Real-rooted atlas

A007318 → D → D+1

nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2426
3912324
4163624480
5258090405240
636150240180606672
7492525255603158471792

Show all 15 stored rows

Generated b-files: A001788 (row sums), A171631 ?

Computed in your browser ?

Other recurrences

Derivative · proved equal for all n (proof step)

\(\left(1 - x - 2\,n + n x + n^{2}\right)\,P_{n}(x) = \left(n x + n x^{2} + n^{2} + n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-922a52ffa8f2e057 · revision 2 · JSON
Definition source
derived:one_plus_derivative
derived:derivative (equal by e-77049924460718df)
Fingerprint
f6-e99c3824c4864868

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:  0;
  1:  1;
  2:  4,   2;
  3:  9,  12,    3;
  4: 16,  36,   24,    4;
  5: 25,  80,   90,   40,    5;
  6: 36, 150,  240,  180,   60,    6;
  7: 49, 252,  525,  560,  315,   84,   7;
  8: 64, 392, 1008, 1400, 1120,  504, 112,   8;
  9: 81, 576, 1764, 3024, 3150, 2016, 756, 144, 9;
%F Formula
The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n+(2*n^2-2*n)*x+(n^2-n)*x^2)*P(n-2,x) + (2*n^2-4*n+(2*n^2-4*n)*x)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (2*n^2 - 2*n)*x + (n^2 - n)*x^2)*P[n - 2, x] + (2*n^2 - 4*n + (2*n^2 - 4*n)*x)*P[n - 1, x])/(n^2 - 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] = 0; v[2] = 1; v[3] = 4 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = (-(n^2 - n + (2*n^2 - 2*n)*x + (n^2 - n)*x^2)*v[i - 2] + (2*n^2 - 4*n + (2*n^2 - 4*n)*x)*v[i - 1])/(n^2 - 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.

A171631: triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1:   1;
   2:   4,   2;
   3:   9,  12,    3;
   4:  16,  36,   24,    4;
   5:  25,  80,   90,   40,    5;
   6:  36, 150,  240,  180,   60,    6;
   7:  49, 252,  525,  560,  315,   84,    7;
   8:  64, 392, 1008, 1400, 1120,  504,  112,    8;
   9:  81, 576, 1764, 3024, 3150, 2016,  756,  144,   9;
  10: 100, 810, 2880, 5880, 7560, 6300, 3360, 1080, 180, 10;
%F Formula
The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n+(2*n^2-2*n)*x+(n^2-n)*x^2)*P(n-2,x) + (2*n^2-4*n+(2*n^2-4*n)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (2*n^2 - 2*n)*x + (n^2 - n)*x^2)*P[n - 2, x] + (2*n^2 - 4*n + (2*n^2 - 4*n)*x)*P[n - 1, x])/(n^2 - 3*n + 2)]];
Table[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] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(n^2 - n + (2*n^2 - 2*n)*x + (n^2 - n)*x^2)*v[i - 2] + (2*n^2 - 4*n + (2*n^2 - 4*n)*x)*v[i - 1])/(n^2 - 3*n + 2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #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-9a80f6183c7ede37, e-d70ffbaedd962540
  • 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-9a80f6183c7ede37, e-d70ffbaedd962540
  • 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-9a80f6183c7ede37, e-d70ffbaedd962540
  • 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-a110aefd975957cb
  • 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-a110aefd975957cb
  • 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-a110aefd975957cb

Proof chains

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

Same fingerprint (27)

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.