Real-rooted atlas

A114656 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(2 - x - 2\,x^{2} + 2\,n + 5\,n x + n x^{2}\right)\,P_{n-1}(x) + \left(4\,x - x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2235
34161131
486010245215
5161925846081971597
63256026404760353090312425
764153610320281603534020256427999955
812840323651213944026012024805211559820793824675
9256102401200646092801568224219251216780966580481030496939769
105122534437324824245768197056156340801735440011064240374229051885959334605
1110246144011097608970240384518409542400014318640013125120071591700212786402646723513972967
12204814643231820803129984016562304051699648010015420801231744800959145000456687660121023606136488694501041935
134096344064885350410407936066566016025475788806166237440972702720010099148400682430496028810750326886860487103937339784038517
148192798720240148483324006402526025216116164139523427233638467164734208889539668167972536686447529925872180153239763921503898372693519354455513105
151638418350086373785610257694729131938816496439623681750999810564161855160326814469101447754316810246104126028003253300377601118402539802234551200019688015193179928556219
Technical details
Node id
n-a971c5dff9b54269 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-a9ed9375aa220a32

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1;
   2:   2,     3;
   3:   4,    16,     11;
   4:   8,    60,    102,      45;
   5:  16,   192,    584,     608,     197;
   6:  32,   560,   2640,    4760,    3530,      903;
   7:  64,  1536,  10320,   28160,   35340,    20256,     4279;
   8: 128,  4032,  36512,  139440,  260120,   248052,   115598,    20793;
   9: 256, 10240, 120064,  609280, 1568224,  2192512,  1678096,   658048,  103049;
  10: 512, 25344, 373248, 2424576, 8197056, 15634080, 17354400, 11064240, 3742290, 518859;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (2*n+2+(5*n-1)*x+(n-2)*x^2)*P(n-1,x) + (4*x-x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (5*n - 1)*x + (n - 2)*x^2)*P[n - 1, x] + (4*x - x^3)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n + 2 + (5*n - 1)*x + (n - 2)*x^2)*v[i - 1] + (4*x - x^3)*deriv(v[i - 1], x))/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023