Real-rooted atlas

A094638 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-16\,n x^{2} + 8\,n^{2} x + 32\,n^{2} x^{2} - n^{3} - 8\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 8\,n x + 2\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
2349232158
3208111216163843320
41424118323329633920614486616
51094412555254169610563208381441228802695536
69388813868968189056242672003658854423924736294912097399440
78922241621177612251820849380480011244669441390317568776060928825753604006847808
8932025620171808018671986569589767424295661591045496354816057811779584282195394562642411520184871442240
91062662402675175552294035301121843666455047218886702081810694464512285380727603226183083622401137454350336951268147209453831555456
1013140806403778307827248189661555235790711191041706538344396854218306829312114986768228352157665851801600128496853057536503496415641603805072588800530687942407296
111752530688056727969638482484109480967095238950246439993349163110415441379755806724139274065629184762077808875929692627506427330566799209523052544242846829379584016742319390720032441760880542720
122508210662409034073436672147664369138176144587749708134494243855843420164294605570109644813966971374898790432446164560153804852859947823634841657769349456540467238609917664506675212677929293053952080363133075456002145312383081591808
1338353000089601522386139207682765599450813440304041723979950082253026004301578241185691325798064128454504028908619366412791910398340907008262580711823773532163840838178692818534438166106362699841536234299310150506250247122919910686064640417888291992371200152584569009152501760
1462411484441600270825082550784054167087923448832661174767036767232549527142135669555232843084883754246144145233976059650686976481238426265391972352119540673356845599948822004958577601019904002923056099040969097216266489699724410486784015135807739310993571844285535837298779750402340174435157278720011614920660773605840896
Technical details
Node id
n-e92c4975928f400f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a9c2fd330558da12

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         6,          4;
  2:        34,         92,          32;
  3:       208,       1112,        1616,          384;
  4:      1424,      11832,       33296,        33920,         6144;
  5:     10944,     125552,      541696,      1056320,       838144,        122880;
  6:     93888,    1386896,     8189056,     24267200,     36588544,      23924736,       2949120;
  7:    892224,   16211776,   122518208,    493804800,   1124466944,    1390317568,     776060928,      82575360;
  8:   9320256,  201718080,  1867198656,   9589767424,  29566159104,   54963548160,   57811779584,   28219539456,    2642411520;
  9: 106266240, 2675175552, 29403530112, 184366645504, 721888670208, 1810694464512, 2853807276032, 2618308362240, 1137454350336, 95126814720;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(8*n^3-8*n^2)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (2*n^2-1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (8*n^2 - 8*n)*x)*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] = 1; v[2] = 6 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (8*n^2 - 8*n)*x)*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

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.