Real-rooted atlas

A088996 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n - 24\,n x - 16\,n x^{2} + 12\,n^{2} + 40\,n^{2} x + 32\,n^{2} x^{2} - 4\,n^{3} - 16\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 - 6\,n - 8\,n x + 4\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1549
2238432139
315193615523843023
41297111443224033664614484489
51377315007661256011295368514561228802880281
61741832289260119869603147782442164224247357442949120115777315
725554393922356824968636884680601616243688961699311616813514752825753605358042015

All 15 rows

Technical details
Node id
n-52357e53bac241a2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ee19bf4a3c044177

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,           4;
  2:        23,          84,           32;
  3:       151,         936,         1552,          384;
  4:      1297,       11144,        32240,        33664,          6144;
  5:     13773,      150076,       612560,      1129536,        851456,        122880;
  6:    174183,     2289260,     11986960,     31477824,      42164224,      24735744,       2949120;
  7:   2555439,    39223568,    249686368,    846806016,    1624368896,    1699311616,     813514752,      82575360;
  8:  42655905,   747577296,   5589301920,  23159091200,   57621014784,   86633902080,   74116284416,   29913513984,    2642411520;
  9: 798123285, 15707066292, 134655986784, 657749352064, 2006008872192, 3923459306496, 4835879649280, 3492587962368, 1216934313984, 95126814720;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(16*n^3-40*n^2+24*n)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (4*n^2-6*n+1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (16*n^3 - 40*n^2 + 24*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (4*n^2 - 6*n + 1 + (8*n^2 - 8*n)*x)*P[n - 1, 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] = 1; v[2] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (16*n^3 - 40*n^2 + 24*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (4*n^2 - 6*n + 1 + (8*n^2 - 8*n)*x)*v[i - 1])/(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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.