Real-rooted atlas

A106246 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(x^{5}\right)\,P_{n-5}(x) + \left(-5\,x^{4}\right)\,P_{n-4}(x) + \left(10\,x^{3}\right)\,P_{n-3}(x) + \left(-10\,x^{2}\right)\,P_{n-2}(x) + \left(5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)
\(P_{2}(x) = 9 + 14\,x + 3\,x^{2}\)
\(P_{3}(x) = 6 + 42\,x + 30\,x^{2} + 4\,x^{3}\)
\(P_{4}(x) = 36\,x + 114\,x^{2} + 52\,x^{3} + 5\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
2914326
364230482
4036114525207
500120240806446
60003004351147856
7000063071415481506

All 15 rows

Technical details
Node id
n-44fffa453c434887 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f08f60cfe94bcd8c

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: 4,  2;
  2: 9, 14,   3;
  3: 6, 42,  30,   4;
  4: 0, 36, 114,  52,   5;
  5: 0,  0, 120, 240,  80,    6;
  6: 0,  0,   0, 300, 435,  114,    7;
  7: 0,  0,   0,   0, 630,  714,  154,    8;
  8: 0,  0,   0,   0,   0, 1176, 1092,  200,   9;
  9: 0,  0,   0,   0,   0,    0, 2016, 1584, 252, 10;
%F Formula
The row polynomials satisfy P(n,x) = x^5*P(n-5,x) - 5*x^4*P(n-4,x) + 10*x^3*P(n-3,x) - 10*x^2*P(n-2,x) + 5*x*P(n-1,x) for n >= 5, with P(0,x) = 1, P(1,x) = 4 + 2*x, P(2,x) = 9 + 14*x + 3*x^2, P(3,x) = 6 + 42*x + 30*x^2 + 4*x^3, P(4,x) = 36*x + 114*x^2 + 52*x^3 + 5*x^4.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[2, x_] := 9 + 14*x + 3*x^2;
P[3, x_] := 6 + 42*x + 30*x^2 + 4*x^3;
P[4, x_] := 36*x + 114*x^2 + 52*x^3 + 5*x^4;
P[n_, x_] := P[n, x] = Expand[x^5*P[n - 5, x] - 5*x^4*P[n - 4, x] + 10*x^3*P[n - 3, x] - 10*x^2*P[n - 2, x] + 5*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] = 4 + 2*x; v[3] = 9 + 14*x + 3*x^2; v[4] = 6 + 42*x + 30*x^2 + 4*x^3; v[5] = 36*x + 114*x^2 + 52*x^3 + 5*x^4; for(n = 5, N, my(i = n + 1); v[i] = x^5*v[i - 5] - 5*x^4*v[i - 4] + 10*x^3*v[i - 3] - 10*x^2*v[i - 2] + 5*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.