Real-rooted atlas

A034870 → D → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
14812
232192384256864
3192192076801536015360614446656
4102414336860162867205734406881284587521310722239488
5512092160737280344064010321920206438402752512023592960117964802621440100776960
62457654067254067203244032012976128036333158472666316810380902401038090240692060160276824064503316484353564672
7114688298188835782656262406144131203072047233105921259549491225190989824377864847364198498304033587986432183207198726106906624939524096182849716224

All 15 rows

Technical details
Node id
n-baf5264cc15878a7 · revision 0 · JSON
Definition source
derived:basis_map_A013611
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:      32,      192,        384,         256;
  3:     192,     1920,       7680,       15360,       15360,         6144;
  4:    1024,    14336,      86016,      286720,      573440,       688128,        458752,        131072;
  5:    5120,    92160,     737280,     3440640,    10321920,     20643840,      27525120,      23592960,       11796480,        2621440;
  6:   24576,   540672,    5406720,    32440320,   129761280,    363331584,     726663168,    1038090240,     1038090240,      692060160,      276824064,       50331648;
  7:  114688,  2981888,   35782656,   262406144,  1312030720,   4723310592,   12595494912,   25190989824,    37786484736,    41984983040,    33587986432,    18320719872,     6106906624,      939524096;
  8:  524288, 15728640,  220200960,  1908408320, 11450449920,  50381979648,  167939932160,  431845539840,   863691079680,  1343519457280,  1612223348736,  1465657589760,   977105059840,   450971566080,   128849018880,    17179869184;
  9: 2359296, 80216064, 1283457024, 12834570240, 89841991680, 467178356736, 1868713426944, 5873099341824, 14682748354560, 29365496709120, 46984794734592, 59798829662208, 59798829662208, 45999099740160, 26285199851520, 10514079940608, 2628519985152, 309237645312;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n+16*n*x+16*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[((4*n + 16*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] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 16*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.