Real-rooted atlas

A274496 → (θ+1)²

nonnegative coefficients

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

Recurrence
\(\left(64 + 56\,x + 4\,x^{2} - 12\,x^{3} + 80\,n x + 24\,n x^{2} - 6\,n x^{3} + 56\,n^{2} x^{2} + 22\,n^{2} x^{3} + n^{2} x^{4} + 12\,n^{3} x^{3} + 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(192 - 200\,x - 220\,x^{2} - 132\,x^{3} - 20\,x^{4} - 4\,x^{5} - 64\,n + 608\,n x - 64\,n x^{2} - 142\,n x^{3} - 62\,n x^{4} - 8\,n x^{5} - 136\,n^{2} x + 532\,n^{2} x^{2} + 160\,n^{2} x^{3} + 11\,n^{2} x^{4} - n^{2} x^{5} - 80\,n^{3} x^{2} + 162\,n^{3} x^{3} + 59\,n^{3} x^{4} + 7\,n^{3} x^{5} - 16\,n^{4} x^{3} + 21\,n^{4} x^{4} + 5\,n^{4} x^{5} - n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(32 - 112\,x - 312\,x^{2} - 288\,x^{3} - 160\,x^{4} - 48\,x^{5} - 8\,x^{6} + 96\,n x - 184\,n x^{2} - 564\,n x^{3} - 416\,n x^{4} - 156\,n x^{5} - 24\,n x^{6} + 88\,n^{2} x^{2} - 52\,n^{2} x^{3} - 254\,n^{2} x^{4} - 140\,n^{2} x^{5} - 26\,n^{2} x^{6} + 24\,n^{3} x^{3} - 4\,n^{3} x^{4} - 40\,n^{3} x^{5} - 12\,n^{3} x^{6} + 2\,n^{4} x^{4} - 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
22810
341620
44323672
586472144
6896216128448
716192432256896
81625686410244002560
932512172820488005120
1032640288051204000115213824
116412805760102408000230427648
126415368640204802400013824313671680
131283072172804096048000276486272143360
141283584241927168011200096768439048192360448

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Technical details
Node id
n-44e08c411ecb806d · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4f7c344a4c25ac22

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:  2;
  2:  2,   8;
  3:  4,  16;
  4:  4,  32,   36;
  5:  8,  64,   72;
  6:  8,  96,  216,  128;
  7: 16, 192,  432,  256;
  8: 16, 256,  864, 1024, 400;
  9: 32, 512, 1728, 2048, 800;
%F Formula
The row polynomials satisfy (64+(80*n+56)*x+(56*n^2+24*n+4)*x^2+(12*n^3+22*n^2-6*n-12)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = -(64*n-192+(136*n^2-608*n+200)*x+(80*n^3-532*n^2+64*n+220)*x^2+(16*n^4-162*n^3-160*n^2+142*n+132)*x^3+(n^5-21*n^4-59*n^3-11*n^2+62*n+20)*x^4-(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^5)*P(n-1,x) + (32+(96*n-112)*x+(88*n^2-184*n-312)*x^2+(24*n^3-52*n^2-564*n-288)*x^3+(2*n^4-4*n^3-254*n^2-416*n-160)*x^4-(40*n^3+140*n^2+156*n+48)*x^5-(2*n^4+12*n^3+26*n^2+24*n+8)*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[(-(64*n - 192 + (136*n^2 - 608*n + 200)*x + (80*n^3 - 532*n^2 + 64*n + 220)*x^2 + (16*n^4 - 162*n^3 - 160*n^2 + 142*n + 132)*x^3 + (n^5 - 21*n^4 - 59*n^3 - 11*n^2 + 62*n + 20)*x^4 - (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*P[n - 1, x] + (32 + (96*n - 112)*x + (88*n^2 - 184*n - 312)*x^2 + (24*n^3 - 52*n^2 - 564*n - 288)*x^3 + (2*n^4 - 4*n^3 - 254*n^2 - 416*n - 160)*x^4 - (40*n^3 + 140*n^2 + 156*n + 48)*x^5 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*D[P[n - 1, x], x])/(64 + (80*n + 56)*x + (56*n^2 + 24*n + 4)*x^2 + (12*n^3 + 22*n^2 - 6*n - 12)*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] = (-(64*n - 192 + (136*n^2 - 608*n + 200)*x + (80*n^3 - 532*n^2 + 64*n + 220)*x^2 + (16*n^4 - 162*n^3 - 160*n^2 + 142*n + 132)*x^3 + (n^5 - 21*n^4 - 59*n^3 - 11*n^2 + 62*n + 20)*x^4 - (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*v[i - 1] + (32 + (96*n - 112)*x + (88*n^2 - 184*n - 312)*x^2 + (24*n^3 - 52*n^2 - 564*n - 288)*x^3 + (2*n^4 - 4*n^3 - 254*n^2 - 416*n - 160)*x^4 - (40*n^3 + 140*n^2 + 156*n + 48)*x^5 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*deriv(v[i - 1], x))/(64 + (80*n + 56)*x + (56*n^2 + 24*n + 4)*x^2 + (12*n^3 + 22*n^2 - 6*n - 12)*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-e6a7da242cad039d
  • 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-e6a7da242cad039d
  • 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-e6a7da242cad039d

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.