Real-rooted atlas

A113953 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(72\,n + 368\,n x + 320\,n x^{2} + 144\,n^{2} x + 448\,n^{2} x^{2} + 128\,n^{3} x^{2}\right)\,P_{n}(x) = \left(360 + 2272\,x + 3968\,x^{2} + 2560\,x^{3} - 288\,n - 320\,n x + 2688\,n x^{2} + 3584\,n x^{3} - 864\,n^{2} x - 1152\,n^{2} x^{2} + 1024\,n^{2} x^{3} - 512\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(81 + 864\,x + 3104\,x^{2} + 4096\,x^{3} + 1280\,x^{4} + 324\,n x + 2448\,n x^{2} + 5056\,n x^{3} + 1792\,n x^{4} + 288\,n^{2} x^{2} + 1280\,n^{2} x^{3} + 512\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1189
23324883
3588336256685
4112401344256012805435
52158445121382416640614441725
643140813584573441139209830428672313275
7853256380642055685721608171525447681310722312125
81717440101664668160240128048537605332992288358458982416838875
93411668026150420275208942080235069443670016032505856147456002621440121328125
1068337024654192584294430563840991887362023669762548039681881538567340032011534336866546875
1113658130415995041617638497937920378753024958046208158046617616568156161045954560357564416503316486143728125
12273117720038400004337100829837696013406453764056055808831992627211438456832102288588805628755968171127603221810380843287846875
135461383496907785611326515287287168044705648641576352153638782107648664159518727788560384060555264000294943457288069840896939524096303372328125
141092382528021185232289382400246966144014206058496572509593601645677117443386916864004952005017605045233909763462817382401511459389443758096384040265318402116256546875
Technical details
Node id
n-38914bba12a80347 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-fd333723d620d6e6

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,     8;
  2:   3,    32,     48;
  3:   5,    88,    336,     256;
  4:  11,   240,   1344,    2560,    1280;
  5:  21,   584,   4512,   13824,   16640,     6144;
  6:  43,  1408,  13584,   57344,  113920,    98304,    28672;
  7:  85,  3256,  38064,  205568,  572160,   817152,   544768,   131072;
  8: 171,  7440, 101664,  668160, 2401280,  4853760,  5332992,  2883584,   589824;
  9: 341, 16680, 261504, 2027520, 8942080, 23506944, 36700160, 32505856, 14745600, 2621440;
%F Formula
The row polynomials satisfy (72*n+(144*n^2+368*n)*x+(128*n^3+448*n^2+320*n)*x^2)*P(n,x) = -(288*n-360+(864*n^2+320*n-2272)*x+(512*n^3+1152*n^2-2688*n-3968)*x^2-(1024*n^2+3584*n+2560)*x^3)*P(n-1,x) + (81+(324*n+864)*x+(288*n^2+2448*n+3104)*x^2+(1280*n^2+5056*n+4096)*x^3+(512*n^2+1792*n+1280)*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[(-(288*n - 360 + (864*n^2 + 320*n - 2272)*x + (512*n^3 + 1152*n^2 - 2688*n - 3968)*x^2 - (1024*n^2 + 3584*n + 2560)*x^3)*P[n - 1, x] + (81 + (324*n + 864)*x + (288*n^2 + 2448*n + 3104)*x^2 + (1280*n^2 + 5056*n + 4096)*x^3 + (512*n^2 + 1792*n + 1280)*x^4)*D[P[n - 1, x], x])/(72*n + (144*n^2 + 368*n)*x + (128*n^3 + 448*n^2 + 320*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] = (-(288*n - 360 + (864*n^2 + 320*n - 2272)*x + (512*n^3 + 1152*n^2 - 2688*n - 3968)*x^2 - (1024*n^2 + 3584*n + 2560)*x^3)*v[i - 1] + (81 + (324*n + 864)*x + (288*n^2 + 2448*n + 3104)*x^2 + (1280*n^2 + 5056*n + 4096)*x^3 + (512*n^2 + 1792*n + 1280)*x^4)*deriv(v[i - 1], x))/(72*n + (144*n^2 + 368*n)*x + (128*n^3 + 448*n^2 + 320*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. [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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.