Real-rooted atlas

A136572 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1549
2188032130
37864814403842550
440849922073630720614463000
52520408002496006912007680001228801875000
6180003628802937600119808002488320022118400294912065250000
7146160352800035562240191923200587059200975421440722534400825753602598750000
81330560374169604515840003034644480122830848003005743104041617981440264241152002642411520116550000000
913426560431101440606154752048771072000245806202880795943895040162310127616019263179980801070176665600951268147205811750000000
10148780800537062400086220288000808206336000487710720000019664496230400530629263360009274864435200096315899904000475634073600003805072588800318937500000000
11179625600072009907200129969100800013910206464000977929666560004721039769600001586269362585600366892233523200056112929832960005179655061504000230206891622400016742319390720019100812500000000
12234710784001034643456000207388532736002495406735360002003069730816000112657497587712004532198178816000013052730754990080026416240813670400035912275093094400029834813154263040012054469961318400080363133075456001239603750000000000
13330032102400158664489984003497094881280004673154937651200421723738275840002708150276063232001269274472821555200437680852697088000011029557487966617600198415319889346560002427669796293181440018334848811165286400679068474487603200041788829199237120086655318750000000000
144969162598400258745168281600621964800737280091390746230784000915938367779635200661262821616517120035386496940559564800142158740956014182400428927235643146240000960797007840647577600155557610793247703040017302664729944129536001197876788996132044800409530526152523776000234017443515727872006491547562500000000000
Technical details
Node id
n-716a133f6399cf18 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-236e6d52004bfa9f

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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,         4;
  2:       18,        80,         32;
  3:       78,       648,       1440,         384;
  4:      408,      4992,      20736,       30720,         6144;
  5:     2520,     40800,     249600,      691200,       768000,       122880;
  6:    18000,    362880,    2937600,    11980800,     24883200,     22118400,       2949120;
  7:   146160,   3528000,   35562240,   191923200,    587059200,    975421440,     722534400,      82575360;
  8:  1330560,  37416960,  451584000,  3034644480,  12283084800,  30057431040,   41617981440,   26424115200,    2642411520;
  9: 13426560, 431101440, 6061547520, 48771072000, 245806202880, 795943895040, 1623101276160, 1926317998080, 1070176665600, 95126814720;
%F Formula
The row polynomials satisfy (4*n-3+4*x)*P(n,x) = (4*n^2+n+(16*n^2+8*n)*x+16*n*x^2)*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^2 + n + (16*n^2 + 8*n)*x + 16*n*x^2)*P[n - 1, x])/(4*n - 3 + 4*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^2 + n + (16*n^2 + 8*n)*x + 16*n*x^2)*v[i - 1])/(4*n - 3 + 4*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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.