Real-rooted atlas

A126351 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,x + n\right)\,P_{n}(x) = \left(-3\,x^{2} - 2\,x^{3} - 6\,n - 13\,n x - 13\,n x^{2} - 7\,n x^{3} - 2\,n x^{4} - 5\,n^{2} - 11\,n^{2} x - 9\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(6 + 22\,x + 33\,x^{2} + 26\,x^{3} + 11\,x^{4} + 2\,x^{5} + 5\,n + 17\,n x + 22\,n x^{2} + 13\,n x^{3} + 3\,n x^{4} + n^{2} + 3\,n^{2} x + 3\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(6 + 8\,x + 5\,x^{2} + 2\,x^{3} + 7\,n + 10\,n x + 5\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2 - 6\,x - 6\,x^{2} - 2\,x^{3} - n - 2\,n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
22321448
3108157678340
45341075733193162551
528037261691329335313220473
61562349742614233678810881143364175954
7922493497285342454187781771663884138511281614986

All 15 rows

Technical details
Node id
n-f823c8ac75bd5e22 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-94f7a0350f063e71

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,        2;
  2:      23,       21,        4;
  3:     108,      157,       67,        8;
  4:     534,     1075,      733,      193,       16;
  5:    2803,     7261,     6913,     2933,      531,       32;
  6:   15623,    49742,    61423,    36788,    10881,     1433,      64;
  7:   92249,   349728,   534245,   418778,   177166,    38841,    3851,    128;
  8:  575341,  2537297,  4639171,  4530277,  2533296,   806153,  136109,  10401,   256;
  9: 3778708, 19041277, 40667833, 47792573, 33541154, 14257619, 3549631, 473677, 28387, 512;
%F Formula
The row polynomials satisfy (n+2+2*x)*P(n,x) = -(n^3+5*n^2+6*n+(2*n^3+11*n^2+13*n)*x+(n^3+9*n^2+13*n+3)*x^2+(3*n^2+7*n+2)*x^3+2*n*x^4)*P(n-2,x) + (n^2+5*n+6+(3*n^2+17*n+22)*x+(3*n^2+22*n+33)*x^2+(n^2+13*n+26)*x^3+(3*n+11)*x^4+2*x^5)*d/dx P(n-2,x) + (2*n^2+7*n+6+(2*n^2+10*n+8)*x+(5*n+5)*x^2+2*x^3)*P(n-1,x) - (n+2+(2*n+6)*x+(n+6)*x^2+2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 5*n^2 + 6*n + (2*n^3 + 11*n^2 + 13*n)*x + (n^3 + 9*n^2 + 13*n + 3)*x^2 + (3*n^2 + 7*n + 2)*x^3 + 2*n*x^4)*P[n - 2, x] + (n^2 + 5*n + 6 + (3*n^2 + 17*n + 22)*x + (3*n^2 + 22*n + 33)*x^2 + (n^2 + 13*n + 26)*x^3 + (3*n + 11)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n^2 + 7*n + 6 + (2*n^2 + 10*n + 8)*x + (5*n + 5)*x^2 + 2*x^3)*P[n - 1, x] - (n + 2 + (2*n + 6)*x + (n + 6)*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 2 + 2*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 5*n^2 + 6*n + (2*n^3 + 11*n^2 + 13*n)*x + (n^3 + 9*n^2 + 13*n + 3)*x^2 + (3*n^2 + 7*n + 2)*x^3 + 2*n*x^4)*v[i - 2] + (n^2 + 5*n + 6 + (3*n^2 + 17*n + 22)*x + (3*n^2 + 22*n + 33)*x^2 + (n^2 + 13*n + 26)*x^3 + (3*n + 11)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (2*n^2 + 7*n + 6 + (2*n^2 + 10*n + 8)*x + (5*n + 5)*x^2 + 2*x^3)*v[i - 1] - (n + 2 + (2*n + 6)*x + (n + 6)*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 2 + 2*x)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.