Real-rooted atlas

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

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-4 - n^{2} + n^{3}\right)\,P_{n}(x) = \left(-18 + 2\,x^{2} - 9\,n - 6\,n x - n x^{2} - 3\,n^{2} x - n^{2} x^{2} + 3\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(18 + 30\,x + 14\,x^{2} + 2\,x^{3} + 9\,n + 15\,n x + 7\,n x^{2} + n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 2\,x\)
\(P_{2}(x) = 31 + 20\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1628
23120354
3144138424328
46217923667251856
525384050252076011069984
69963191161498561201365156751712
7379088505080514415801260022262108260096

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(-2 - n + n^{2}\right)\,P_{n}(x) = \left(18 + 12\,x + 2\,x^{2} - 9\,n - 6\,n x - n x^{2} - 9\,n^{2} - 6\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-18 - 6\,x + 6\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 2\,x\)
\(P_{2}(x) = 31 + 20\,x + 3\,x^{2}\)
Technical details
Node id
n-ca15886a6e790e73 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-ff79e0a5be8a3b3e)
Fingerprint
f6-0cd5a224b7200c7b

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:     31,      20,       3;
  3:    144,     138,      42,       4;
  4:    621,     792,     366,      72,      5;
  5:   2538,    4050,    2520,     760,    110,      6;
  6:   9963,   19116,   14985,    6120,   1365,    156,     7;
  7:  37908,   85050,   80514,   41580,  12600,   2226,   210,    8;
  8: 140697,  361584,  401436,  250992,  96390,  23184,  3388,  272,   9;
  9: 511758, 1482786, 1889568, 1388016, 646380, 197316, 39312, 4896, 342, 10;
%F Formula
The row polynomials satisfy (n^3-n^2-4)*P(n,x) = (3*n^3-9*n-18+(n^3-3*n^2-6*n)*x-(n^2+n-2)*x^2)*P(n-1,x) + (9*n+18+(15*n+30)*x+(7*n+14)*x^2+(n+2)*x^3)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 2*x, P(2,x) = 31 + 20*x + 3*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 2*x;
P[2, x_] := 31 + 20*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^3 - 9*n - 18 + (n^3 - 3*n^2 - 6*n)*x - (n^2 + n - 2)*x^2)*P[n - 1, x] + (9*n + 18 + (15*n + 30)*x + (7*n + 14)*x^2 + (n + 2)*x^3)*D[P[n - 1, x], x])/(n^3 - n^2 - 4)]];
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] = 6 + 2*x; v[3] = 31 + 20*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((3*n^3 - 9*n - 18 + (n^3 - 3*n^2 - 6*n)*x - (n^2 + n - 2)*x^2)*v[i - 1] + (9*n + 18 + (15*n + 30)*x + (7*n + 14)*x^2 + (n + 2)*x^3)*deriv(v[i - 1], x))/(n^3 - n^2 - 4)); 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.