Real-rooted atlas

A225117 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-11 + 6\,x + 3\,n\right)\,P_{n}(x) = \left(80 - 460\,x + 1244\,x^{2} - 900\,x^{3} + 144\,x^{4} - 30\,n + 246\,n x - 954\,n x^{2} + 540\,n x^{3} - 54\,n x^{4} - 36\,n^{2} x + 270\,n^{2} x^{2} - 81\,n^{2} x^{3} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(48\,x - 348\,x^{2} + 570\,x^{3} - 324\,x^{4} + 54\,x^{5} - 18\,n x + 189\,n x^{2} - 252\,n x^{3} + 81\,n x^{4} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-71 + 251\,x - 156\,x^{2} + 18\,x^{3} + 21\,n - 123\,n x + 45\,n x^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-33\,x + 51\,x^{2} - 18\,x^{3} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
31715133
4101213631378
55612807188425515508
629653192946532147501023197200
715241323403941427618430109443409512012040
87722930285731614659720371695733284779075516383147764080
9388577269299352454856065484395693701386748152608156337026553511278607680
10194851723136704134255083061265200021214831873244603851635887238577439829284262143138093690880
119755385194250689944955991125260214382872495416949330346135525974918747222669117971520280378053104857511249949232000
12488055971605706806956436811585149135331878531439235573971416162020894810730551410103013341324361149389185937719689380814194303144783895340800
1324409147313133602357568578968974408700268246933137578251469311164251431139151846383859326408014363651561585418765254241639957603723659138077327041677721511739500776921600
14122059662910665540196578134430197156814668582300144269036224272656365225285273938140472461307091677805612126256616533164268315592148582325816418651941597051051109849675479260267108863172804471541401600
15610328624986190942639639475664883048072381352851052301020376816107495834171573156913598621711327793623629457798416648575667803962956907726133406354827339987085993334968709061439099794519812305867768620184726843545513266273336880153600
Technical details
Node id
n-c18c59cf0567612d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-42411ce3f1f92da1

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:       3,         1;
   3:      17,        15,          1;
   4:     101,       213,         63,           1;
   5:     561,      2807,       1884,         255,           1;
   6:    2965,     31929,      46532,       14750,        1023,          1;
   7:   15241,    323403,     941427,      618430,      109443,       4095,         1;
   8:   77229,   3028573,   16146597,    20371695,     7332847,     790755,     16383,        1;
   9:  388577,  26929935,  245485606,   548439569,   370138674,   81526081,   5633702,    65535,      1;
  10: 1948517, 231367041, 3425508306, 12652000212, 14831873244, 6038516358, 872385774, 39829284, 262143, 1;
%F Formula
The row polynomials satisfy (3*n-11+6*x)*P(n,x) = -(30*n-80+(36*n^2-246*n+460)*x+(27*n^3-270*n^2+954*n-1244)*x^2+(81*n^2-540*n+900)*x^3+(54*n-144)*x^4)*P(n-2,x) - ((18*n-48)*x+(27*n^2-189*n+348)*x^2-(27*n^2-252*n+570)*x^3-(81*n-324)*x^4-54*x^5)*d/dx P(n-2,x) + (21*n-71+(18*n^2-123*n+251)*x+(45*n-156)*x^2+18*x^3)*P(n-1,x) + ((9*n-33)*x-(9*n-51)*x^2-18*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n - 80 + (36*n^2 - 246*n + 460)*x + (27*n^3 - 270*n^2 + 954*n - 1244)*x^2 + (81*n^2 - 540*n + 900)*x^3 + (54*n - 144)*x^4)*P[n - 2, x] - ((18*n - 48)*x + (27*n^2 - 189*n + 348)*x^2 - (27*n^2 - 252*n + 570)*x^3 - (81*n - 324)*x^4 - 54*x^5)*D[P[n - 2, x], x] + (21*n - 71 + (18*n^2 - 123*n + 251)*x + (45*n - 156)*x^2 + 18*x^3)*P[n - 1, x] + ((9*n - 33)*x - (9*n - 51)*x^2 - 18*x^3)*D[P[n - 1, x], x])/(3*n - 11 + 6*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] = 3 + x; for(n = 3, N, my(i = n); v[i] = (-(30*n - 80 + (36*n^2 - 246*n + 460)*x + (27*n^3 - 270*n^2 + 954*n - 1244)*x^2 + (81*n^2 - 540*n + 900)*x^3 + (54*n - 144)*x^4)*v[i - 2] - ((18*n - 48)*x + (27*n^2 - 189*n + 348)*x^2 - (27*n^2 - 252*n + 570)*x^3 - (81*n - 324)*x^4 - 54*x^5)*deriv(v[i - 2], x) + (21*n - 71 + (18*n^2 - 123*n + 251)*x + (45*n - 156)*x^2 + 18*x^3)*v[i - 1] + ((9*n - 33)*x - (9*n - 51)*x^2 - 18*x^3)*deriv(v[i - 1], x))/(3*n - 11 + 6*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

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.