Real-rooted atlas

A232535 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-6 - 3\,x + 18\,n + 2\,n x - 12\,n^{2} + 11\,n^{2} x + n^{2} x^{2} - 6\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-24 - 44\,x - 18\,x^{2} - x^{3} + 72\,n + 108\,n x + 30\,n x^{2} - 2\,n x^{3} - 48\,n^{2} - 28\,n^{2} x + 23\,n^{2} x^{2} + 9\,n^{2} x^{3} - 24\,n^{3} x - 10\,n^{3} x^{2} + 6\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-12\,x - 22\,x^{2} - 12\,x^{3} - 2\,x^{4} + 24\,n x + 32\,n x^{2} - 8\,n x^{4} + 12\,n^{2} x^{2} + 8\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
21228949
34816011116335
4192832888304252241
57684096590432646753614743
63072194563532827136938013084995729
712288901121973761945609552022848230364615071
84915240960010506241265664809600279936493923776813917825
91966081835008539443276840966073600277094471702497536585910024775207
10786432812646426935296443023364175360023761920820019216558081793888700121155710257
113145728356515841315307522453667842687795201837301767974220821671936352123231152012463144973464047
1212582912155189248630718464131596288016436428801313538048689217536237273088523036807001280515944173281696057963457
1350331648671088640297900441668744642569645260800883084492854455009282279473152640553472117145600131628648211842349119637547675575
1420132659228856811521389153484835131490304547160064005649806131240086208512198133678086804891648159544320024644505623602176126344431164225231895353841
Technical details
Node id
n-2c6a837009b82db0 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-97e4558a746ca129

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,       4;
  2:     12,      28,       9;
  3:     48,     160,     111,      16;
  4:    192,     832,     888,     304,      25;
  5:    768,    4096,    5904,    3264,     675,      36;
  6:   3072,   19456,   35328,   27136,    9380,    1308,     49;
  7:  12288,   90112,  197376,  194560,   95520,   22848,   2303,    64;
  8:  49152,  409600, 1050624, 1265664,  809600,  279936,  49392,  3776,   81;
  9: 196608, 1835008, 5394432, 7684096, 6073600, 2770944, 717024, 97536, 5859, 100;
%F Formula
The row polynomials satisfy -(12*n^2-18*n+6+(6*n^3-11*n^2-2*n+3)*x-(2*n^3+n^2)*x^2)*P(n,x) = -(48*n^2-72*n+24+(24*n^3+28*n^2-108*n+44)*x+(10*n^3-23*n^2-30*n+18)*x^2-(6*n^3+9*n^2-2*n-1)*x^3)*P(n-1,x) + ((24*n-12)*x+(12*n^2+32*n-22)*x^2+(8*n^2-12)*x^3-(4*n^2+8*n+2)*x^4)*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[(-(48*n^2 - 72*n + 24 + (24*n^3 + 28*n^2 - 108*n + 44)*x + (10*n^3 - 23*n^2 - 30*n + 18)*x^2 - (6*n^3 + 9*n^2 - 2*n - 1)*x^3)*P[n - 1, x] + ((24*n - 12)*x + (12*n^2 + 32*n - 22)*x^2 + (8*n^2 - 12)*x^3 - (4*n^2 + 8*n + 2)*x^4)*D[P[n - 1, x], x])/(-12*n^2 + 18*n - 6 - (6*n^3 - 11*n^2 - 2*n + 3)*x + (2*n^3 + n^2)*x^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] = (-(48*n^2 - 72*n + 24 + (24*n^3 + 28*n^2 - 108*n + 44)*x + (10*n^3 - 23*n^2 - 30*n + 18)*x^2 - (6*n^3 + 9*n^2 - 2*n - 1)*x^3)*v[i - 1] + ((24*n - 12)*x + (12*n^2 + 32*n - 22)*x^2 + (8*n^2 - 12)*x^3 - (4*n^2 + 8*n + 2)*x^4)*deriv(v[i - 1], x))/(-12*n^2 + 18*n - 6 - (6*n^3 - 11*n^2 - 2*n + 3)*x + (2*n^3 + n^2)*x^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

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