Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(n + 6\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6 + 18\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
161824
212108324324768
318270162048607290437418432
4245044536226806804012247212247252488393216
5308109720680403061809185401837080236196017714705904907864320
636118817820160380962280404157612124728259815603897234038972340233834046377292150994944
74216382948432432424324301313512252540488157621464354648294591080490709296588580333572290166786669615662818572288

All 15 rows

Technical details
Node id
n-f4ed0867db54c950 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b0cbff57821dcb89

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:  6,   18;
  2: 12,  108,   324,    324;
  3: 18,  270,  1620,   4860,     7290,     4374;
  4: 24,  504,  4536,  22680,    68040,   122472,    122472,      52488;
  5: 30,  810,  9720,  68040,   306180,   918540,   1837080,    2361960,    1771470,      590490;
  6: 36, 1188, 17820, 160380,   962280,  4041576,  12124728,   25981560,   38972340,    38972340,    23383404,      6377292;
  7: 42, 1638, 29484, 324324,  2432430, 13135122,  52540488,  157621464,  354648294,   591080490,   709296588,    580333572,    290166786,     66961566;
  8: 48, 2160, 45360, 589680,  5307120, 35026992, 175134960,  675520560, 2026561680,  4728643920,  8511559056,  11606671440,  11606671440,   8035387920,   3443737680,    688747536;
  9: 54, 2754, 66096, 991440, 10410120, 81198936, 487193616, 2296769904, 8612887140, 25838661420, 62012787408, 118388048688, 177582073032, 204902391960, 175630621680, 105378373008, 39516889878, 6973568802;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n+6*n*x+9*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6 + 18*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6 + 18*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + 6*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] = 6 + 18*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + 6*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.