Real-rooted atlas

A126216 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(24 + 72\,x + 12\,n + 108\,n x + 54\,n x^{2} + 36\,n^{2} x + 81\,n^{2} x^{2} + 27\,n^{3} x^{2}\right)\,P_{n}(x) = \left(24 + 72\,x + 84\,n + 504\,n x + 702\,n x^{2} + 162\,n x^{3} + 288\,n^{2} x + 891\,n^{2} x^{2} + 567\,n^{2} x^{3} + 189\,n^{3} x^{2} + 243\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-8 - 84\,x - 288\,x^{2} - 378\,x^{3} - 162\,x^{4} - 48\,n x - 306\,n x^{2} - 621\,n x^{3} - 405\,n x^{4} - 36\,n^{2} x^{2} - 162\,n^{2} x^{3} - 162\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
211422780
345252324108729
41971452280819444056806
59038250213302268010125145864746
6427946638151065217620151875481145103624694
720793263256102513618673201786050898128214326174966092505
81030491486104676166414908320181780201265544048580569097925904959919494
951885983957584371181211320041616809282015027314480586576245643843720087196830593260686
10264672347483730278414415828471240145180350015893191441101635640472916880117802755147622506495395905905816
1113648869268877268175325634058952658601191556575015460375392131883360847361442000260140369554128250057159432212576459058738954
1271039373152438835610942385256410325102009400653652514117398178414329946584898889281568456853255651357536510024006961442168279286908733592824712380
133726935198652791394678030890942805518430007186616172451226831027994144597552699611917083372486835339226432668587678906780537621010335464568808321761223205225969921100084
14196880151949171395798417657422637189031444383653548158613951024544332211413763451506805132104425424409079882947135441832666275014817796803813263409119884337993450729686294267174453560286015807266
Technical details
Node id
n-8b0f4c9b4460ebb0 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-31e7b5ad0ec8251b

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,       6;
  2:     11,      42,       27;
  3:     45,     252,      324,       108;
  4:    197,    1452,     2808,      1944,       405;
  5:    903,    8250,    21330,     22680,     10125,      1458;
  6:   4279,   46638,   151065,    217620,    151875,     48114,     5103;
  7:  20793,  263256,  1025136,   1867320,   1786050,    898128,   214326,    17496;
  8: 103049, 1486104,  6761664,  14908320,  18178020,  12655440,  4858056,   909792,   59049;
  9: 518859, 8395758, 43711812, 113200416, 168092820, 150273144, 80586576, 24564384, 3720087, 196830;
%F Formula
The row polynomials satisfy (12*n+24+(36*n^2+108*n+72)*x+(27*n^3+81*n^2+54*n)*x^2)*P(n,x) = (84*n+24+(288*n^2+504*n+72)*x+(189*n^3+891*n^2+702*n)*x^2+(243*n^3+567*n^2+162*n)*x^3)*P(n-1,x) - (8+(48*n+84)*x+(36*n^2+306*n+288)*x^2+(162*n^2+621*n+378)*x^3+(162*n^2+405*n+162)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((84*n + 24 + (288*n^2 + 504*n + 72)*x + (189*n^3 + 891*n^2 + 702*n)*x^2 + (243*n^3 + 567*n^2 + 162*n)*x^3)*P[n - 1, x] - (8 + (48*n + 84)*x + (36*n^2 + 306*n + 288)*x^2 + (162*n^2 + 621*n + 378)*x^3 + (162*n^2 + 405*n + 162)*x^4)*D[P[n - 1, x], x])/(12*n + 24 + (36*n^2 + 108*n + 72)*x + (27*n^3 + 81*n^2 + 54*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((84*n + 24 + (288*n^2 + 504*n + 72)*x + (189*n^3 + 891*n^2 + 702*n)*x^2 + (243*n^3 + 567*n^2 + 162*n)*x^3)*v[i - 1] - (8 + (48*n + 84)*x + (36*n^2 + 306*n + 288)*x^2 + (162*n^2 + 621*n + 378)*x^3 + (162*n^2 + 405*n + 162)*x^4)*deriv(v[i - 1], x))/(12*n + 24 + (36*n^2 + 108*n + 72)*x + (27*n^3 + 81*n^2 + 54*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023