Real-rooted atlas

A007318 → D → D

palindromic γ-positive nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
00
10
222
36612
412241248
520606020160
63012018012030480
742210420420210421344
8563368401120840336563584
9725041512252025201512504729216
1090720252050406300504025207209023040
111109903960924013860138609240396099011056320
1213213205940158402772033264277201584059401320132135168
131561716858025740514807207272072514802574085801716156319488
1418221841201240040900901441441681681441449009040040120122184182745472

Generated b-files: A001815 (row sums), A140880, A140880 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-4330f616ae0ed2bc · revision 2 · JSON
Definition source
derived:derivative
Fingerprint
f6-77a7ed3b1d5f257e

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -2)
  • 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:  0;
  1:  0;
  2:  2;
  3:  6,   6;
  4: 12,  24,   12;
  5: 20,  60,   60,   20;
  6: 30, 120,  180,  120,   30;
  7: 42, 210,  420,  420,  210,   42;
  8: 56, 336,  840, 1120,  840,  336,  56;
  9: 72, 504, 1512, 2520, 2520, 1512, 504, 72;
%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) = 0, P(2,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
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] = 0; v[3] = 2; 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.

A140880: triangle, offset 0, from the row P2
%e Example
Triangle begins:
  0:   2;
  1:   6,   6;
  2:  12,  24,   12;
  3:  20,  60,   60,   20;
  4:  30, 120,  180,  120,    30;
  5:  42, 210,  420,  420,   210,    42;
  6:  56, 336,  840, 1120,   840,   336,   56;
  7:  72, 504, 1512, 2520,  2520,  1512,  504,   72;
  8:  90, 720, 2520, 5040,  6300,  5040, 2520,  720,  90;
  9: 110, 990, 3960, 9240, 13860, 13860, 9240, 3960, 990, 110;
%F Formula
The row polynomials satisfy (n^2+n)*P(n,x) = -(n^2+3*n+2+(2*n^2+6*n+4)*x+(n^2+3*n+2)*x^2)*P(n-2,x) + (2*n^2+4*n+(2*n^2+4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 6 + 6*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 6 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 3*n + 2 + (2*n^2 + 6*n + 4)*x + (n^2 + 3*n + 2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + (2*n^2 + 4*n)*x)*P[n - 1, x])/(n^2 + n)]];
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] = 2; v[2] = 6 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + 3*n + 2 + (2*n^2 + 6*n + 4)*x + (n^2 + 3*n + 2)*x^2)*v[i - 2] + (2*n^2 + 4*n + (2*n^2 + 4*n)*x)*v[i - 1])/(n^2 + n)); 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.

A140880: reversed triangle, offset 0, from the row P2
%e Example
Triangle begins:
  0:   2;
  1:   6,   6;
  2:  12,  24,   12;
  3:  20,  60,   60,   20;
  4:  30, 120,  180,  120,    30;
  5:  42, 210,  420,  420,   210,    42;
  6:  56, 336,  840, 1120,   840,   336,   56;
  7:  72, 504, 1512, 2520,  2520,  1512,  504,   72;
  8:  90, 720, 2520, 5040,  6300,  5040, 2520,  720,  90;
  9: 110, 990, 3960, 9240, 13860, 13860, 9240, 3960, 990, 110;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n^2+n)*P(n,x) = -(n^2+3*n+2+(2*n^2+6*n+4)*x+(n^2+3*n+2)*x^2)*P(n-2,x) + (2*n^2+4*n+(2*n^2+4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 6 + 6*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 6 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 3*n + 2 + (2*n^2 + 6*n + 4)*x + (n^2 + 3*n + 2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + (2*n^2 + 4*n)*x)*P[n - 1, x])/(n^2 + n)]];
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] = 2; v[2] = 6 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + 3*n + 2 + (2*n^2 + 6*n + 4)*x + (n^2 + 3*n + 2)*x^2)*v[i - 2] + (2*n^2 + 4*n + (2*n^2 + 4*n)*x)*v[i - 1])/(n^2 + n)); 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-d8d4324b02c6b5ad
  • 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-d8d4324b02c6b5ad
  • 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-d8d4324b02c6b5ad
  • 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-f6bd14f39c5efd23
  • 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-f6bd14f39c5efd23
  • 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-f6bd14f39c5efd23

Proof chains

2 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 (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. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.