Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-3 + 2\,x + 4\,n\right)\,P_{n}(x) = \left(1 + 6\,x + 12\,x^{2} + 8\,x^{3} + 4\,n + 16\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1512421
29561209616297
31313254011201200576643645
4172401456492810080125448960307225641553
521380306014400436808870412096010752057600153601024452709
625552554433440134640380160768768111513611404807884803379207372840964782969
7297569100669763363361217216326726465894409993984112752649225216521830418636803440641638449424013
833992139201209607280003214848107627522782208056010240878592001066024969840230467092480321126409830400157286465536502211745
9371260201962023681419840740275229702400936898562352430084729753607617392649776332809884958727720796164512153601871708165013504070778882621445036466357
10411560281203192002558160153799687193856026790912080620800019778969603972993024653565952087715430409525657600825556992055883857922857697280104595456024903680031457280104857649977243081
114518923788448048043304802949408015758265667671859223739724806876334080165541376003322866892855621902336774224281608906489856083825786880635678883843796815052817258250240565182464012111052801384120324194304491636600541
1249225649680696256697065653044992318723328155054592062132590082075270348858241458176138026778624276905574400470313467904674797584384814041530368819397066752680468152320458685939712245127708672100279517184297124495365788139520603979776167772164801302120177
135326526370097760010764000905132806041235203283737600147978688005599193600017949414912049051844608011471802368002300446310400395579392000058229286502407310307164160778127933440069621972992005173149696000313837748224015174572441605643436032001526726656002726297600026172456966710886446600873519365
145730808013613366081605240014781312010850112006516956160326226700801379169792004972083916801539269222400411204777984095085114163201906084085760033126426869760498430033920006473726558208072243035504640688028909568005540443324416037246677811200205421071564809068756336640309162147840076947023462412683575296011274289152268435456449910251614233
Technical details
Node id
n-525a6d09fe1dc277 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ec29035d385d8da2

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  5,   12,     4;
  2:  9,   56,   120,     96,      16;
  3: 13,  132,   540,   1120,    1200,     576,       64;
  4: 17,  240,  1456,   4928,   10080,   12544,     8960,     3072,       256;
  5: 21,  380,  3060,  14400,   43680,   88704,   120960,   107520,     57600,     15360,      1024;
  6: 25,  552,  5544,  33440,  134640,  380160,   768768,  1115136,   1140480,    788480,    337920,     73728,      4096;
  7: 29,  756,  9100,  66976,  336336, 1217216,  3267264,  6589440,   9993984,  11275264,   9225216,   5218304,   1863680,    344064,     16384;
  8: 33,  992, 13920, 120960,  728000, 3214848, 10762752, 27822080,  56010240,  87859200, 106602496,  98402304,  67092480,  32112640,   9830400,   1572864,    65536;
  9: 37, 1260, 20196, 202368, 1419840, 7402752, 29702400, 93689856, 235243008, 472975360, 761739264, 977633280, 988495872, 772079616, 451215360, 187170816, 50135040, 7077888, 262144;
%F Formula
The row polynomials satisfy (4*n-3+2*x)*P(n,x) = (4*n+1+(16*n+6)*x+(16*n+12)*x^2+8*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 1 + (16*n + 6)*x + (16*n + 12)*x^2 + 8*x^3)*P[n - 1, x])/(4*n - 3 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 1 + (16*n + 6)*x + (16*n + 12)*x^2 + 8*x^3)*v[i - 1])/(4*n - 3 + 2*x)); 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.