Real-rooted atlas

A008278 → x ↦ x+1 (shift +1) → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 4\,n x - 2\,n x^{2} + 8\,n^{2} + 10\,n^{2} x + 3\,n^{2} x^{2} - 4\,n^{3} - 4\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-4\,n - 8\,n x - 5\,n x^{2} - n x^{3} + 8\,n^{2} + 12\,n^{2} x + 6\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(3 + 2\,x - 6\,n - 4\,n x + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(4 + 4\,x + x^{2} - 4\,n - 4\,n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2729
34626375
4322278694673
524272858113416456588
61964529488163143964375669792
717023631164422253477044128708467795181

All 15 rows

Technical details
Node id
n-70517fb9c6ff9739 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-dbd8e65f8a4216d7

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:        7,        2;
  3:       46,       26,        3;
  4:      322,      278,       69,        4;
  5:     2427,     2858,     1134,      164,       5;
  6:    19645,    29488,    16314,     3964,     375,       6;
  7:   170236,   311644,   222534,    77044,   12870,     846,      7;
  8:  1573732,  3407828,  2992938,  1354684,  329690,   40242,   1897,    8;
  9: 15459637, 38722316, 40476552, 22732516, 7332245, 1335774, 123550, 4232, 9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-8*n^2+n+(4*n^3-10*n^2+4*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) + (8*n^2-4*n+(12*n^2-8*n)*x+(6*n^2-5*n)*x^2+(n^2-n)*x^3)*d/dx P(n-2,x) + (4*n^2-6*n+3+(2*n^2-4*n+2)*x)*P(n-1,x) - (4*n-4+(4*n-4)*x+(n-1)*x^2)*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[(-(4*n^3 - 8*n^2 + n + (4*n^3 - 10*n^2 + 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] + (8*n^2 - 4*n + (12*n^2 - 8*n)*x + (6*n^2 - 5*n)*x^2 + (n^2 - n)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 6*n + 3 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] - (4*n - 4 + (4*n - 4)*x + (n - 1)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
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] = (-(4*n^3 - 8*n^2 + n + (4*n^3 - 10*n^2 + 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] + (8*n^2 - 4*n + (12*n^2 - 8*n)*x + (6*n^2 - 5*n)*x^2 + (n^2 - n)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 6*n + 3 + (2*n^2 - 4*n + 2)*x)*v[i - 1] - (4*n - 4 + (4*n - 4)*x + (n - 1)*x^2)*deriv(v[i - 1], x))/(n - 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. [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.