Real-rooted atlas

A274498 → (θ+1)²

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(4 + 7\,x + x^{2} - 6\,x^{3} + 10\,n x + 6\,n x^{2} - 3\,n x^{3} + 14\,n^{2} x^{2} + 11\,n^{2} x^{3} + n^{2} x^{4} + 6\,n^{3} x^{3} + 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(20 - 25\,x - 55\,x^{2} - 74\,x^{3} - 20\,x^{4} - 8\,x^{5} - 8\,n + 110\,n x - 14\,n x^{2} - 73\,n x^{3} - 62\,n x^{4} - 16\,n x^{5} - 34\,n^{2} x + 172\,n^{2} x^{2} + 97\,n^{2} x^{3} + 13\,n^{2} x^{4} - 2\,n^{2} x^{5} - 40\,n^{3} x^{2} + 90\,n^{3} x^{3} + 62\,n^{3} x^{4} + 14\,n^{3} x^{5} - 16\,n^{4} x^{3} + 21\,n^{4} x^{4} + 10\,n^{4} x^{5} - 2\,n^{5} x^{4} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(2 - 6\,x - 42\,x^{2} - 76\,x^{3} - 84\,x^{4} - 48\,x^{5} - 16\,x^{6} + 12\,n x - 12\,n x^{2} - 144\,n x^{3} - 210\,n x^{4} - 156\,n x^{5} - 48\,n x^{6} + 22\,n^{2} x^{2} + 14\,n^{2} x^{3} - 116\,n^{2} x^{4} - 138\,n^{2} x^{5} - 52\,n^{2} x^{6} + 12\,n^{3} x^{3} + 12\,n^{3} x^{4} - 36\,n^{3} x^{5} - 24\,n^{3} x^{6} + 2\,n^{4} x^{4} + 2\,n^{4} x^{5} - 4\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
133
232427
397281
49144324477
5274329721431
627648291634567047
781194487481036821141

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-33cdc2fbd5134453 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-5367d60e6d4311ec

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   3;
  2:   3,   24;
  3:   9,   72;
  4:   9,  144,   324;
  5:  27,  432,   972;
  6:  27,  648,  2916,   3456;
  7:  81, 1944,  8748,  10368;
  8:  81, 2592, 17496,  41472, 32400;
  9: 243, 7776, 52488, 124416, 97200;
%F Formula
The row polynomials satisfy (4+(10*n+7)*x+(14*n^2+6*n+1)*x^2+(6*n^3+11*n^2-3*n-6)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = -(8*n-20+(34*n^2-110*n+25)*x+(40*n^3-172*n^2+14*n+55)*x^2+(16*n^4-90*n^3-97*n^2+73*n+74)*x^3+(2*n^5-21*n^4-62*n^3-13*n^2+62*n+20)*x^4-(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) + (2+(12*n-6)*x+(22*n^2-12*n-42)*x^2+(12*n^3+14*n^2-144*n-76)*x^3+(2*n^4+12*n^3-116*n^2-210*n-84)*x^4+(2*n^4-36*n^3-138*n^2-156*n-48)*x^5-(4*n^4+24*n^3+52*n^2+48*n+16)*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[(-(8*n - 20 + (34*n^2 - 110*n + 25)*x + (40*n^3 - 172*n^2 + 14*n + 55)*x^2 + (16*n^4 - 90*n^3 - 97*n^2 + 73*n + 74)*x^3 + (2*n^5 - 21*n^4 - 62*n^3 - 13*n^2 + 62*n + 20)*x^4 - (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] + (2 + (12*n - 6)*x + (22*n^2 - 12*n - 42)*x^2 + (12*n^3 + 14*n^2 - 144*n - 76)*x^3 + (2*n^4 + 12*n^3 - 116*n^2 - 210*n - 84)*x^4 + (2*n^4 - 36*n^3 - 138*n^2 - 156*n - 48)*x^5 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(4 + (10*n + 7)*x + (14*n^2 + 6*n + 1)*x^2 + (6*n^3 + 11*n^2 - 3*n - 6)*x^3 + (n^4 + 2*n^3 + 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] = (-(8*n - 20 + (34*n^2 - 110*n + 25)*x + (40*n^3 - 172*n^2 + 14*n + 55)*x^2 + (16*n^4 - 90*n^3 - 97*n^2 + 73*n + 74)*x^3 + (2*n^5 - 21*n^4 - 62*n^3 - 13*n^2 + 62*n + 20)*x^4 - (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] + (2 + (12*n - 6)*x + (22*n^2 - 12*n - 42)*x^2 + (12*n^3 + 14*n^2 - 144*n - 76)*x^3 + (2*n^4 + 12*n^3 - 116*n^2 - 210*n - 84)*x^4 + (2*n^4 - 36*n^3 - 138*n^2 - 156*n - 48)*x^5 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(4 + (10*n + 7)*x + (14*n^2 + 6*n + 1)*x^2 + (6*n^3 + 11*n^2 - 3*n - 6)*x^3 + (n^4 + 2*n^3 + 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-be9753f82aa1d431
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-be9753f82aa1d431
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-be9753f82aa1d431

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.