Real-rooted atlas

A114687 → basis A013611 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(4 + 2\,n\right)\,P_{n}(x) = \left(4 + 62\,n + 48\,n x\right)\,P_{n-1}(x) + \left(-45 - 76\,x - 32\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111819
213118464379
31661340823045127885
4221015928058880256004096169957
530515110081201315200844800266240327683772279
6433571116996968274147202323200010874880265420826214485770631
7630098812856875845484093445765580803575398401302855682569011220971521989277561
89324499614798524448106786570241339788083210340614144501140684814826864642432696321677721646902266569
91400469445180628278760204042965760297423974400274636357632166149881856656198860801623195648022649241601342177281121137137307
1021294403305113559451324403844896340800638446763520068629465497604978966659072246162653184081568727040017238589440020803747840107374182427111743536627
11327161829666122827704214128716853961894401335687576960001638256962355201386193559224328236509010329634256285859840973036584960017864379596801889785610248589934592662134276822213
12507115641523813847494183471841325392319015424273821612601856037736520621465603648830734467072253305458314444812687364514119684534627309977601122503124582401814194185830417008070492166871947673616308967770543853
137920881045935116492159346217688243406874853725445522013709350195284475304757628928918934633732177927300356003436953642791391008391168184544225226915845761798351355904125937675298406418112736080691215187004358656549755813888404681252951304127
14124548017695545511096681303810184404445487019820246401098604731967142400184727321278801100822329746424529223681998113939179438080134209967659941888067840569954533376025604807252036812870747584989233152138006318651801601782239629148160134690174402560439804651110410106661153889707391

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(2 + n\right)\,P_{n}(x) = \left(81 + 144\,x + 64\,x^{2} - 81\,n - 144\,n x - 64\,n x^{2}\right)\,P_{n-2}(x) + \left(11 + 8\,x + 22\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 11 + 8\,x\)
Technical details
Node id
n-d80ddf0140a13e4f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-888868a927928b8d)
Fingerprint
f6-abfb6509f30d68de

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:          11,           8;
  2:         131,         184,           64;
  3:        1661,        3408,         2304,          512;
  4:       22101,       59280,        58880,        25600,         4096;
  5:      305151,     1008120,      1315200,       844800,       266240,        32768;
  6:     4335711,    16996968,     27414720,     23232000,     10874880,      2654208,      262144;
  7:    63009881,   285687584,    548409344,    576558080,    357539840,    130285568,    25690112,     2097152;
  8:   932449961,  4798524448,  10678657024,  13397880832,  10340614144,   5011406848,  1482686464,   243269632,   16777216;
  9: 14004694451, 80628278760, 204042965760, 297423974400, 274636357632, 166149881856, 65619886080, 16231956480, 2264924160, 134217728;
%F Formula
The row polynomials satisfy (2*n+4)*P(n,x) = (62*n+4+48*n*x)*P(n-1,x) - (45+76*x+32*x^2)*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[((62*n + 4 + 48*n*x)*P[n - 1, x] - (45 + 76*x + 32*x^2)*D[P[n - 1, x], x])/(2*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; for(n = 1, N, my(i = n + 1); v[i] = ((62*n + 4 + 48*n*x)*v[i - 1] - (45 + 76*x + 32*x^2)*deriv(v[i - 1], x))/(2*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

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