Real-rooted atlas

A007318 → ×(1+2n+n²) → (θ+1)²

nonnegative coefficients

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

Recurrence
\(\left(n^{2} - 2\,n^{3} + n^{4}\right)\,P_{n}(x) = \left(-n^{2} - 4\,n^{2} x + 9\,n^{2} x^{2} - 2\,n^{3} - 8\,n^{3} x + 18\,n^{3} x^{2} - n^{4} - 4\,n^{4} x + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(4\,n^{2} x^{2} + 4\,n^{2} x^{3} + 8\,n^{3} x^{2} + 8\,n^{3} x^{3} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(2 + 4\,x - 4\,n^{2} - 8\,n^{2} x + 2\,n^{4} + 4\,n^{4} x\right)\,P_{n-1}(x) + \left(-4\,x^{2} + 8\,n^{2} x^{2} - 4\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141620
297281162
316192432256896
425400135016006254000
536720324057604500129615552
64911766615156801837510584240154880
764179212096358405600048384219524096180224
88125922041272576141750163296111132414726561559872
910036003240013440031500045360041160023040072900100001664000
1012148404900523232063525010977121245090929280441045121000146414770304
11144633671280380160118800023950083259872304128019245607920001916642073613271040
12169811210038659488020913754818528765164485662726776055371800013496342920322856135995648
13196101921375928968963503500908107216480464215255042043241214014000678277622014724306123841695535104
14225126001842751310400563062516216200331080754942080054729675450450002725222511793600346027561740050625248832000

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Technical details
Node id
n-4dd6644b7960d1ef · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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

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:   4,   16;
  2:   9,   72,    81;
  3:  16,  192,   432,    256;
  4:  25,  400,  1350,   1600,    625;
  5:  36,  720,  3240,   5760,   4500,   1296;
  6:  49, 1176,  6615,  15680,  18375,  10584,   2401;
  7:  64, 1792, 12096,  35840,  56000,  48384,  21952,   4096;
  8:  81, 2592, 20412,  72576, 141750, 163296, 111132,  41472,  6561;
  9: 100, 3600, 32400, 134400, 315000, 453600, 411600, 230400, 72900, 10000;
%F Formula
The row polynomials satisfy (n^4-2*n^3+n^2)*P(n,x) = -(n^4+2*n^3+n^2+(4*n^4+8*n^3+4*n^2)*x-(9*n^4+18*n^3+9*n^2)*x^2)*P(n-2,x) + ((4*n^4+8*n^3+4*n^2)*x^2+(4*n^4+8*n^3+4*n^2)*x^3)*d/dx P(n-2,x) + (2*n^4-4*n^2+2+(4*n^4-8*n^2+4)*x)*P(n-1,x) - (4*n^4-8*n^2+4)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 + 2*n^3 + n^2 + (4*n^4 + 8*n^3 + 4*n^2)*x - (9*n^4 + 18*n^3 + 9*n^2)*x^2)*P[n - 2, x] + ((4*n^4 + 8*n^3 + 4*n^2)*x^2 + (4*n^4 + 8*n^3 + 4*n^2)*x^3)*D[P[n - 2, x], x] + (2*n^4 - 4*n^2 + 2 + (4*n^4 - 8*n^2 + 4)*x)*P[n - 1, x] - (4*n^4 - 8*n^2 + 4)*x^2*D[P[n - 1, x], x])/(n^4 - 2*n^3 + n^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 + 2*n^3 + n^2 + (4*n^4 + 8*n^3 + 4*n^2)*x - (9*n^4 + 18*n^3 + 9*n^2)*x^2)*v[i - 2] + ((4*n^4 + 8*n^3 + 4*n^2)*x^2 + (4*n^4 + 8*n^3 + 4*n^2)*x^3)*deriv(v[i - 2], x) + (2*n^4 - 4*n^2 + 2 + (4*n^4 - 8*n^2 + 4)*x)*v[i - 1] - (4*n^4 - 8*n^2 + 4)*x^2*deriv(v[i - 1], x))/(n^4 - 2*n^3 + n^2)); 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-b639f023f58d36b2
  • 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-b639f023f58d36b2
  • 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-b639f023f58d36b2

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 (12)

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

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  4. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  5. [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
  6. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.