Real-rooted atlas

A334218 → (θ+1)²

nonnegative coefficients

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

Recurrence
\(\left(24 - 44\,n + 30\,n^{2} - 9\,n^{3} + n^{4}\right)\,P_{n}(x) = \left(-24\,x + 264\,x^{2} + 44\,n x - 964\,n x^{2} - 24\,n^{2} x + 1426\,n^{2} x^{2} + 4\,n^{3} x - 1095\,n^{3} x^{2} + 461\,n^{4} x^{2} - 101\,n^{5} x^{2} + 9\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(2\,x + 2\,x^{2} + 248\,x^{3} - 3\,n x - 25\,n x^{2} - 860\,n x^{3} + n^{2} x + 42\,n^{2} x^{2} + 1190\,n^{2} x^{3} - 23\,n^{3} x^{2} - 837\,n^{3} x^{3} + 4\,n^{4} x^{2} + 313\,n^{4} x^{3} - 58\,n^{5} x^{3} + 4\,n^{6} x^{3}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} - 38\,x^{3} + 40\,x^{4} + 9\,n x^{2} + 123\,n x^{3} - 132\,n x^{4} - 12\,n^{2} x^{2} - 158\,n^{2} x^{3} + 170\,n^{2} x^{4} + 6\,n^{3} x^{2} + 101\,n^{3} x^{3} - 107\,n^{3} x^{4} - n^{4} x^{2} - 32\,n^{4} x^{3} + 33\,n^{4} x^{4} + 4\,n^{5} x^{3} - 4\,n^{5} x^{4}\right)\,P_{n-2}''(x) + \left(-6 - 96\,x + 17\,n + 272\,n x - 17\,n^{2} - 296\,n^{2} x + 7\,n^{3} + 156\,n^{3} x - n^{4} - 40\,n^{4} x + 4\,n^{5} x\right)\,P_{n-1}(x) + \left(15\,x + 48\,x^{2} - 41\,n x - 160\,n x^{2} + 39\,n^{2} x + 204\,n^{2} x^{2} - 15\,n^{3} x - 124\,n^{3} x^{2} + 2\,n^{4} x + 36\,n^{4} x^{2} - 4\,n^{5} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
111
211
3088
40122739
501614464224
60204958801251520
702414046336390021611880
80283591338245285014364343105000
90328640152448483200343008470405121034880
100362000761819235142755060556189321314227272911249280
1104045180233728022058500562284004323466093491204066201000133660800
12044100287841984012517780051890018470628080632045516842625440111430013311723075200
130482198882930630466104640042068652489245857824747887155221417903361831644002956272172823950080000
1405247771199480368331099762531037045832103520326182135210221952698333531221324399050075232028376433762197357015859200
150561030428331458624159367075002133338936401032856354488203855411558417073747900725925941490007713366430029831276161934914827445681118643200

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Technical details
Node id
n-73486cfabd571954 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-bd6f537d6d33e339

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1;
   3: 0,  8;
   4: 0, 12,    27;
   5: 0, 16,   144,     64;
   6: 0, 20,   495,    880,     125;
   7: 0, 24,  1404,   6336,    3900,     216;
   8: 0, 28,  3591,  33824,   52850,   14364,     343;
   9: 0, 32,  8640, 152448,  483200,  343008,   47040,    512;
  10: 0, 36, 20007, 618192, 3514275, 5060556, 1893213, 142272, 729;
%F Formula
The row polynomials satisfy (n^4-9*n^3+30*n^2-44*n+24)*P(n,x) = ((4*n^3-24*n^2+44*n-24)*x+(9*n^6-101*n^5+461*n^4-1095*n^3+1426*n^2-964*n+264)*x^2)*P(n-2,x) + ((n^2-3*n+2)*x+(4*n^4-23*n^3+42*n^2-25*n+2)*x^2+(4*n^6-58*n^5+313*n^4-837*n^3+1190*n^2-860*n+248)*x^3)*d/dx P(n-2,x) - ((n^4-6*n^3+12*n^2-9*n+2)*x^2-(4*n^5-32*n^4+101*n^3-158*n^2+123*n-38)*x^3+(4*n^5-33*n^4+107*n^3-170*n^2+132*n-40)*x^4)*d^2/dx^2 P(n-2,x) - (n^4-7*n^3+17*n^2-17*n+6-(4*n^5-40*n^4+156*n^3-296*n^2+272*n-96)*x)*P(n-1,x) + ((2*n^4-15*n^3+39*n^2-41*n+15)*x-(4*n^5-36*n^4+124*n^3-204*n^2+160*n-48)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 8*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(((4*n^3 - 24*n^2 + 44*n - 24)*x + (9*n^6 - 101*n^5 + 461*n^4 - 1095*n^3 + 1426*n^2 - 964*n + 264)*x^2)*P[n - 2, x] + ((n^2 - 3*n + 2)*x + (4*n^4 - 23*n^3 + 42*n^2 - 25*n + 2)*x^2 + (4*n^6 - 58*n^5 + 313*n^4 - 837*n^3 + 1190*n^2 - 860*n + 248)*x^3)*D[P[n - 2, x], x] - ((n^4 - 6*n^3 + 12*n^2 - 9*n + 2)*x^2 - (4*n^5 - 32*n^4 + 101*n^3 - 158*n^2 + 123*n - 38)*x^3 + (4*n^5 - 33*n^4 + 107*n^3 - 170*n^2 + 132*n - 40)*x^4)*D[P[n - 2, x], {x, 2}] - (n^4 - 7*n^3 + 17*n^2 - 17*n + 6 - (4*n^5 - 40*n^4 + 156*n^3 - 296*n^2 + 272*n - 96)*x)*P[n - 1, x] + ((2*n^4 - 15*n^3 + 39*n^2 - 41*n + 15)*x - (4*n^5 - 36*n^4 + 124*n^3 - 204*n^2 + 160*n - 48)*x^2)*D[P[n - 1, x], x])/(n^4 - 9*n^3 + 30*n^2 - 44*n + 24)]];
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; v[3] = 8*x; for(n = 4, N, my(i = n); v[i] = (((4*n^3 - 24*n^2 + 44*n - 24)*x + (9*n^6 - 101*n^5 + 461*n^4 - 1095*n^3 + 1426*n^2 - 964*n + 264)*x^2)*v[i - 2] + ((n^2 - 3*n + 2)*x + (4*n^4 - 23*n^3 + 42*n^2 - 25*n + 2)*x^2 + (4*n^6 - 58*n^5 + 313*n^4 - 837*n^3 + 1190*n^2 - 860*n + 248)*x^3)*deriv(v[i - 2], x) - ((n^4 - 6*n^3 + 12*n^2 - 9*n + 2)*x^2 - (4*n^5 - 32*n^4 + 101*n^3 - 158*n^2 + 123*n - 38)*x^3 + (4*n^5 - 33*n^4 + 107*n^3 - 170*n^2 + 132*n - 40)*x^4)*deriv(deriv(v[i - 2], x), x) - (n^4 - 7*n^3 + 17*n^2 - 17*n + 6 - (4*n^5 - 40*n^4 + 156*n^3 - 296*n^2 + 272*n - 96)*x)*v[i - 1] + ((2*n^4 - 15*n^3 + 39*n^2 - 41*n + 15)*x - (4*n^5 - 36*n^4 + 124*n^3 - 204*n^2 + 160*n - 48)*x^2)*deriv(v[i - 1], x))/(n^4 - 9*n^3 + 30*n^2 - 44*n + 24)); 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, 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-404a8e815c57c4e9
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-404a8e815c57c4e9
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-404a8e815c57c4e9
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-404a8e815c57c4e9

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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.