Real-rooted atlas

A064861 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(8 + 4\,x + 57\,n + 60\,n x + 16\,n x^{2} + 49\,n^{2} + 56\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(28 + 30\,x + 8\,x^{2} + 1460\,n + 3235\,n x + 2660\,n x^{2} + 960\,n x^{3} + 128\,n x^{4} + 1463\,n^{2} + 3580\,n^{2} x + 3248\,n^{2} x^{2} + 1296\,n^{2} x^{3} + 192\,n^{2} x^{4} + 343\,n^{3} + 931\,n^{3} x + 924\,n^{3} x^{2} + 400\,n^{3} x^{3} + 64\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-1968 - 5776\,x - 6732\,x^{2} - 3896\,x^{3} - 1120\,x^{4} - 128\,x^{5} - 2268\,n - 7050\,n x - 8678\,n x^{2} - 5292\,n x^{3} - 1600\,n x^{4} - 192\,n x^{5} - 588\,n^{2} - 1946\,n^{2} x - 2530\,n^{2} x^{2} - 1620\,n^{2} x^{3} - 512\,n^{2} x^{4} - 64\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 7 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
07411
133401285
2841468416330
3324774690272402100
4756219625411464420487425
52700961214211111684920115211243875
660482462442840412842380082081568128148500
7207369806420239223809217461081744238563968288843750
8453602376005518807460406468353730201430803520050403202784375
915163289424023684403709800380560526714641299620432640943201216070415440625
103265922091744607824010577520122484609909654571594823505606753601291201478476850118750
111073088758160024509952479412006318918059125266402703702011670473148401888160328416345601664273375000
12228614417309376603923041285623361863048601940775481495202178625668837260972119039202733808427392407681792877078125
13741830461072704233105040546733152880320420102936430890140826360053056030586046411851392034390048724684810483209318438404727109375
14156764161369820165578191361404273024244405728031151131203003741552223124199212873178085768728001991454085201011299473921315328107520409615035625000
Technical details
Node id
n-7a412f8307bbe357 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-88e2a299ce8e2b61

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, 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:      7,      4;
  1:     33,     40,      12;
  2:     84,    146,      84,      16;
  3:    324,    774,     690,     272,      40;
  4:    756,   2196,    2541,    1464,     420,      48;
  5:   2700,   9612,   14211,   11168,    4920,    1152,     112;
  6:   6048,  24624,   42840,   41284,   23800,    8208,    1568,    128;
  7:  20736,  98064,  202392,  238092,  174610,   81744,   23856,   3968,   288;
  8:  45360, 237600,  551880,  746040,  646835,  373020,  143080,  35200,  5040,   320;
  9: 151632, 894240, 2368440, 3709800, 3805605, 2671464, 1299620, 432640, 94320, 12160, 704;
%F Formula
The row polynomials satisfy (49*n^2+57*n+8+(56*n^2+60*n+4)*x+(16*n^2+16*n)*x^2)*P(n,x) = (343*n^3+1463*n^2+1460*n+28+(931*n^3+3580*n^2+3235*n+30)*x+(924*n^3+3248*n^2+2660*n+8)*x^2+(400*n^3+1296*n^2+960*n)*x^3+(64*n^3+192*n^2+128*n)*x^4)*P(n-1,x) - (588*n^2+2268*n+1968+(1946*n^2+7050*n+5776)*x+(2530*n^2+8678*n+6732)*x^2+(1620*n^2+5292*n+3896)*x^3+(512*n^2+1600*n+1120)*x^4+(64*n^2+192*n+128)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 7 + 4*x.
%t Mathematica
P[0, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((343*n^3 + 1463*n^2 + 1460*n + 28 + (931*n^3 + 3580*n^2 + 3235*n + 30)*x + (924*n^3 + 3248*n^2 + 2660*n + 8)*x^2 + (400*n^3 + 1296*n^2 + 960*n)*x^3 + (64*n^3 + 192*n^2 + 128*n)*x^4)*P[n - 1, x] - (588*n^2 + 2268*n + 1968 + (1946*n^2 + 7050*n + 5776)*x + (2530*n^2 + 8678*n + 6732)*x^2 + (1620*n^2 + 5292*n + 3896)*x^3 + (512*n^2 + 1600*n + 1120)*x^4 + (64*n^2 + 192*n + 128)*x^5)*D[P[n - 1, x], x])/(49*n^2 + 57*n + 8 + (56*n^2 + 60*n + 4)*x + (16*n^2 + 16*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 7 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = ((343*n^3 + 1463*n^2 + 1460*n + 28 + (931*n^3 + 3580*n^2 + 3235*n + 30)*x + (924*n^3 + 3248*n^2 + 2660*n + 8)*x^2 + (400*n^3 + 1296*n^2 + 960*n)*x^3 + (64*n^3 + 192*n^2 + 128*n)*x^4)*v[i - 1] - (588*n^2 + 2268*n + 1968 + (1946*n^2 + 7050*n + 5776)*x + (2530*n^2 + 8678*n + 6732)*x^2 + (1620*n^2 + 5292*n + 3896)*x^3 + (512*n^2 + 1600*n + 1120)*x^4 + (64*n^2 + 192*n + 128)*x^5)*deriv(v[i - 1], x))/(49*n^2 + 57*n + 8 + (56*n^2 + 60*n + 4)*x + (16*n^2 + 16*n)*x^2)); 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

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.