Real-rooted atlas

A033282 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
220301161
311225218645595
46722016224010921976217
54224158402352017280628090368047
627456123552229680225720123660358024279770149
7183040960960214614426426401937320845460203350207938939707
8124467274680321947545628828800264904641547145656080641153416103049105843409
98599552580469761731225602994001923308497922422405441175079363641414465408765188591273241431
10601968644514764801515671040299909376038731929603411023616207445056086021208023275434037106550264672315517824973
1142600857635145707521311597056029224419840431953843204446696038432530891008169118611206122534880146997774021064936213648869191202877411
123042918400273862656001124662640642786552524804639069670405466685593604675361433602923915484161327017225604262325408091966406401196815500710393732377843390873
1321909012480213612871680957302128640261082398720048343757496326417731713536628371414528045943407544322508342051072100994724230429148335673657102948864680576769637269351929807864423071
14158840340480166782357504080994880512002411155289088049154200775680725890656691208007641933491267032497930112427873346060162072365927632074970223167841964305737576352366829724387359772661968801519376255282112629
Technical details
Node id
n-1dd418d798e58c37 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-9a4b425d2a954ae8

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:       4,        3;
  2:      20,       30,        11;
  3:     112,      252,       186,        45;
  4:     672,     2016,      2240,      1092,       197;
  5:    4224,    15840,     23520,     17280,      6280,       903;
  6:   27456,   123552,    229680,    225720,    123660,     35802,      4279;
  7:  183040,   960960,   2146144,   2642640,   1937320,    845460,    203350,    20793;
  8: 1244672,  7468032,  19475456,  28828800,  26490464,  15471456,   5608064,  1153416,  103049;
  9: 8599552, 58046976, 173122560, 299400192, 330849792, 242240544, 117507936, 36414144, 6540876, 518859;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (8*n+4+9*n*x+(2*n-2)*x^2)*P(n-1,x) - (4*x+6*x^2+2*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 4 + 9*n*x + (2*n - 2)*x^2)*P[n - 1, x] - (4*x + 6*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + 4 + 9*n*x + (2*n - 2)*x^2)*v[i - 1] - (4*x + 6*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 2)); 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. [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