Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(n + 4\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
14812
28489664216
3121204809609603842916
416224134444808960107527168204834992
5203602880134404032080640107520921604608010240393660
62452852803168012672035481670963210137601013760675840270336491524251528
72872887366406432032011531523075072615014492252161025024082001924472832149094422937644641044

All 15 rows

Technical details
Node id
n-619cc8ea95273502 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-177c65847e3c5511

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, intercept -1)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  4,    8;
  2:  8,   48,    96,     64;
  3: 12,  120,   480,    960,     960,     384;
  4: 16,  224,  1344,   4480,    8960,   10752,     7168,     2048;
  5: 20,  360,  2880,  13440,   40320,   80640,   107520,    92160,     46080,     10240;
  6: 24,  528,  5280,  31680,  126720,  354816,   709632,  1013760,   1013760,    675840,    270336,     49152;
  7: 28,  728,  8736,  64064,  320320, 1153152,  3075072,  6150144,   9225216,  10250240,   8200192,   4472832,   1490944,    229376;
  8: 32,  960, 13440, 116480,  698880, 3075072, 10250240, 26357760,  52715520,  82001920,  98402304,  89456640,  59637760,  27525120,   7864320,   1048576;
  9: 36, 1224, 19584, 195840, 1370880, 7128576, 28514304, 89616384, 224040960, 448081920, 716931072, 912457728, 912457728, 701890560, 401080320, 160432128, 40108032, 4718592;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n+4*n*x+4*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4 + 8*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4*n*x + 4*n*x^2)*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] = 0; v[2] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + 4*n*x + 4*n*x^2)*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

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.