Real-rooted atlas

A007318 → (1/2+(1/2)n)θ+1+n → reverse

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(-3 - 6\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(3 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)
\(P_{2}(x) = 6 + 9\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
269318
3102418456
4155060305160
52190150120456432
6281473153502106371120
7362245888407003368482816
845324100817641890126050410896912
955450162033604410378021007201351016640
10666052475594092409702693033009901651139424
11787923630990017820221761940411880495013201981292160
1291101451481573032175463324804836036193057150171623413212992
131051274709824024550559009010810896096630633003010010218427314487424
1412015759555354909009016516522522523166018018010510545045136502730315151105920

Generated b-files: A001793 (row sums), A124932 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-2c11b65ec5b65118 · revision 1 · JSON
Definition source
derived:reversal
Fingerprint
f6-72ae8e6419635e63

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): yes

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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;
  2:  6,   9,    3;
  3: 10,  24,   18,    4;
  4: 15,  50,   60,   30,    5;
  5: 21,  90,  150,  120,   45,    6;
  6: 28, 147,  315,  350,  210,   63,    7;
  7: 36, 224,  588,  840,  700,  336,   84,   8;
  8: 45, 324, 1008, 1764, 1890, 1260,  504, 108,   9;
  9: 55, 450, 1620, 3360, 4410, 3780, 2100, 720, 135, 10;
%F Formula
The row polynomials satisfy P(n,x) = (1+3*x+3*x^2+x^3)*P(n-3,x) - (3+6*x+3*x^2)*P(n-2,x) + (3+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 6 + 9*x + 3*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 6 + 9*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x + 3*x^2 + x^3)*P[n - 3, x] - (3 + 6*x + 3*x^2)*P[n - 2, x] + (3 + 3*x)*P[n - 1, x]];
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] = 3 + 2*x; v[3] = 6 + 9*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (1 + 3*x + 3*x^2 + x^3)*v[i - 3] - (3 + 6*x + 3*x^2)*v[i - 2] + (3 + 3*x)*v[i - 1]); 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.

A124932: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:  1;
   2:  2,   3;
   3:  3,   9,   6;
   4:  4,  18,  24,   10;
   5:  5,  30,  60,   50,   15;
   6:  6,  45, 120,  150,   90,   21;
   7:  7,  63, 210,  350,  315,  147,   28;
   8:  8,  84, 336,  700,  840,  588,  224,   36;
   9:  9, 108, 504, 1260, 1890, 1764, 1008,  324,  45;
  10: 10, 135, 720, 2100, 3780, 4410, 3360, 1620, 450, 55;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+3*x+3*x^2+x^3)*P(n-3,x) - (3+6*x+3*x^2)*P(n-2,x) + (3+3*x)*P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 3 + 2*x, P(3,x) = 6 + 9*x + 3*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[3, x_] := 6 + 9*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x + 3*x^2 + x^3)*P[n - 3, x] - (3 + 6*x + 3*x^2)*P[n - 2, x] + (3 + 3*x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; v[3] = 6 + 9*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (1 + 3*x + 3*x^2 + x^3)*v[i - 3] - (3 + 6*x + 3*x^2)*v[i - 2] + (3 + 3*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); 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 >= 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-e9ad51cd66b1f493
  • 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-e9ad51cd66b1f493
  • 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-e9ad51cd66b1f493

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 (3)

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