Real-rooted atlas

A008288 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\,x^{2} - 5\,x^{3} - x^{4} - 4\,n - 2\,n x - 11\,n x^{2} + 4\,n x^{3} + n x^{4} + 4\,n^{2} + 7\,n^{2} x^{2} + n^{2} x^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(-32 - 56\,x + 20\,x^{2} + 53\,x^{3} + 14\,x^{4} + x^{5} + 4\,n + 66\,n x + 49\,n x^{2} - 5\,n x^{3} - 2\,n x^{4} + 28\,n^{2} + 4\,n^{2} x - 19\,n^{2} x^{2} - 3\,n^{2} x^{3} - 8\,n^{3} - 6\,n^{3} x\right)\,P_{n-1}(x) + \left(32\,x + 120\,x^{2} + 156\,x^{3} + 83\,x^{4} + 16\,x^{5} + x^{6} + 16\,n + 88\,n x + 138\,n x^{2} + 81\,n x^{3} + 16\,n x^{4} + n x^{5} + 16\,n^{2} + 32\,n^{2} x + 18\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
3107118
4303411176
58713873151314
62455033771271911272
7676170916728011962315078
81836552466984206146428027120036
949251720424950194908906242037931178306
101307952061880068249046982167623723493351303632
1134446153971297539325974223208993382895354276223911169518
129009044693497242912207279787925242801906374684775867664314479132
13234227127731030912874378057402900325299961107765339521720011025492547117070394
14605865360286796028711515392115756937113693555839941215556356984310616113485109951164777960
151560200100489852925437050919055590731004821218528477300123351253926560911315151262173331288551244888134
Technical details
Node id
n-9fed93b7289c027d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8df4cfbffdd1234a

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:    10,     7,     1;
   4:    30,    34,    11,     1;
   5:    87,   138,    73,    15,     1;
   6:   245,   503,   377,   127,    19,     1;
   7:   676,  1709,  1672,   801,   196,    23,    1;
   8:  1836,  5524,  6698,  4206,  1464,   280,   27,   1;
   9:  4925, 17204, 24950, 19490,  8906,  2420,  379,  31,  1;
  10: 13079, 52061, 88006, 82490, 46982, 16762, 3723, 493, 35, 1;
%F Formula
The row polynomials satisfy (4*n^2-4*n+(2*n^3-2*n)*x+(7*n^2-11*n+4)*x^2+(n^2+4*n-5)*x^3+(n-1)*x^4)*P(n,x) = -(8*n^3-28*n^2-4*n+32+(6*n^3-4*n^2-66*n+56)*x+(19*n^2-49*n-20)*x^2+(3*n^2+5*n-53)*x^3+(2*n-14)*x^4-x^5)*P(n-1,x) + (16*n^2+16*n+(32*n^2+88*n+32)*x+(18*n^2+138*n+120)*x^2+(2*n^2+81*n+156)*x^3+(16*n+83)*x^4+(n+16)*x^5+x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 28*n^2 - 4*n + 32 + (6*n^3 - 4*n^2 - 66*n + 56)*x + (19*n^2 - 49*n - 20)*x^2 + (3*n^2 + 5*n - 53)*x^3 + (2*n - 14)*x^4 - x^5)*P[n - 1, x] + (16*n^2 + 16*n + (32*n^2 + 88*n + 32)*x + (18*n^2 + 138*n + 120)*x^2 + (2*n^2 + 81*n + 156)*x^3 + (16*n + 83)*x^4 + (n + 16)*x^5 + x^6)*D[P[n - 1, x], x])/(4*n^2 - 4*n + (2*n^3 - 2*n)*x + (7*n^2 - 11*n + 4)*x^2 + (n^2 + 4*n - 5)*x^3 + (n - 1)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(8*n^3 - 28*n^2 - 4*n + 32 + (6*n^3 - 4*n^2 - 66*n + 56)*x + (19*n^2 - 49*n - 20)*x^2 + (3*n^2 + 5*n - 53)*x^3 + (2*n - 14)*x^4 - x^5)*v[i - 1] + (16*n^2 + 16*n + (32*n^2 + 88*n + 32)*x + (18*n^2 + 138*n + 120)*x^2 + (2*n^2 + 81*n + 156)*x^3 + (16*n + 83)*x^4 + (n + 16)*x^5 + x^6)*deriv(v[i - 1], x))/(4*n^2 - 4*n + (2*n^3 - 2*n)*x + (7*n^2 - 11*n + 4)*x^2 + (n^2 + 4*n - 5)*x^3 + (n - 1)*x^4)); 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.