Real-rooted atlas

A007318 → 2θ+1 → 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
\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(1 - 3\,x + 2\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
256112
37159132
49283012180
511457050151192
6136613514075181448
715912313152451052111024
8171203646166303921402412304
9191535401092138611345881802715120
1021190765180027302772189084022530111264
1123231104528054950600650822970115527533124576
1225276138641808415118801201287124455154033036153248
132732517946006135852187925740223081415764352002390391114688
14293782275837221021380385105151480390392202290092548455421245760

Generated b-files: A116666 (reversed triangle), A134083 ?

Computed in your browser ?

Technical details
Node id
n-c1f6a6363c73beb4 · revision 1 · JSON
Definition source
derived:reversal
Fingerprint
f6-cbfa13cb29880cff

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,   1;
  2:  5,   6,   1;
  3:  7,  15,   9,    1;
  4:  9,  28,  30,   12,    1;
  5: 11,  45,  70,   50,   15,    1;
  6: 13,  66, 135,  140,   75,   18,   1;
  7: 15,  91, 231,  315,  245,  105,  21,   1;
  8: 17, 120, 364,  616,  630,  392, 140,  24,  1;
  9: 19, 153, 540, 1092, 1386, 1134, 588, 180, 27, 1;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+(4*n-3)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1 + (4*n - 3)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 1 + (4*n - 3)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x))/(2*n - 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.

A116666: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,  3;
   3: 1,  6,   5;
   4: 1,  9,  15,   7;
   5: 1, 12,  30,  28,    9;
   6: 1, 15,  50,  70,   45,   11;
   7: 1, 18,  75, 140,  135,   66,   13;
   8: 1, 21, 105, 245,  315,  231,   91,  15;
   9: 1, 24, 140, 392,  630,  616,  364, 120,  17;
  10: 1, 27, 180, 588, 1134, 1386, 1092, 540, 153, 19;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n-3)*P(n,x) = (2*n-1+(4*n-7)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (4*n - 7)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
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; for(n = 2, N, my(i = n); v[i] = ((2*n - 1 + (4*n - 7)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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.

A134083: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:  1;
  1:  3,   1;
  2:  5,   6,   1;
  3:  7,  15,   9,    1;
  4:  9,  28,  30,   12,    1;
  5: 11,  45,  70,   50,   15,    1;
  6: 13,  66, 135,  140,   75,   18,   1;
  7: 15,  91, 231,  315,  245,  105,  21,   1;
  8: 17, 120, 364,  616,  630,  392, 140,  24,  1;
  9: 19, 153, 540, 1092, 1386, 1134, 588, 180, 27, 1;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+(4*n-3)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1 + (4*n - 3)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 1 + (4*n - 3)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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.

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-c51898909065bf38
  • 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-c51898909065bf38
  • 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-c51898909065bf38

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

Candidate equalities (1)

Definitions that agree on the stored rows without an all-n certificate.

Same fingerprint (1)

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.