Real-rooted atlas

A135065: A127733 * A007318 as infinite lower triangular matrices.

A007318 → ×(1+2n+n²)

A135065 palindromic γ-positive nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · Hadamard kernel what this means

Recurrence from OEIS A135065
\(P_{n}(x) = \left(8 + 8\,x - 4\,n - 4\,n x\right)\,P_{n-1}(x) + \left(1 + 2\,x + x^{2} + 4\,n + 8\,n x + 4\,n x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
2918936
316484816128
42510015010025400
536180360360180361152
649294735980735294493136
7644481344224022401344448648192

Show all 15 stored rows

Generated b-files: A014477 (row sums), A135065, A135065 (reversed triangle) ?

Computed in your browser ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

\(\left(n^{2}\right)\,P_{n}(x) = \left(1 + x + 2\,n + 2\,n x + n^{2} + n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-375ba152c4aebf61 · revision 3 · JSON
Definition source
oeis_catalog:A135065
derived:shift_x_plus_one (equal by e-7fde857125b8da55)
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • 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:   4,   4;
  2:   9,  18,    9;
  3:  16,  48,   48,   16;
  4:  25, 100,  150,  100,    25;
  5:  36, 180,  360,  360,   180,    36;
  6:  49, 294,  735,  980,   735,   294,   49;
  7:  64, 448, 1344, 2240,  2240,  1344,  448,   64;
  8:  81, 648, 2268, 4536,  5670,  4536, 2268,  648,  81;
  9: 100, 900, 3600, 8400, 12600, 12600, 8400, 3600, 900, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(4*n-8+(4*n-8)*x)*P(n-1,x) + (4*n+1+(8*n+2)*x+(4*n+1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(4*n - 8 + (4*n - 8)*x)*P[n - 1, x] + (4*n + 1 + (8*n + 2)*x + (4*n + 1)*x^2)*P[n - 2, 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] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n - 8 + (4*n - 8)*x)*v[i - 1] + (4*n + 1 + (8*n + 2)*x + (4*n + 1)*x^2)*v[i - 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.

A135065: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   4,   4;
  2:   9,  18,    9;
  3:  16,  48,   48,   16;
  4:  25, 100,  150,  100,    25;
  5:  36, 180,  360,  360,   180,    36;
  6:  49, 294,  735,  980,   735,   294,   49;
  7:  64, 448, 1344, 2240,  2240,  1344,  448,   64;
  8:  81, 648, 2268, 4536,  5670,  4536, 2268,  648,  81;
  9: 100, 900, 3600, 8400, 12600, 12600, 8400, 3600, 900, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(4*n-8+(4*n-8)*x)*P(n-1,x) + (4*n+1+(8*n+2)*x+(4*n+1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(4*n - 8 + (4*n - 8)*x)*P[n - 1, x] + (4*n + 1 + (8*n + 2)*x + (4*n + 1)*x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n - 8 + (4*n - 8)*x)*v[i - 1] + (4*n + 1 + (8*n + 2)*x + (4*n + 1)*x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A135065: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   4,   4;
  2:   9,  18,    9;
  3:  16,  48,   48,   16;
  4:  25, 100,  150,  100,    25;
  5:  36, 180,  360,  360,   180,    36;
  6:  49, 294,  735,  980,   735,   294,   49;
  7:  64, 448, 1344, 2240,  2240,  1344,  448,   64;
  8:  81, 648, 2268, 4536,  5670,  4536, 2268,  648,  81;
  9: 100, 900, 3600, 8400, 12600, 12600, 8400, 3600, 900, 100;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = -(4*n-8+(4*n-8)*x)*P(n-1,x) + (4*n+1+(8*n+2)*x+(4*n+1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(4*n - 8 + (4*n - 8)*x)*P[n - 1, x] + (4*n + 1 + (8*n + 2)*x + (4*n + 1)*x^2)*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n - 8 + (4*n - 8)*x)*v[i - 1] + (4*n + 1 + (8*n + 2)*x + (4*n + 1)*x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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-b78deffc64bc1942, e-f15d78d6731b57c8
  • 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-7157ac2ddeb657bf, e-b78deffc64bc1942, e-f15d78d6731b57c8
  • 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-7157ac2ddeb657bf, e-b78deffc64bc1942, e-f15d78d6731b57c8
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-7157ac2ddeb657bf

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

Used in 7 proof steps

Same fingerprint (59)

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.