Real-rooted atlas

A094638 → D → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n x^{2} + 6\,n^{2} x + 18\,n^{2} x^{2} - n^{3} - 6\,n^{3} x - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 6\,n x + 2\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
271219
346174162382
43261974394225928834
5255621624680679439248600235239
62221224338410603172294352246402010497607134045
721297628799761614913248043692799389007049138425719120243435180
82239344361426322490213249491654522160760860293666277622052308327054387209244661940
92565936048235089639539898361845642218453641942605993752796241145571857647510151001621427701120387022041405
103185409606845421456651996785163611802374641282177200555302423800964447382638302444753791550656277038067536071425670400017716651848855
11426157632010316876083211212445042647202948704272302848112020508706507325770617329232933422223574573449136020972907942242411015170521120025927518355200880627874819850
126114851136016480273858562014729164036014745806936020871783523731889024401497962787385909533358168538101944360981385281227418027939159298211353446425604699065264981120101824435722240047243894273040150
139370304294402784196728806437847168503761631120721363416321723827420377460067756786254426384193760282745042111406118449338022800619119407717310204669377040265298160487113619994901504214179288638492160430208240926464002721203296069011075
1415275952518400496340065393920743111921471227267875496407230976421957900214554680188516081368663560062264867676425244811539227031183236332828475887846981230360389328946307872819203823192356018029755225531349736616505856103904699944410892801945864966652006400167510084760674345025
Technical details
Node id
n-260658076b4835f1 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-2708d889240ab808

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:        1;
  2:        7,        12;
  3:       46,       174,        162;
  4:      326,      1974,       3942,        2592;
  5:     2556,     21624,      68067,       94392,       48600;
  6:    22212,    243384,    1060317,     2294352,     2464020,     1049760;
  7:   212976,   2879976,   16149132,    48043692,    79938900,    70491384,     25719120;
  8:  2239344,  36142632,  249021324,   949165452,  2160760860,  2936662776,   2205230832,   705438720;
  9: 25659360, 482350896, 3953989836, 18456422184, 53641942605, 99375279624, 114557185764, 75101510016, 21427701120;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(6*n^3-6*n^2)*x+(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + (2*n^2-1+(6*n^2-6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (6*n^3 - 6*n^2)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (6*n^2 - 6*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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (6*n^3 - 6*n^2)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (6*n^2 - 6*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

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. [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.