Real-rooted atlas

A272866 → Veronese 2 → D

nonnegative coefficients

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

Recurrence

\(\left(16 - 12\,x + 6\,x^{2} - 25\,n + 25\,n x - 10\,n x^{2} + 9\,n^{2} - 13\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-7\,n - 193\,n x + 107\,n x^{2} - 22\,n x^{3} + 9\,n^{2} + 59\,n^{2} x - 55\,n^{2} x^{2} + 12\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(18\,n x - 134\,n x^{2} + 74\,n x^{3} - 8\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
211213
33060393
4583901744626
5951370205538053905
61413540119677080705623439
7196760246599621321900511767136717

All 15 rows

Technical details
Node id
n-82f81d3f074b1e67 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-3ec3919013c8a27a

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:   0;
  1:   1;
  2:  11,     2;
  3:  30,    60,      3;
  4:  58,   390,    174,       4;
  5:  95,  1370,   2055,     380,       5;
  6: 141,  3540,  11967,    7080,     705,      6;
  7: 196,  7602,  46599,   62132,   19005,   1176,     7;
  8: 260, 14420, 140784,  346060,  234640,  43260,  1820,    8;
  9: 333, 25020, 357462, 1425132, 1781415, 714924, 87570, 2664, 9;
%F Formula
The row polynomials satisfy (9*n^2-25*n+16-(13*n^2-25*n+12)*x+(4*n^2-10*n+6)*x^2)*P(n,x) = (9*n^2-7*n+(59*n^2-193*n)*x-(55*n^2-107*n)*x^2+(12*n^2-22*n)*x^3)*P(n-1,x) + (18*n*x-134*n*x^2+74*n*x^3-8*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^2 - 7*n + (59*n^2 - 193*n)*x - (55*n^2 - 107*n)*x^2 + (12*n^2 - 22*n)*x^3)*P[n - 1, x] + (18*n*x - 134*n*x^2 + 74*n*x^3 - 8*n*x^4)*D[P[n - 1, x], x])/(9*n^2 - 25*n + 16 - (13*n^2 - 25*n + 12)*x + (4*n^2 - 10*n + 6)*x^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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^2 - 7*n + (59*n^2 - 193*n)*x - (55*n^2 - 107*n)*x^2 + (12*n^2 - 22*n)*x^3)*v[i - 1] + (18*n*x - 134*n*x^2 + 74*n*x^3 - 8*n*x^4)*deriv(v[i - 1], x))/(9*n^2 - 25*n + 16 - (13*n^2 - 25*n + 12)*x + (4*n^2 - 10*n + 6)*x^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)

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.