Real-rooted atlas

A272866 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 15\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 15 + 18\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
1151833
21506308103241914
311258100222752916018225437483259
4750076500324000737100972000743580306180524883219348
546875618750354375011542500235406253116475026790750144342004428675590490116701365
62812504556250329062501397250003872812507353787509759487509054180005757277502391484505845851063772924061207502
716406253150000027464062514387625005049843750125342437502262861562530082185000294506887502097715725010571247225357128352072541696566961566137404187151

All 15 rows

Technical details
Node id
n-a8f6492efcb0f220 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9de613d7fd734864

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:       15,         18;
  2:      150,        630,         810,          324;
  3:     1125,       8100,       22275,        29160,        18225,          4374;
  4:     7500,      76500,      324000,       737100,       972000,        743580,        306180,          52488;
  5:    46875,     618750,     3543750,     11542500,     23540625,      31164750,      26790750,       14434200,        4428675,         590490;
  6:   281250,    4556250,    32906250,    139725000,    387281250,     735378750,     975948750,      905418000,      575727750,      239148450,       58458510,        6377292;
  7:  1640625,   31500000,   274640625,   1438762500,   5049843750,   12534243750,   22628615625,    30082185000,    29450688750,    20977157250,    10571247225,     3571283520,      725416965,      66961566;
  8:  9375000,  208125000,  2126250000,  13253625000,  56345625000,  172991700000,  396120375000,   688740975000,   916735725000,   934155180000,   722936907000,   417393761400,   174100071600,   49551558840,    8609344200,    688747536;
  9: 52734375, 1328906250, 15567187500, 112539375000, 562127343750, 2058103687500, 5717911500000, 12310240275000, 20789861203125, 27698040618750, 29112337980000, 24005721411000, 15342568809750, 7442778060900, 2647373341500, 650866421520, 98792224695, 6973568802;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n+15*n*x+9*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 15 + 18*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 15 + 18*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 15*n*x + 9*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] = 15 + 18*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 15*n*x + 9*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.