Real-rooted atlas

A130534 → D²∘x (shift +1) → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 + 32\,x - 24\,x^{2} + 8\,x^{3} - x^{4} + 32\,n - 48\,n x + 24\,n x^{2} - 4\,n x^{3} - 24\,n^{2} + 24\,n^{2} x - 6\,n^{2} x^{2} + 8\,n^{3} - 4\,n^{3} x - n^{4}\right)\,P_{n-4}(x) + \left(-15 + 28\,x - 18\,x^{2} + 4\,x^{3} + 28\,n - 36\,n x + 12\,n x^{2} - 18\,n^{2} + 12\,n^{2} x + 4\,n^{3}\right)\,P_{n-3}(x) + \left(-7 + 12\,x - 6\,x^{2} + 12\,n - 12\,n x - 6\,n^{2}\right)\,P_{n-2}(x) + \left(-2 + 4\,x + 4\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 3\,x\)
\(P_{2}(x) = 29 + 30\,x + 6\,x^{2}\)
\(P_{3}(x) = 155 + 225\,x + 90\,x^{2} + 10\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1639
22930665
31552259010480
49491695960210153829
5663613692966030104202133439
6524641206249941438920777075628319976
746395611617801077174496650125370175141260363343740
845446761220226012421830648963019413453446103570019804537982076
9488893681391747281529649008829260030096990636243384084067320297055466692204
105732977681715850840201142035612603294004770366151154357821824068418691201192954290666173604216
1172811215362276285731228200294408189362773607815026505210846936338281042847057868385456520091560067887537976344
1299592910208323555188320420613631256299744164720133090812855392959336777981773800113091778811144133074724653243248190911325124579024
1314599360408324908341518272665758373280049970180624002362929408840753080015688167920432680266952371403040177140245830585137537405050501092010521336804726192
14228366158457607918140129004811154637459094487648459880800437894187276001490689718918435979938834846295890744008065075590075544669205107622522422056076258014280120364225491464832
Technical details
Node id
n-e150fdce28bb71f3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-002dc50a5aa6294d

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,         3;
  2:       29,        30,         6;
  3:      155,       225,        90,       10;
  4:      949,      1695,       960,      210,       15;
  5:     6636,     13692,      9660,     3010,      420,      21;
  6:    52464,    120624,     99414,    38920,     7770,     756,     28;
  7:   463956,   1161780,   1077174,   496650,   125370,   17514,   1260,    36;
  8:  4544676,  12202260,  12421830,  6489630,  1941345,  344610,  35700,  1980,   45;
  9: 48889368, 139174728, 152964900, 88292600, 30096990, 6362433, 840840, 67320, 2970, 55;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-8*n^3+24*n^2-32*n+16+(4*n^3-24*n^2+48*n-32)*x+(6*n^2-24*n+24)*x^2+(4*n-8)*x^3+x^4)*P(n-4,x) + (4*n^3-18*n^2+28*n-15+(12*n^2-36*n+28)*x+(12*n-18)*x^2+4*x^3)*P(n-3,x) - (6*n^2-12*n+7+(12*n-12)*x+6*x^2)*P(n-2,x) + (4*n-2+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 6 + 3*x, P(2,x) = 29 + 30*x + 6*x^2, P(3,x) = 155 + 225*x + 90*x^2 + 10*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[2, x_] := 29 + 30*x + 6*x^2;
P[3, x_] := 155 + 225*x + 90*x^2 + 10*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (4*n^3 - 24*n^2 + 48*n - 32)*x + (6*n^2 - 24*n + 24)*x^2 + (4*n - 8)*x^3 + x^4)*P[n - 4, x] + (4*n^3 - 18*n^2 + 28*n - 15 + (12*n^2 - 36*n + 28)*x + (12*n - 18)*x^2 + 4*x^3)*P[n - 3, x] - (6*n^2 - 12*n + 7 + (12*n - 12)*x + 6*x^2)*P[n - 2, x] + (4*n - 2 + 4*x)*P[n - 1, 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] = 6 + 3*x; v[3] = 29 + 30*x + 6*x^2; v[4] = 155 + 225*x + 90*x^2 + 10*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (4*n^3 - 24*n^2 + 48*n - 32)*x + (6*n^2 - 24*n + 24)*x^2 + (4*n - 8)*x^3 + x^4)*v[i - 4] + (4*n^3 - 18*n^2 + 28*n - 15 + (12*n^2 - 36*n + 28)*x + (12*n - 18)*x^2 + 4*x^3)*v[i - 3] - (6*n^2 - 12*n + 7 + (12*n - 12)*x + 6*x^2)*v[i - 2] + (4*n - 2 + 4*x)*v[i - 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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.