Real-rooted atlas

A125092 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(8 + 12\,x + 6\,x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(-12 - 12\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + x\)
\(P_{2}(x) = 18 + 10\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1516
21810129
35654151126
4160224108201513
54328005601802511998
611202592240011202703017533
72816784090725600196037835127702
8691222528313602419211200313650440199873
916640622081013769408054432201604704648451354294
10394241664003110403379202352001088643360067208105011240029
119216043366491520011404809292805174401995845280092409905514290894
122129921105920260198436608003421440223027210348803421447920012320118860114703201
1348742427688967188480112752641189760088957444832256192192055598411440016016140465149955454
1411059206823936193822723354624039463424333132802075673696645123363360864864160160203841638701168466797
Technical details
Node id
n-365173a2200e21cc · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-12c97c7d57838f47

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     5,     1;
  2:    18,    10,      1;
  3:    56,    54,     15,     1;
  4:   160,   224,    108,    20,     1;
  5:   432,   800,    560,   180,    25,     1;
  6:  1120,  2592,   2400,  1120,   270,    30,    1;
  7:  2816,  7840,   9072,  5600,  1960,   378,   35,   1;
  8:  6912, 22528,  31360, 24192, 11200,  3136,  504,  40,  1;
  9: 16640, 62208, 101376, 94080, 54432, 20160, 4704, 648, 45, 1;
%F Formula
The row polynomials satisfy P(n,x) = (8+12*x+6*x^2+x^3)*P(n-3,x) - (12+12*x+3*x^2)*P(n-2,x) + (6+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + x, P(2,x) = 18 + 10*x + x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + x;
P[2, x_] := 18 + 10*x + x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 12*x + 6*x^2 + x^3)*P[n - 3, x] - (12 + 12*x + 3*x^2)*P[n - 2, x] + (6 + 3*x)*P[n - 1, 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] = 5 + x; v[3] = 18 + 10*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 12*x + 6*x^2 + x^3)*v[i - 3] - (12 + 12*x + 3*x^2)*v[i - 2] + (6 + 3*x)*v[i - 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)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [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
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.