186432003490618779648003173289799680001586644899840003661488230400026153487360001231571119812608
Technical details
Node id
n-c4d9ef50e4651641 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-269631c8dfab89fe

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:    2;
  1:    4,     4;
  2:    8,    24,     12;
  3:   16,    96,    144,      48;
  4:   32,   320,    960,     960,      240;
  5:   64,   960,   4800,    9600,     7200,      1440;
  6:  128,  2688,  20160,   67200,   100800,     60480,     10080;
  7:  256,  7168,  75264,  376320,   940800,   1128960,    564480,     80640;
  8:  512, 18432, 258048, 1806336,  6773760,  13547520,  13547520,   5806080,   725760;
  9: 1024, 46080, 829440, 7741440, 40642560, 121927680, 203212800, 174182400, 65318400, 7257600;
%F Formula
The row polynomials satisfy 2*P(n,x) = (4+4*n*x+(n^2-n)*x^2)*P(n-1,x) - (4*x^2+(2*n-2)*x^3)*d/dx P(n-1,x) + x^4*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4 + 4*n*x + (n^2 - n)*x^2)*P[n - 1, x] - (4*x^2 + (2*n - 2)*x^3)*D[P[n - 1, x], x] + x^4*D[P[n - 1, x], {x, 2}])/(2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((4 + 4*n*x + (n^2 - n)*x^2)*v[i - 1] - (4*x^2 + (2*n - 2)*x^3)*deriv(v[i - 1], x) + x^4*deriv(deriv(v[i - 1], x), x))/(2)); 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.