Real-rooted atlas

A269951 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-20 + 9\,n - n^{2}\right)\,P_{n-4}(x) + \left(20 - 8\,x - 13\,n + 2\,n x + 2\,n^{2}\right)\,P_{n-3}(x) + \left(3 + 4\,x - x^{2} + 2\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-3 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)
\(P_{3}(x) = 2 + 4\,x + x^{2}\)
\(P_{4}(x) = 5 + 15\,x + 8\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2112
32417
45158129
5166250131142
665297311121191814
73261664207910502452615391
8195710815153319294281544234140689
9137008045012541887017317596475735431345598
101096016757451135270874194367227904751334211505313267058
1198641063330521130817394651404439924126411622577125296171664134049663
1298641016555468712317884611043525156645975180895673745098509133449252465761388070125
131085051127430539261458464402138552830976545206126847594462555240988214710596637568034328914803056006
1413020613459155609705186653982531863594132310964582288415948792610681050051904152912377478420662181220454655103164167568942
15169267974861218660114482568963866032678551117581663854352816748246505018804415916370475963052316993353032694381638418972261751181920501598199
Technical details
Node id
n-616044f9b3dd62e4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2acd4555bb2b374f

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:      1,      1;
   3:      2,      4,       1;
   4:      5,     15,       8,      1;
   5:     16,     62,      50,     13,      1;
   6:     65,    297,     311,    121,     19,     1;
   7:    326,   1664,    2079,   1050,    245,    26,     1;
   8:   1957,  10815,   15331,   9294,   2815,   442,    34,    1;
   9:  13700,  80450,  125418,  87017,  31759,  6475,   735,   43,  1;
  10: 109601, 675745, 1135270, 874194, 367227, 90475, 13342, 1150, 53, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-9*n+20)*P(n-4,x) + (2*n^2-13*n+20+(2*n-8)*x)*P(n-3,x) - (n^2-2*n-3+(2*n-4)*x+x^2)*P(n-2,x) + (2*n-3+2*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 2 + 4*x + x^2, P(4,x) = 5 + 15*x + 8*x^2 + x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 2 + 4*x + x^2;
P[4, x_] := 5 + 15*x + 8*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 9*n + 20)*P[n - 4, x] + (2*n^2 - 13*n + 20 + (2*n - 8)*x)*P[n - 3, x] - (n^2 - 2*n - 3 + (2*n - 4)*x + x^2)*P[n - 2, x] + (2*n - 3 + 2*x)*P[n - 1, 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] = 1 + x; v[3] = 2 + 4*x + x^2; v[4] = 5 + 15*x + 8*x^2 + x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 9*n + 20)*v[i - 4] + (2*n^2 - 13*n + 20 + (2*n - 8)*x)*v[i - 3] - (n^2 - 2*n - 3 + (2*n - 4)*x + x^2)*v[i - 2] + (2*n - 3 + 2*x)*v[i - 1]); 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.