Real-rooted atlas

A081580 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(12\,n + 12\,n x + 11\,n x^{2} + 22\,n x^{3} + n x^{4} + 12\,n^{2} + 18\,n^{2} x + 23\,n^{2} x^{2} + n^{2} x^{3} + 6\,n^{3} x\right)\,P_{n}(x) = \left(-120 + 24\,x + 442\,x^{2} + 333\,x^{3} + 36\,x^{4} + x^{5} + 252\,n + 408\,n x - 57\,n x^{2} - 219\,n x^{3} - 10\,n x^{4} + 12\,n^{2} - 234\,n^{2} x - 247\,n^{2} x^{2} - 11\,n^{2} x^{3} - 72\,n^{3} - 66\,n^{3} x\right)\,P_{n-1}(x) + \left(288 + 1416\,x + 2556\,x^{2} + 2063\,x^{3} + 683\,x^{4} + 49\,x^{5} + x^{6} + 432\,n + 1416\,n x + 1578\,n x^{2} + 641\,n x^{3} + 48\,n x^{4} + n x^{5} + 144\,n^{2} + 288\,n^{2} x + 150\,n^{2} x^{2} + 6\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
21811130
37074191164
4303450193271974
51197233515053753515448
64756114979920372562043130562
7183965325258202300307600928511168460

All 15 rows

Technical details
Node id
n-d4bfb604a96a3d62 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-34e939c6b6951736

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,       1;
  2:     18,      11,       1;
  3:     70,      74,      19,       1;
  4:    303,     450,     193,      27,      1;
  5:   1197,    2335,    1505,     375,     35,      1;
  6:   4756,   11497,    9920,    3725,    620,     43,     1;
  7:  18396,   53252,   58202,   30030,   7600,    928,    51,    1;
  8:  70605,  238308,  316038,  209322,  73410,  13620,  1299,   59,  1;
  9: 267751, 1033485, 1615926, 1315434, 599262, 154578, 22275, 1733, 67, 1;
%F Formula
The row polynomials satisfy (12*n^2+12*n+(6*n^3+18*n^2+12*n)*x+(23*n^2+11*n)*x^2+(n^2+22*n)*x^3+n*x^4)*P(n,x) = -(72*n^3-12*n^2-252*n+120+(66*n^3+234*n^2-408*n-24)*x+(247*n^2+57*n-442)*x^2+(11*n^2+219*n-333)*x^3+(10*n-36)*x^4-x^5)*P(n-1,x) + (144*n^2+432*n+288+(288*n^2+1416*n+1416)*x+(150*n^2+1578*n+2556)*x^2+(6*n^2+641*n+2063)*x^3+(48*n+683)*x^4+(n+49)*x^5+x^6)*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[(-(72*n^3 - 12*n^2 - 252*n + 120 + (66*n^3 + 234*n^2 - 408*n - 24)*x + (247*n^2 + 57*n - 442)*x^2 + (11*n^2 + 219*n - 333)*x^3 + (10*n - 36)*x^4 - x^5)*P[n - 1, x] + (144*n^2 + 432*n + 288 + (288*n^2 + 1416*n + 1416)*x + (150*n^2 + 1578*n + 2556)*x^2 + (6*n^2 + 641*n + 2063)*x^3 + (48*n + 683)*x^4 + (n + 49)*x^5 + x^6)*D[P[n - 1, x], x])/(12*n^2 + 12*n + (6*n^3 + 18*n^2 + 12*n)*x + (23*n^2 + 11*n)*x^2 + (n^2 + 22*n)*x^3 + n*x^4)]];
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] = (-(72*n^3 - 12*n^2 - 252*n + 120 + (66*n^3 + 234*n^2 - 408*n - 24)*x + (247*n^2 + 57*n - 442)*x^2 + (11*n^2 + 219*n - 333)*x^3 + (10*n - 36)*x^4 - x^5)*v[i - 1] + (144*n^2 + 432*n + 288 + (288*n^2 + 1416*n + 1416)*x + (150*n^2 + 1578*n + 2556)*x^2 + (6*n^2 + 641*n + 2063)*x^3 + (48*n + 683)*x^4 + (n + 49)*x^5 + x^6)*deriv(v[i - 1], x))/(12*n^2 + 12*n + (6*n^3 + 18*n^2 + 12*n)*x + (23*n^2 + 11*n)*x^2 + (n^2 + 22*n)*x^3 + n*x^4)); 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. [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.