Real-rooted atlas

A114687 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-6 - 20\,x + 16\,x^{2} - 3\,n + 22\,n x + 8\,n x^{2} + 16\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(-6 - 20\,x + 16\,x^{2} - 9\,n - 66\,n x + 32\,n x^{2} + 96\,n x^{3} + 64\,n^{2} + 176\,n^{2} x + 192\,n^{2} x^{2} + 56\,n^{3} + 96\,n^{3} x\right)\,P_{n-1}(x) + \left(-3 - 24\,x - 68\,x^{2} - 96\,x^{3} - 64\,x^{4} - 12\,n - 88\,n x - 192\,n x^{2} - 128\,n x^{3} - 8\,n^{2} - 48\,n^{2} x - 64\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
2396016115
3213552384641213
4116542965120217625613013
56403307805296039040115201024141727
6353712101324764005129602649605836840961562287
7196297139048039271685622400428288016629762867201638417385305
810937859004560305105925479577655931904322846729863168137625665536194926249
96116031574037642271484804920867846339594244936458242257059845603328064880642621442198849779
103430254336162846016378240804161678720649883136064216965123984261120148881408030769152030146560104857624927923531
111929003812257184952115194393603362074848061823393280743732183045884079308829991075840937525248016436428801384120324194304283780255381
1210872982771398501180079431061248262001224320555176413440789279879168763250442240498237898752213424865280568354406408581545984629145600167772163241937003965
1361412211158612774430053893844035219833134280964762478333184782544717926489723993948167184970743808395731894272014500188651523337806151684394791731228353495046710886437147785283655
14347497320515277823203243608059706160146592763175043936328402406473480355300352976431917260809298986098688063037499965440298054018662409474961047552190802663833622137536512012683575296268435456426766777006855
Technical details
Node id
n-cca2636b43e84f12 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-34bd435130f77bfa

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:       7,        4;
  2:      39,       60,        16;
  3:     213,      552,       384,        64;
  4:    1165,     4296,      5120,      2176,       256;
  5:    6403,    30780,     52960,     39040,     11520,      1024;
  6:   35371,   210132,    476400,    512960,    264960,     58368,      4096;
  7:  196297,  1390480,   3927168,   5622400,   4282880,   1662976,    286720,    16384;
  8: 1093785,  9004560,  30510592,  54795776,  55931904,  32284672,   9863168,  1376256,   65536;
  9: 6116031, 57403764, 227148480, 492086784, 633959424, 493645824, 225705984, 56033280, 6488064, 262144;
%F Formula
The row polynomials satisfy (8*n^3+16*n^2-3*n-6+(16*n^2+22*n-20)*x+(8*n+16)*x^2)*P(n,x) = (56*n^3+64*n^2-9*n-6+(96*n^3+176*n^2-66*n-20)*x+(192*n^2+32*n+16)*x^2+96*n*x^3)*P(n-1,x) - (8*n^2+12*n+3+(48*n^2+88*n+24)*x+(64*n^2+192*n+68)*x^2+(128*n+96)*x^3+64*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[((56*n^3 + 64*n^2 - 9*n - 6 + (96*n^3 + 176*n^2 - 66*n - 20)*x + (192*n^2 + 32*n + 16)*x^2 + 96*n*x^3)*P[n - 1, x] - (8*n^2 + 12*n + 3 + (48*n^2 + 88*n + 24)*x + (64*n^2 + 192*n + 68)*x^2 + (128*n + 96)*x^3 + 64*x^4)*D[P[n - 1, x], x])/(8*n^3 + 16*n^2 - 3*n - 6 + (16*n^2 + 22*n - 20)*x + (8*n + 16)*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] = ((56*n^3 + 64*n^2 - 9*n - 6 + (96*n^3 + 176*n^2 - 66*n - 20)*x + (192*n^2 + 32*n + 16)*x^2 + 96*n*x^3)*v[i - 1] - (8*n^2 + 12*n + 3 + (48*n^2 + 88*n + 24)*x + (64*n^2 + 192*n + 68)*x^2 + (128*n + 96)*x^3 + 64*x^4)*deriv(v[i - 1], x))/(8*n^3 + 16*n^2 - 3*n - 6 + (16*n^2 + 22*n - 20)*x + (8*n + 16)*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