Real-rooted atlas

A142459 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-15 + 8\,x + 4\,n\right)\,P_{n}(x) = \left(55 - 414\,x + 2615\,x^{2} - 2188\,x^{3} + 352\,x^{4} - 20\,n + 224\,n x - 2236\,n x^{2} + 1312\,n x^{3} - 128\,n x^{4} - 32\,n^{2} x + 656\,n^{2} x^{2} - 192\,n^{2} x^{3} - 64\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(44\,x - 736\,x^{2} + 1348\,x^{3} - 784\,x^{4} + 128\,x^{5} - 16\,n x + 432\,n x^{2} - 608\,n x^{3} + 192\,n x^{4} - 64\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-86 + 438\,x - 284\,x^{2} + 32\,x^{3} + 24\,n - 224\,n x + 80\,n x^{2} + 32\,n^{2} x\right)\,P_{n-1}(x) + \left(-60\,x + 92\,x^{2} - 32\,x^{3} + 16\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2213
31112124
460177621300
53092432198631215040
6155828681542481979015621105840
778073048601206363946720187557781212661120
8390563040393228274143554457514612260173499939062177837760
91953052916009638627222810907423848613528302108243201584952419531212594592000
1097655427298731360554528962881907594840268890164185503767262925186832143817804976562197037740800
11488280325174321409004156043768329076764815728350738381262425739736370589584890396452926081300219887488281214022655436800
122441405222998685441129006704291014972050782765536620884028807003587364590635460375397188704601078933052957752366011731276205244140621183030822374400
1312207030120889246499218000996053286309502630035560165370971162253533171025296377282709211854020196922639273837488129574105838715700938892544010572797064612207031219067065354547200
14610351550189053023077724644028034039261248618247808264716599680791059013907780916283849517326234840509609694706176422768396227278654744726982329075247497735922263328026809522841582026103515621485735643993600000
1530517577991707214512265233275433037404311172582579732648321267288145500220331530426654169480201296358335996720941317998307370844684768307161971523945338153802983731627886841207679206107389120875081254179285742195338013051757812127978373094031360000
Technical details
Node id
n-84cea0d001714dc2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-84d0fce31cd67972

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,         1;
   3:     11,        12,          1;
   4:     60,       177,         62,           1;
   5:    309,      2432,       1986,         312,           1;
   6:   1558,     28681,      54248,       19790,        1562,           1;
   7:   7807,    304860,    1206363,      946720,      187557,        7812,          1;
   8:  39056,   3040393,   22827414,    35544575,    14612260,     1734999,      39062,         1;
   9: 195305,  29160096,  386272228,  1090742384,   861352830,   210824320,   15849524,    195312,      1;
  10: 976554, 272987313, 6055452896, 28819075948, 40268890164, 18550376726, 2925186832, 143817804, 976562, 1;
%F Formula
The row polynomials satisfy (4*n-15+8*x)*P(n,x) = -(20*n-55+(32*n^2-224*n+414)*x+(64*n^3-656*n^2+2236*n-2615)*x^2+(192*n^2-1312*n+2188)*x^3+(128*n-352)*x^4)*P(n-2,x) - ((16*n-44)*x+(64*n^2-432*n+736)*x^2-(64*n^2-608*n+1348)*x^3-(192*n-784)*x^4-128*x^5)*d/dx P(n-2,x) + (24*n-86+(32*n^2-224*n+438)*x+(80*n-284)*x^2+32*x^3)*P(n-1,x) + ((16*n-60)*x-(16*n-92)*x^2-32*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(20*n - 55 + (32*n^2 - 224*n + 414)*x + (64*n^3 - 656*n^2 + 2236*n - 2615)*x^2 + (192*n^2 - 1312*n + 2188)*x^3 + (128*n - 352)*x^4)*P[n - 2, x] - ((16*n - 44)*x + (64*n^2 - 432*n + 736)*x^2 - (64*n^2 - 608*n + 1348)*x^3 - (192*n - 784)*x^4 - 128*x^5)*D[P[n - 2, x], x] + (24*n - 86 + (32*n^2 - 224*n + 438)*x + (80*n - 284)*x^2 + 32*x^3)*P[n - 1, x] + ((16*n - 60)*x - (16*n - 92)*x^2 - 32*x^3)*D[P[n - 1, x], x])/(4*n - 15 + 8*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (-(20*n - 55 + (32*n^2 - 224*n + 414)*x + (64*n^3 - 656*n^2 + 2236*n - 2615)*x^2 + (192*n^2 - 1312*n + 2188)*x^3 + (128*n - 352)*x^4)*v[i - 2] - ((16*n - 44)*x + (64*n^2 - 432*n + 736)*x^2 - (64*n^2 - 608*n + 1348)*x^3 - (192*n - 784)*x^4 - 128*x^5)*deriv(v[i - 2], x) + (24*n - 86 + (32*n^2 - 224*n + 438)*x + (80*n - 284)*x^2 + 32*x^3)*v[i - 1] + ((16*n - 60)*x - (16*n - 92)*x^2 - 32*x^3)*deriv(v[i - 1], x))/(4*n - 15 + 8*x)); 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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.