Real-rooted atlas

A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.

A007318 → θ+1

A103406 nonnegative coefficients only rational roots

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

Recurrence from OEIS A103406
\(P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-1}(x) + \left(-1 - 2\,x - x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
21438
3169420
4181816548
51103040256112
6112458075367256
711463140175126498576
8116842243503361966491280
911810833663075658828881102816
10120135480105015121470960405100116144
11122165660165027723234264014855501211213312
121241988802475475264686336445522007261441328672
13126234114435757722120121372811583715031469361691461440
14128273145650051201221021274562702720020110114368118319615131072

Generated b-files: A001792 (row sums), A093375 (reversed triangle), A103406 ?

Computed in your browser ?

Other recurrences

Image under θ+1 · proved equal for all n (proof step)

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

Reversed rows · proved equal for all n (proof step)

\(P_{n}(x) = \left(1 + 2\,x - x^{2} + n x^{2}\right)\,P_{n-1}(x) + \left(-x^{2} - x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-801895951aa20f29 · revision 3 · JSON
Definition source
oeis_catalog:A103406
derived:theta_plus_one (equal by e-2526de52e9387fae)
derived:reversal (equal by e-3dbd2ded62b9bb8b)
Fingerprint
f6-e99c3824c4864868

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: 1,  2;
  2: 1,  4,   3;
  3: 1,  6,   9,   4;
  4: 1,  8,  18,  16,   5;
  5: 1, 10,  30,  40,  25,   6;
  6: 1, 12,  45,  80,  75,  36,   7;
  7: 1, 14,  63, 140, 175, 126,  49,   8;
  8: 1, 16,  84, 224, 350, 336, 196,  64,  9;
  9: 1, 18, 108, 336, 630, 756, 588, 288, 81, 10;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1+2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + 2*x + 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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 + 2*x + 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.

A093375: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:  1;
   2:  2,  1;
   3:  3,  4,   1;
   4:  4,  9,   6,   1;
   5:  5, 16,  18,   8,   1;
   6:  6, 25,  40,  30,  10,   1;
   7:  7, 36,  75,  80,  45,  12,   1;
   8:  8, 49, 126, 175, 140,  63,  14,   1;
   9:  9, 64, 196, 336, 350, 224,  84,  16,  1;
  10: 10, 81, 288, 588, 756, 630, 336, 108, 18, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*x)*P(n-1,x) - (1+2*x+x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + 2*x + x^2)*P[n - 2, 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] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = (2 + 2*x)*v[i - 1] - (1 + 2*x + x^2)*v[i - 2]); 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.

A103406: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  4,   3;
  3: 1,  6,   9,   4;
  4: 1,  8,  18,  16,   5;
  5: 1, 10,  30,  40,  25,   6;
  6: 1, 12,  45,  80,  75,  36,   7;
  7: 1, 14,  63, 140, 175, 126,  49,   8;
  8: 1, 16,  84, 224, 350, 336, 196,  64,  9;
  9: 1, 18, 108, 336, 630, 756, 588, 288, 81, 10;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1+2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + 2*x + 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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 + 2*x + 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.

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-a112169ce63438eb, e-ac2f606fc4a9c0c0, e-becc9d9cf02b45c7
  • 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-a112169ce63438eb, e-ac2f606fc4a9c0c0, e-becc9d9cf02b45c7
  • 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-a112169ce63438eb, e-ac2f606fc4a9c0c0, e-becc9d9cf02b45c7
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-a112169ce63438eb, e-becc9d9cf02b45c7

Proof chains

5 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 9 proof steps

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.