Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(n + 8\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 8 + 32\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
183240
21619276810242000
324480384015360307202457675000
43289610752716802867206881289175045242882500000
540144023040215040129024051609601376256023592960235929601048576078125000
648211242240506880405504022708224908328962595225605190451206920601605536481282013265922343750000
756291269888102502410250240738017283936092161574436864472331059210496245760167939932161832071987212213813248375809638468359375000

All 15 rows

Technical details
Node id
n-a9faf912d2e8db2a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-abfa4e49247086e7

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:  8,   32;
  2: 16,  192,    768,    1024;
  3: 24,  480,   3840,   15360,    30720,     24576;
  4: 32,  896,  10752,   71680,   286720,    688128,     917504,      524288;
  5: 40, 1440,  23040,  215040,  1290240,   5160960,   13762560,    23592960,     23592960,     10485760;
  6: 48, 2112,  42240,  506880,  4055040,  22708224,   90832896,   259522560,    519045120,    692060160,     553648128,     201326592;
  7: 56, 2912,  69888, 1025024, 10250240,  73801728,  393609216,  1574436864,   4723310592,  10496245760,   16793993216,   18320719872,   12213813248,     3758096384;
  8: 64, 3840, 107520, 1863680, 22364160, 196804608, 1312030720,  6747586560,  26990346240,  83969966080,  201527918592,  366414397440,  488552529920,   450971566080,   257698037760,    68719476736;
  9: 72, 4896, 156672, 3133440, 43868160, 456228864, 3649830912, 22941794304, 114708971520, 458835886080, 1468274835456, 3737426853888, 7474853707776, 11499774935040, 13142599925760, 10514079940608, 5257039970304, 1236950581248;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n+8*n*x+16*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 8 + 32*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 8 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + 8*n*x + 16*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] = 8 + 32*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + 8*n*x + 16*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.