Real-rooted atlas

A059364 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 12\,n x + 81\,n x^{2} + 4\,n^{2} + 36\,n^{2} x - 162\,n^{2} x^{2} - 4\,n^{3} - 24\,n^{3} x + 81\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(12\,n x^{2} + 36\,n x^{3} - 36\,n^{2} x^{2} - 72\,n^{2} x^{3} + 24\,n^{3} x^{2} + 36\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-1 - 2\,n - 12\,n x + 4\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(12\,n x^{2} - 12\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131215
215132162309
31051464380725927968
4945182167379110454448600246096
51039525707614264913117744298485010497608846316
6135135409042828955961861783841264896009101419225719120362582820
7202702572707760627612138237369830447250722255190258384298727578870543872016684090344
8344594251430626320145956253866713098905616933991422525153288862422033212894010563440947221427701120851458742568
9654729075309035179803642274312741976078810592604954697107511363060748924134301441067929777133755648401685215745671425670400047722858932816
1013749310575727541366580973476352040460923118023712219669180639075499964704877844746024673490362731562190195968455579442591336163771692912000259275183552002913898575283056
113162341432251854836042724027791950743785719723378410757448193739427796375219378676761592563961531548437481948981116301275520410324792795042042232739289440080071357366201457601018244357222400192511013983963200
127905853580625509246727305400845040488055286567095294502916448315787489255419975973210143840945480206533840324117401930872562111237162243257514093464470980238185056811576240011509746499837428483311722960283750404302082409264640013682187531110608704
132134580466768751498297479582330027284733209597620523974676899637427361261488171866781037544011299280289913756107425375356401887743188561726975645855424240337917569475396954221153236505933446800143475843858943884552623620952515157114881631959421663353536019458649666520064001040893347155061667968
1461902833536293754703672209205907009328463748988611795898991638054480959845231653869952371179252039275725473436670404563131516085889481426511361339244702061934912169930339090639615140481887540995486709054000015399806951140910418384897992145368976235852835471998440550011375368513584839815902156809381848946357888000084395348789350236529536
Technical details
Node id
n-89aecd7a8c2a861e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-177861d0ae33fc4e

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,          12;
  2:        15,         132,          162;
  3:       105,        1464,         3807,          2592;
  4:       945,       18216,        73791,        104544,         48600;
  5:     10395,      257076,      1426491,       3117744,       2984850,        1049760;
  6:    135135,     4090428,     28955961,      86178384,     126489600,       91014192,       25719120;
  7:   2027025,    72707760,    627612138,    2373698304,    4725072225,     5190258384,     2987275788,     705438720;
  8:  34459425,  1430626320,  14595625386,   67130989056,  169339914225,   251532888624,   220332128940,  105634409472,   21427701120;
  9: 654729075, 30903517980, 364227431274, 1976078810592, 6049546971075, 11363060748924, 13430144106792, 9777133755648, 4016852157456, 714256704000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(24*n^3-36*n^2+12*n)*x-(81*n^3-162*n^2+81*n)*x^2)*P(n-2,x) + ((24*n^3-36*n^2+12*n)*x^2+(36*n^3-72*n^2+36*n)*x^3)*d/dx P(n-2,x) + (4*n^2-2*n-1+(12*n^2-12*n)*x)*P(n-1,x) - (12*n^2-12*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (24*n^3 - 36*n^2 + 12*n)*x - (81*n^3 - 162*n^2 + 81*n)*x^2)*P[n - 2, x] + ((24*n^3 - 36*n^2 + 12*n)*x^2 + (36*n^3 - 72*n^2 + 36*n)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 2*n - 1 + (12*n^2 - 12*n)*x)*P[n - 1, x] - (12*n^2 - 12*n)*x^2*D[P[n - 1, x], 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] = 3 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (24*n^3 - 36*n^2 + 12*n)*x - (81*n^3 - 162*n^2 + 81*n)*x^2)*v[i - 2] + ((24*n^3 - 36*n^2 + 12*n)*x^2 + (36*n^3 - 72*n^2 + 36*n)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 2*n - 1 + (12*n^2 - 12*n)*x)*v[i - 1] - (12*n^2 - 12*n)*x^2*deriv(v[i - 1], x))/(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.