Real-rooted atlas

A007318 → −D²+(−1+n)D+1+n → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4 + 4\,x - 2\,x^{2} + 2\,n + 4\,n x + 7\,n x^{2} + 2\,n^{2} + 3\,n^{2} x + n^{2} x^{3} + n^{3} x + n^{3} x^{2}\right)\,P_{n}(x) = \left(16 + 32\,x + 24\,x^{2} + 8\,x^{3} + x^{4} + 12\,n + 32\,n x + 33\,n x^{2} + 14\,n x^{3} + 2\,n x^{4} + 4\,n^{2} + 14\,n^{2} x + 12\,n^{2} x^{2} + 5\,n^{2} x^{3} + n^{2} x^{4} + 2\,n^{3} x + 3\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
21428951
3441329016282
4128512540208251413
5352176025201520400366588
6928556810080832034506844929079
72368165763628838080217006804107864122958

All 15 rows

Technical details
Node id
n-cd148496166ada8f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b47be3954087bc09

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,      4;
  2:    14,     28,      9;
  3:    44,    132,     90,     16;
  4:   128,    512,    540,    208,     25;
  5:   352,   1760,   2520,   1520,    400,     36;
  6:   928,   5568,  10080,   8320,   3450,    684,    49;
  7:  2368,  16576,  36288,  38080,  21700,   6804,  1078,    64;
  8:  5888,  47104, 120960, 154112, 112000,  48384, 12152,  1600,   81;
  9: 14336, 129024, 380160, 569856, 504000, 278208, 96432, 20160, 2268, 100;
%F Formula
The row polynomials satisfy (2*n^2+2*n+4+(n^3+3*n^2+4*n+4)*x+(n^3+7*n-2)*x^2+n^2*x^3)*P(n,x) = (4*n^2+12*n+16+(2*n^3+14*n^2+32*n+32)*x+(3*n^3+12*n^2+33*n+24)*x^2+(n^3+5*n^2+14*n+8)*x^3+(n^2+2*n+1)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 12*n + 16 + (2*n^3 + 14*n^2 + 32*n + 32)*x + (3*n^3 + 12*n^2 + 33*n + 24)*x^2 + (n^3 + 5*n^2 + 14*n + 8)*x^3 + (n^2 + 2*n + 1)*x^4)*P[n - 1, x])/(2*n^2 + 2*n + 4 + (n^3 + 3*n^2 + 4*n + 4)*x + (n^3 + 7*n - 2)*x^2 + n^2*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 12*n + 16 + (2*n^3 + 14*n^2 + 32*n + 32)*x + (3*n^3 + 12*n^2 + 33*n + 24)*x^2 + (n^3 + 5*n^2 + 14*n + 8)*x^3 + (n^2 + 2*n + 1)*x^4)*v[i - 1])/(2*n^2 + 2*n + 4 + (n^3 + 3*n^2 + 4*n + 4)*x + (n^3 + 7*n - 2)*x^2 + n^2*x^3)); 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.