Real-rooted atlas

A099040 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(2 + 6\,x - 3\,n - 6\,n x + n^{2}\right)\,P_{n}(x) = \left(-28 - 90\,x - 72\,x^{2} + 18\,n + 30\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-6 - 36\,x - 66\,x^{2} - 36\,x^{3} + 2\,n + 8\,n x + 6\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
166
23072102
31205766481344
443230967128518415840
5146413968486007257638880175488
64776568802650326013446609602799361868928
7151682165761264896385689664540805598720195955219365888
84723278624055209602115590447666880629856004506969613436928196669440
9144864275443222597056104312448295837920526839552574148736349360128906992641966694400
104390089386496881020804758808321628022240365288486453672129284971663360263027865660466176019428532224
111317504312883203306666242045979648819916992022211241984411407942405138281267241358864384193491763203990767616190042079232
1239221761024277761203610752839387059238568337920122484037248274064902272431621001216468370999296333168629760139676866560261213880321843779993600
13115971843303083524271999616331539194881718338924806261166149121643278347648312741811814442797354711044103234703360261395278848099261274521616978902220817765739528192
143409267210517045761484584588812690046310473241288640030120973632009074891064960202655748096003351972699648040526849802240347875213086722006122600857669613499105281097098297344170181580554240
Technical details
Node id
n-3ef784a499ce7f14 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-26250b2c4c73936c

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, 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:      6;
  2:     30,      72;
  3:    120,     576,      648;
  4:    432,    3096,     7128,      5184;
  5:   1464,   13968,    48600,     72576,     38880;
  6:   4776,   56880,   265032,    601344,    660960,    279936;
  7:  15168,  216576,  1264896,   3856896,   6454080,   5598720,   1959552;
  8:  47232,  786240,  5520960,  21155904,  47666880,  62985600,  45069696,  13436928;
  9: 144864, 2754432, 22597056, 104312448, 295837920, 526839552, 574148736, 349360128, 90699264;
%F Formula
The row polynomials satisfy (n^2-3*n+2-(6*n-6)*x)*P(n,x) = -(2*n^2-18*n+28-(30*n-90)*x+72*x^2)*P(n-1,x) + (2*n-6+(8*n-36)*x+(6*n-66)*x^2-36*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 18*n + 28 - (30*n - 90)*x + 72*x^2)*P[n - 1, x] + (2*n - 6 + (8*n - 36)*x + (6*n - 66)*x^2 - 36*x^3)*D[P[n - 1, x], x])/(n^2 - 3*n + 2 - (6*n - 6)*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] = 6; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 18*n + 28 - (30*n - 90)*x + 72*x^2)*v[i - 1] + (2*n - 6 + (8*n - 36)*x + (6*n - 66)*x^2 - 36*x^3)*deriv(v[i - 1], x))/(n^2 - 3*n + 2 - (6*n - 6)*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. [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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.