Real-rooted atlas

A132393 → D+1 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 + 4\,x - 4\,x^{2} - 2\,n x + 4\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(2 - 3\,x + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
21315
3168217
41102928674
51157515512424394
61211605659496681202484
7128301161045646636424872018108
81365183892165694020852464311765040149904
9145834835849665179529387260463956259488403201389456
10155127516410129423648963207030540674204544676241603236288014257440
1116618703003030297320064668829260254941504639157648889368248860803628800160460640
12178265151920651453549694231802441126032940335236726571950280573297768281004480399168001965444480
131913653856571307163136718011004033035210017671893627736470004906875876191047281121536345188736047900160026029779840
141105491413586324764743141438328506335018712349378872720857299744164727010227187610785172944099592910208458325388806227020800370643938560
Technical details
Node id
n-a0ea8889d977e0cb · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-b29c1c14d7373850

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: 1,  1;
  2: 1,  3,   1;
  3: 1,  6,   8,    2;
  4: 1, 10,  29,   28,     6;
  5: 1, 15,  75,  155,   124,     24;
  6: 1, 21, 160,  565,   949,    668,    120;
  7: 1, 28, 301, 1610,  4564,   6636,   4248,    720;
  8: 1, 36, 518, 3892, 16569,  40208,  52464,  31176,   5040;
  9: 1, 45, 834, 8358, 49665, 179529, 387260, 463956, 259488, 40320;
%F Formula
The row polynomials satisfy P(n,x) = -(1+(2*n-4)*x+(n^2-4*n+4)*x^2)*P(n-2,x) + (2+(2*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[-(1 + (2*n - 4)*x + (n^2 - 4*n + 4)*x^2)*P[n - 2, x] + (2 + (2*n - 3)*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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + (2*n - 4)*x + (n^2 - 4*n + 4)*x^2)*v[i - 2] + (2 + (2*n - 3)*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

Same fingerprint ?

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.