Real-rooted atlas

A225470 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 + 24\,x - 16\,x^{2} + 18\,n - 24\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-3 + 8\,x + 6\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13811
2187248138
31627928642562074
4194410800151207680128036824
529160177552291600195840576006144757896
652488034292166314112508032020160003870722867217780272
7110224807621257615317164814036198467737600178053122408448131072468851120
826453952019172816644133395008418560307223275123207524679681408942081415577658982413736439360
9714256704053882921088123105355200135056056320837708480003148869427273156608001026293760796262402621440442870644160
10214277011200167362816896040164581825284708243584000318840572160013467608985603664170639366423183360070071091204325376001153433615585873645440
117071141369600569439456652801425848890371841767931485788161287586067328005974745711616018376647303168379469994393652027785216045416448000228379852850331648594638564177280
12254561089305600210655117490688054747196793303047125006090369024551927558527488027689497684992009403807694561282206167285104643580615242547239473381376002820491182081177760563221810380824450108421043200
1399278824829184008419198453581312022615337454332313630707374248082022425087677827756544013448156378078822449596615595118592129029359294218242389215222890496313084642590720283682034155521690086408192595423395849395240961077937296262937600
144169710642825728003615486410367498240100035936380344504321410324851509217280012072193400061849600685243421452541952027106384859719925767686506946134016001584102510437990402376828931276800025690369708523521947721531392009824486031360295950090240402653184050728168896892569600
Technical details
Node id
n-63acb15e95adcd5e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4f9c3b61d65ed120

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:          3,           8;
  2:         18,          72,           48;
  3:        162,         792,          864,          256;
  4:       1944,       10800,        15120,         7680,        1280;
  5:      29160,      177552,       291600,       195840,       57600,        6144;
  6:     524880,     3429216,      6314112,      5080320,     2016000,      387072,      28672;
  7:   11022480,    76212576,    153171648,    140361984,    67737600,    17805312,    2408448,     131072;
  8:  264539520,  1917281664,   4133395008,   4185603072,  2327512320,   752467968,  140894208,   14155776,   589824;
  9: 7142567040, 53882921088, 123105355200, 135056056320, 83770848000, 31488694272, 7315660800, 1026293760, 79626240, 2621440;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(24*n-24)*x+16*x^2)*P(n-2,x) + (6*n-3+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (24*n - 24)*x + 16*x^2)*P[n - 2, x] + (6*n - 3 + 8*x)*P[n - 1, x]];
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] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (24*n - 24)*x + 16*x^2)*v[i - 2] + (6*n - 3 + 8*x)*v[i - 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.