Real-rooted atlas

A109983 → D

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x - 2\,n - 2\,n x + n^{2}\right)\,P_{n}(x) = \left(-6\,x - 14\,x^{2} - 8\,x^{3} + 6\,n x + 8\,n x^{2} - n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x^{2} - 7\,x^{3} - 4\,x^{4} + 2\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27Pn(1)
00
1145
202182444
300348150120321
400041005409805602184
500005180147044805670252014325
600000629433601512031500304921108891860
7000000744868044200012705019958415615648048580097
8000000086481260010164041580093693611771767722002059203622928
9000000009900217802217601171170353152863063006589440371943087516022437477
1000000000010121035640446160294294011351340269068803967392035443980175518203695120138049020
11000000000011158455770840840675675032288256972011041890345602369495701847560008147739615519504844881345
1200000000000122028840841501500144144008331523230875644875823862412471030001357956600938929992373173528648997445148375192
1300000000000013254812285025625602892890019848628888461172826604864005499724230782441660075256661484672781568169009750027041560031258302933
14000000000000014315017472042078405513508044230586423279256008380532160211259248203762833074047117214144405623400002285011820075820374001123264800189199514532
Technical details
Node id
n-8faa874efd7868bc · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5fca0e1aa4d16710

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, intercept -1)
  • 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: 0;
  1: 1, 4;
  2: 0, 2, 18, 24;
  3: 0, 0,  3, 48, 150, 120;
  4: 0, 0,  0,  4, 100, 540,  980,  560;
  5: 0, 0,  0,  0,   5, 180, 1470, 4480,  5670,  2520;
  6: 0, 0,  0,  0,   0,   6,  294, 3360, 15120, 31500,  30492,  11088;
  7: 0, 0,  0,  0,   0,   0,    7,  448,  6804, 42000, 127050, 199584,  156156,   48048;
  8: 0, 0,  0,  0,   0,   0,    0,    8,   648, 12600, 101640, 415800,  936936, 1177176,  772200,  205920;
  9: 0, 0,  0,  0,   0,   0,    0,    0,     9,   900,  21780, 221760, 1171170, 3531528, 6306300, 6589440, 3719430, 875160;
%F Formula
The row polynomials satisfy (n^2-2*n+1-(2*n-2)*x)*P(n,x) = -((n^2-6*n+6)*x-(8*n-14)*x^2+8*x^3)*P(n-1,x) + ((2*n-3)*x^2+(2*n-7)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((n^2 - 6*n + 6)*x - (8*n - 14)*x^2 + 8*x^3)*P[n - 1, x] + ((2*n - 3)*x^2 + (2*n - 7)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(n^2 - 2*n + 1 - (2*n - 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-((n^2 - 6*n + 6)*x - (8*n - 14)*x^2 + 8*x^3)*v[i - 1] + ((2*n - 3)*x^2 + (2*n - 7)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(n^2 - 2*n + 1 - (2*n - 2)*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

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.