Real-rooted atlas

A125092 → reverse → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-1 - 4\,x - 6\,x^{2} - 4\,x^{3} - x^{4}\right)\,P_{n-4}(x) + \left(4 + 12\,x + 12\,x^{2} + 4\,x^{3}\right)\,P_{n-3}(x) + \left(-6 - 12\,x - 6\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + x\)
\(P_{2}(x) = 17 + 10\,x + x^{2}\)
\(P_{3}(x) = 43 + 51\,x + 15\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1516
21710128
34351151110
489172102201384
51614454301702511232
626596613358602553013712
7407185533813115150535735110656
85933256742090166230240847640129440
9829533714652222602028611214361261245178848
101121829026685488405565040572186905160765501205824
11147512331455959784513431012243074382293707095935551525824
121897177007398618238029353532234424486012751244055946011226011318912
13239324661115050320606592735763191698412454740207207636351229813266513256320
142969335021726275369001122121165965817807791396824795795322322890891565215477017929856
Technical details
Node id
n-5cb043694daa7f13 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2908e4d2ac2400a9

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:  17,   10,     1;
  3:  43,   51,    15,     1;
  4:  89,  172,   102,    20,     1;
  5: 161,  445,   430,   170,    25,     1;
  6: 265,  966,  1335,   860,   255,    30,    1;
  7: 407, 1855,  3381,  3115,  1505,   357,   35,   1;
  8: 593, 3256,  7420,  9016,  6230,  2408,  476,  40,  1;
  9: 829, 5337, 14652, 22260, 20286, 11214, 3612, 612, 45, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(1+4*x+6*x^2+4*x^3+x^4)*P(n-4,x) + (4+12*x+12*x^2+4*x^3)*P(n-3,x) - (6+12*x+6*x^2)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 5 + x, P(2,x) = 17 + 10*x + x^2, P(3,x) = 43 + 51*x + 15*x^2 + x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + x;
P[2, x_] := 17 + 10*x + x^2;
P[3, x_] := 43 + 51*x + 15*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*P[n - 4, x] + (4 + 12*x + 12*x^2 + 4*x^3)*P[n - 3, x] - (6 + 12*x + 6*x^2)*P[n - 2, x] + (4 + 4*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] = 17 + 10*x + x^2; v[4] = 43 + 51*x + 15*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = -(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*v[i - 4] + (4 + 12*x + 12*x^2 + 4*x^3)*v[i - 3] - (6 + 12*x + 6*x^2)*v[i - 2] + (4 + 4*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

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.