Real-rooted atlas

A113413 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - 17\,x - 11\,x^{2} + 3\,x^{3} + 2\,x^{4} - 6\,n - 4\,n x^{2} + 3\,n x^{3} + 2\,n x^{4} + 4\,n^{2} + 4\,n^{2} x^{2} + 2\,n^{2} x^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(-22 - 111\,x - 135\,x^{2} + x^{3} + 34\,x^{4} + 8\,x^{5} - 18\,n - 8\,n x + 50\,n x^{2} + 23\,n x^{3} + 2\,n x^{4} + 24\,n^{2} + 42\,n^{2} x - 6\,n^{2} x^{3} - 4\,n^{3} - 6\,n^{3} x\right)\,P_{n-1}(x) + \left(-4 - 26\,x - 48\,x^{2} - 5\,x^{3} + 51\,x^{4} + 28\,x^{5} + 4\,x^{6} - 4\,n - 8\,n x + 18\,n x^{2} + 48\,n x^{3} + 26\,n x^{4} + 4\,n x^{5} + 4\,n^{2} + 16\,n^{2} x + 18\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
21920443
363118608249
4193536480160161385
5565209827841616400327495
616037484133321142448809606439747
7444725062563646485640720137282240128207545

All 15 rows

Technical details
Node id
n-d620b336088bf4bf · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-270b1b93697059d5

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:     5,      2;
  2:    19,     20,      4;
  3:    63,    118,     60,       8;
  4:   193,    536,    480,     160,      16;
  5:   565,   2098,   2784,    1616,     400,     32;
  6:  1603,   7484,  13332,   11424,    4880,    960,     64;
  7:  4447,  25062,  56364,   64856,   40720,  13728,   2240,   128;
  8: 12129,  80176, 218464,  317888,  267680, 132096,  36736,  5120,   256;
  9: 32645, 247778, 794400, 1403296, 1494688, 987456, 400512, 94720, 11520, 512;
%F Formula
The row polynomials satisfy (4*n^2-6*n-2+(2*n^3-17)*x+(4*n^2-4*n-11)*x^2+(2*n^2+3*n+3)*x^3+(2*n+2)*x^4)*P(n,x) = -(4*n^3-24*n^2+18*n+22+(6*n^3-42*n^2+8*n+111)*x-(50*n-135)*x^2+(6*n^2-23*n-1)*x^3-(2*n+34)*x^4-8*x^5)*P(n-1,x) + (4*n^2-4*n-4+(16*n^2-8*n-26)*x+(18*n^2+18*n-48)*x^2+(4*n^2+48*n-5)*x^3+(26*n+51)*x^4+(4*n+28)*x^5+4*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[(-(4*n^3 - 24*n^2 + 18*n + 22 + (6*n^3 - 42*n^2 + 8*n + 111)*x - (50*n - 135)*x^2 + (6*n^2 - 23*n - 1)*x^3 - (2*n + 34)*x^4 - 8*x^5)*P[n - 1, x] + (4*n^2 - 4*n - 4 + (16*n^2 - 8*n - 26)*x + (18*n^2 + 18*n - 48)*x^2 + (4*n^2 + 48*n - 5)*x^3 + (26*n + 51)*x^4 + (4*n + 28)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(4*n^2 - 6*n - 2 + (2*n^3 - 17)*x + (4*n^2 - 4*n - 11)*x^2 + (2*n^2 + 3*n + 3)*x^3 + (2*n + 2)*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] = (-(4*n^3 - 24*n^2 + 18*n + 22 + (6*n^3 - 42*n^2 + 8*n + 111)*x - (50*n - 135)*x^2 + (6*n^2 - 23*n - 1)*x^3 - (2*n + 34)*x^4 - 8*x^5)*v[i - 1] + (4*n^2 - 4*n - 4 + (16*n^2 - 8*n - 26)*x + (18*n^2 + 18*n - 48)*x^2 + (4*n^2 + 48*n - 5)*x^3 + (26*n + 51)*x^4 + (4*n + 28)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(4*n^2 - 6*n - 2 + (2*n^3 - 17)*x + (4*n^2 - 4*n - 11)*x^2 + (2*n^2 + 3*n + 3)*x^3 + (2*n + 2)*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.