Real-rooted atlas

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

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,x + 2\,x^{2} + 3\,n + 7\,n x + 2\,n x^{2} + n x^{3} + 2\,n^{2} + 2\,n^{2} x + 2\,n^{2} x^{2} + n^{3} + n^{3} x\right)\,P_{n}(x) = \left(16 + 32\,x + 24\,x^{2} + 8\,x^{3} + x^{4} + 20\,n + 38\,n x + 26\,n x^{2} + 8\,n x^{3} + n x^{4} + 10\,n^{2} + 15\,n^{2} x + 9\,n^{2} x^{2} + 2\,n^{2} x^{3} + 2\,n^{3} + 3\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1628
22820351
3110126424282
43846163367251413
512322560198070011066588
637129504960048401260156729079
7106563248040656268801001020582108122958

All 15 rows

Technical details
Node id
n-2aa22a279dec278f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-216aa5511ef953b2

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:     6,      2;
  2:    28,     20,      3;
  3:   110,    126,     42,      4;
  4:   384,    616,    336,     72,      5;
  5:  1232,   2560,   1980,    700,    110,      6;
  6:  3712,   9504,   9600,   4840,   1260,    156,     7;
  7: 10656,  32480,  40656,  26880,  10010,   2058,   210,    8;
  8: 29440, 104192, 155904, 128128,  62720,  18480,  3136,  272,   9;
  9: 78848, 317952, 554112, 545664, 332640, 129024, 31416, 4536, 342, 10;
%F Formula
The row polynomials satisfy (n^3+2*n^2+3*n+2+(n^3+2*n^2+7*n+2)*x+(2*n^2+2*n+2)*x^2+n*x^3)*P(n,x) = (2*n^3+10*n^2+20*n+16+(3*n^3+15*n^2+38*n+32)*x+(n^3+9*n^2+26*n+24)*x^2+(2*n^2+8*n+8)*x^3+(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[((2*n^3 + 10*n^2 + 20*n + 16 + (3*n^3 + 15*n^2 + 38*n + 32)*x + (n^3 + 9*n^2 + 26*n + 24)*x^2 + (2*n^2 + 8*n + 8)*x^3 + (n + 1)*x^4)*P[n - 1, x])/(n^3 + 2*n^2 + 3*n + 2 + (n^3 + 2*n^2 + 7*n + 2)*x + (2*n^2 + 2*n + 2)*x^2 + n*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] = ((2*n^3 + 10*n^2 + 20*n + 16 + (3*n^3 + 15*n^2 + 38*n + 32)*x + (n^3 + 9*n^2 + 26*n + 24)*x^2 + (2*n^2 + 8*n + 8)*x^3 + (n + 1)*x^4)*v[i - 1])/(n^3 + 2*n^2 + 3*n + 2 + (n^3 + 2*n^2 + 7*n + 2)*x + (2*n^2 + 2*n + 2)*x^2 + n*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.