Real-rooted atlas

A094816 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 + 7\,n - n^{2}\right)\,P_{n-4}(x) + \left(9 - 24\,x - 9\,n + 8\,n x + 2\,n^{2}\right)\,P_{n-3}(x) + \left(4 + 8\,x - 16\,x^{2} - 8\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-1 + 8\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 8\,x\)
\(P_{3}(x) = 5 + 40\,x + 48\,x^{2}\)
\(P_{4}(x) = 16 + 184\,x + 432\,x^{2} + 256\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
22810
35404893
416184432256888
5659283120358412808977
63265352225603712025600614497102
7195735432174432362240358400165888286721127021
8137002670001472688360550445248003010560100352013107214028844
9109601226457613635168377681925624192047652864228802565767168589824186909569
10986410213865681381576324215587847150617607224053764431544321612185603185049626214402658401458
119864101222762344152533684850340736009467296000109498798088040001536374734848010705305601703936001153433640249021213
12108505112253819525618259039920643284252161316273024001696175247361431479132168011028889629390929920677642240088814387250331648646843022592
13130206134531414193440235817985168878276264448192865680128027163901952002564031303680165460338278473080255283221684551680041235251200452984832021810380811004123460305
141692679748641975685584832708284739521278162252057629813048911360452516050821124685515087052833924409851904173242598031366202531512320152075606425624274953830422682796032939524096197647268601910
152369751648056023382702888485153344371841977511648985604860313078566407865636384686088806690131599367015512792104964030233081937921672156912025604956642135244810218381508608138932125696011180336742440265318403738871049312749
Technical details
Node id
n-947462a675f7df3c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-de694c98c2b9f251

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      2,        8;
   3:      5,       40,        48;
   4:     16,      184,       432,       256;
   5:     65,      928,      3120,      3584,      1280;
   6:    326,     5352,     22560,     37120,     25600,      6144;
   7:   1957,    35432,    174432,    362240,    358400,    165888,     28672;
   8:  13700,   267000,   1472688,   3605504,   4524800,   3010560,   1003520,    131072;
   9: 109601,  2264576,  13635168,  37768192,  56241920,  47652864,  22880256,   5767168,   589824;
  10: 986410, 21386568, 138157632, 421558784, 715061760, 722405376, 443154432, 161218560, 31850496, 2621440;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(8*n-24)*x)*P(n-3,x) - (n^2-4+(8*n-8)*x+16*x^2)*P(n-2,x) + (2*n-1+8*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2 + 8*x, P(3,x) = 5 + 40*x + 48*x^2, P(4,x) = 16 + 184*x + 432*x^2 + 256*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 8*x;
P[3, x_] := 5 + 40*x + 48*x^2;
P[4, x_] := 16 + 184*x + 432*x^2 + 256*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (8*n - 24)*x)*P[n - 3, x] - (n^2 - 4 + (8*n - 8)*x + 16*x^2)*P[n - 2, x] + (2*n - 1 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 8*x; v[3] = 5 + 40*x + 48*x^2; v[4] = 16 + 184*x + 432*x^2 + 256*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (8*n - 24)*x)*v[i - 3] - (n^2 - 4 + (8*n - 8)*x + 16*x^2)*v[i - 2] + (2*n - 1 + 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, v[i])))};
T(10)

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