Real-rooted atlas

A088996 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n - 12\,n x - 4\,n x^{2} + 12\,n^{2} + 20\,n^{2} x + 8\,n^{2} x^{2} - 4\,n^{3} - 8\,n^{3} x - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 - 6\,n - 4\,n x + 4\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21326847
38326824448643
470130844652267238411493
57359407268514881928340163840253017
692289613262163121221949521535104496512460806609411
7134528710410968335110005774380856774896310265608195328645120199652967
822341465197051544742847016155652294419679521441516990848678226688151123968103219206843378537
941629135541176825381776596100043759007120674952349926709208361642406644736160110627843081498624185794560262331261325
108598724245941984339224572654158401292727259920234988791668828561755666242332621321344124706872985640695668121668875960320371589120011118091901175
111949772487952342106939492126237381051964024412504753684138392538064120746578585792120781994371648835570530982403865467378176011097217216512167505444864081749606400516137660988075
12481446017608562897384096676372487090982580132093097945451231194201825924325156681381954240609885862868883251742676565844483100886984332288126308727316070432347887619276844043538268160196199055360026043816427037325
131285835921259751814564443849470117066983159076844568309728881570412017197178785598422467449189429430430644871220349356830773655773058470422639331081714432011956500836110131243474791423270912100445301799649281245193513205760510117543936001419138523306008225
143693120887658825559725572427327903906030728315503081662588892633824984481821221513306475210045101082663084576155111361994650784641798008367824750809615669991875848327168101670750214687744004796440916666335232157417764250376601633116420966606438437671064861409280142832912302080083045339879242200075
Technical details
Node id
n-93c4bc52c48c6619 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7c9f7ca6f2f9d8f8

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:         3,          2;
  2:        13,         26,           8;
  3:        83,        268,         244,          48;
  4:       701,       3084,        4652,        2672,         384;
  5:      7359,      40726,       85148,       81928,       34016,        3840;
  6:     92289,     613262,     1631212,     2194952,     1535104,      496512,       46080;
  7:   1345287,   10410968,    33511000,    57743808,    56774896,    31026560,     8195328,      645120;
  8:  22341465,  197051544,   742847016,  1556522944,  1967952144,  1516990848,   678226688,   151123968,   10321920;
  9: 416291355, 4117682538, 17765961000, 43759007120, 67495234992, 67092083616, 42406644736, 16011062784, 3081498624, 185794560;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(8*n^3-20*n^2+12*n)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (4*n^2-6*n+1+(4*n^2-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (8*n^3 - 20*n^2 + 12*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (4*n^2 - 6*n + 1 + (4*n^2 - 4*n)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (8*n^3 - 20*n^2 + 12*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (4*n^2 - 6*n + 1 + (4*n^2 - 4*n)*x)*v[i - 1])/(n - 1)); 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. [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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.