Real-rooted atlas

A269944 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 - 20\,x - 4\,x^{2} + 40\,n + 16\,n x - 26\,n^{2} - 4\,n^{2} x + 8\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(7 + 4\,x - 6\,n + 2\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
2814426
34498568206
44601136820208162640
58020209521698854647203252176
6211920573432498224182488307682336641499232
77929440219762641998686479454161545408151872716812859542560

All 15 rows

Technical details
Node id
n-2f64718ead4942ef · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-da18dc44dfcf69a8

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:           3,           2;
  2:           8,          14,           4;
  3:          44,          98,          56,           8;
  4:         460,        1136,         820,         208,         16;
  5:        8020,       20952,       16988,        5464,        720,        32;
  6:      211920,      573432,      498224,      182488,      30768,      2336,       64;
  7:     7929440,    21976264,    19986864,     7945416,    1545408,    151872,     7168,     128;
  8:   399742800,  1127260160,  1058977952,   445341696,   95109648,  10912896,   674688,   20992,   256;
  9: 26146822000, 74707341600, 71897374560, 31501388992, 7186425648, 914878368, 66764672, 2751744, 59136, 512;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-8*n^3+26*n^2-40*n+25+(4*n^2-16*n+20)*x+4*x^2)*P(n-2,x) + (2*n^2-6*n+7+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (4*n^2 - 16*n + 20)*x + 4*x^2)*P[n - 2, x] + (2*n^2 - 6*n + 7 + 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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (4*n^2 - 16*n + 20)*x + 4*x^2)*v[i - 2] + (2*n^2 - 6*n + 7 + 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.