Real-rooted atlas

A034870 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(4\,n + 16\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 16 + 32\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
1163248
2128768153610243456
3768768030720614406144024576186624
440965734434406411468802293760275251218350085242888957952
5204803686402949120137625604128768082575360110100480943718404718592010485760403107840
69830421626882162688012976128051904512014533263362906652672415236096041523609602768240640110729625620132659217414258688
74587521192755214313062410496245765248122880188932423685038197964810076395929615114593894416793993216013435194572873282879488244276264963758096384731398864896

All 15 rows

Technical details
Node id
n-cb21c99be919a143 · revision 0 · JSON
Definition source
derived:derivative
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:      16,        32;
  2:     128,       768,       1536,        1024;
  3:     768,      7680,      30720,       61440,        61440,         24576;
  4:    4096,     57344,     344064,     1146880,      2293760,       2752512,       1835008,         524288;
  5:   20480,    368640,    2949120,    13762560,     41287680,      82575360,     110100480,       94371840,       47185920,        10485760;
  6:   98304,   2162688,   21626880,   129761280,    519045120,    1453326336,    2906652672,     4152360960,     4152360960,      2768240640,      1107296256,       201326592;
  7:  458752,  11927552,  143130624,  1049624576,   5248122880,   18893242368,   50381979648,   100763959296,   151145938944,    167939932160,    134351945728,     73282879488,     24427626496,      3758096384;
  8: 2097152,  62914560,  880803840,  7633633280,  45801799680,  201527918592,  671759728640,  1727382159360,  3454764318720,   5374077829120,   6448893394944,   5862630359040,   3908420239360,   1803886264320,    515396075520,    68719476736;
  9: 9437184, 320864256, 5133828096, 51338280960, 359367966720, 1868713426944, 7474853707776, 23492397367296, 58730993418240, 117461986836480, 187939178938368, 239195318648832, 239195318648832, 183996398960640, 105140799406080, 42056319762432, 10514079940608, 1236950581248;
%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) = 16 + 32*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 16 + 32*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] = 16 + 32*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.