Real-rooted atlas

A142458 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 12\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(-4 - 400\,x - 60\,x^{2} - 6\,n + 180\,n x - 324\,n x^{2} + 18\,n^{2} + 216\,n^{2} x + 1701\,n^{2} x^{2} - 108\,n^{3} x - 2106\,n^{3} x^{2} + 729\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(84\,x - 380\,x^{2} - 68\,x^{3} + 18\,n x - 27\,n x^{2} - 492\,n x^{3} - 54\,n^{2} x + 216\,n^{2} x^{2} + 2079\,n^{2} x^{3} + 108\,n^{3} x^{2} - 1998\,n^{3} x^{3} + 324\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(72\,x^{2} - 60\,x^{3} - 12\,x^{4} + 27\,n x^{2} + 81\,n x^{3} - 108\,n x^{4} - 81\,n^{2} x^{2} - 324\,n^{2} x^{3} + 405\,n^{2} x^{4} + 324\,n^{3} x^{3} - 324\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(8 - 32\,x - 6\,n + 144\,n x - 9\,n^{2} - 216\,n^{2} x + 108\,n^{3} x\right)\,P_{n-1}(x) + \left(-12\,x - 16\,x^{2} - 45\,n x + 54\,n^{2} x + 108\,n^{2} x^{2} - 108\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
2132942
3115635116524
4166449142656258260
51270849338877121692536156720
61108964245751754240117937598064493467200
714366033642812735480042741875134571245348356487496640
811747282541643236924310411112887508307969841383783522795648812478094080
91699012186564060455094163223677831350340960771441393725874813266777601415499310077790204800
1012796160134526784552806144000442917887250109107446860886811905901021122457600012107410605699040001212679667513600
11111184764959303299558785112200075731437342502957542489288840255439437542193872479596802975996305125106589255500338339111144100471635264000
12144739192679313682546356031084640121413802998375713123991787248152970669862659612677759853995523933807217147353972519427900091329950652616106109121694072465312115200
13117895691647894212596667304084030176185645579893187515780880776537468504185290279790286585277260797260835506981747209303742582319572750050898713547820676630740154567560929701196177425884129894400
141715827824336749304501970199023102681627393966956326025327346390237532688149494035573322370728975219947507947522471227934987573883909295528437590800132586800068617961631791207924134463234036067579350755633762258267399510802432000
Technical details
Node id
n-841d3511dcce72f4 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-34bff62b9cc31312

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: 1,      4;
  2: 1,     32,         9;
  3: 1,    156,       351,         16;
  4: 1,    664,      4914,       2656,          25;
  5: 1,   2708,     49338,      87712,       16925,          36;
  6: 1,  10896,    424575,    1754240,     1179375,       98064,          49;
  7: 1,  43660,   3364281,   27354800,    42741875,    13457124,      534835,         64;
  8: 1, 174728,  25416432,  369243104,  1111288750,   830796984,   138378352,    2795648,       81;
  9: 1, 699012, 186564060, 4550941632, 23677831350, 34096077144, 13937258748, 1326677760, 14154993, 100;
%F Formula
The row polynomials satisfy (9*n^2-12*n+4)*P(n,x) = (18*n^2-6*n-4-(108*n^3-216*n^2-180*n+400)*x+(729*n^4-2106*n^3+1701*n^2-324*n-60)*x^2)*P(n-2,x) - ((54*n^2-18*n-84)*x-(108*n^3+216*n^2-27*n-380)*x^2-(324*n^4-1998*n^3+2079*n^2-492*n-68)*x^3)*d/dx P(n-2,x) - ((81*n^2-27*n-72)*x^2-(324*n^3-324*n^2+81*n-60)*x^3+(324*n^3-405*n^2+108*n+12)*x^4)*d^2/dx^2 P(n-2,x) - (9*n^2+6*n-8-(108*n^3-216*n^2+144*n-32)*x)*P(n-1,x) + ((54*n^2-45*n-12)*x-(108*n^3-108*n^2+16)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n^2 - 6*n - 4 - (108*n^3 - 216*n^2 - 180*n + 400)*x + (729*n^4 - 2106*n^3 + 1701*n^2 - 324*n - 60)*x^2)*P[n - 2, x] - ((54*n^2 - 18*n - 84)*x - (108*n^3 + 216*n^2 - 27*n - 380)*x^2 - (324*n^4 - 1998*n^3 + 2079*n^2 - 492*n - 68)*x^3)*D[P[n - 2, x], x] - ((81*n^2 - 27*n - 72)*x^2 - (324*n^3 - 324*n^2 + 81*n - 60)*x^3 + (324*n^3 - 405*n^2 + 108*n + 12)*x^4)*D[P[n - 2, x], {x, 2}] - (9*n^2 + 6*n - 8 - (108*n^3 - 216*n^2 + 144*n - 32)*x)*P[n - 1, x] + ((54*n^2 - 45*n - 12)*x - (108*n^3 - 108*n^2 + 16)*x^2)*D[P[n - 1, x], x])/(9*n^2 - 12*n + 4)]];
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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((18*n^2 - 6*n - 4 - (108*n^3 - 216*n^2 - 180*n + 400)*x + (729*n^4 - 2106*n^3 + 1701*n^2 - 324*n - 60)*x^2)*v[i - 2] - ((54*n^2 - 18*n - 84)*x - (108*n^3 + 216*n^2 - 27*n - 380)*x^2 - (324*n^4 - 1998*n^3 + 2079*n^2 - 492*n - 68)*x^3)*deriv(v[i - 2], x) - ((81*n^2 - 27*n - 72)*x^2 - (324*n^3 - 324*n^2 + 81*n - 60)*x^3 + (324*n^3 - 405*n^2 + 108*n + 12)*x^4)*deriv(deriv(v[i - 2], x), x) - (9*n^2 + 6*n - 8 - (108*n^3 - 216*n^2 + 144*n - 32)*x)*v[i - 1] + ((54*n^2 - 45*n - 12)*x - (108*n^3 - 108*n^2 + 16)*x^2)*deriv(v[i - 1], x))/(9*n^2 - 12*n + 4)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.