Real-rooted atlas

A114656 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-5 - 6\,x + 7\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-2 - 6\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2325
31114429
44584488185
5197484416144161257
6903275031601680400328925
74279155462238016120600010566465445
8207938775215187213832070560197122688128491825
9103049495368100172811043207181442777606092866562563768209
105188592798586647582483852166640704329817610106881797121612851229324405
1126467231582791041246580613682405735520034882176138163203459840510720384001024231153133
121364886989625756259741680436686360470738400339322368165403392538560001127808014080009011220481841801065
137103937350812945216210941123039445200371383848030984687361797210624723469824198063360353126403784704208896409614810069497
14372693519288426379810044902088207816180002839157006426926332576181349076488718488064296338099269416089610689536099573764792328192120029657805
1519688015191639046526661875173724140023292136211548280944224865697056172616283840966470419203936476544011493081600233602969631440281625718784108953616384979470140661

Other recurrences

Image under the basis map A013609 · proved equal for all n (proof step)

\(\left(1 + n\right)\,P_{n}(x) = \left(2 + 8\,x + 8\,x^{2} - n - 4\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(-3 - 2\,x + 6\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 2\,x\)
Technical details
Node id
n-31db8029f5c1d671 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A013609 (equal by e-d86ba9d5ebdfaca2)
Fingerprint
f6-acc791d1c44ba380

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:      3,       2;
   3:     11,      14,       4;
   4:     45,      84,      48,       8;
   5:    197,     484,     416,     144,      16;
   6:    903,    2750,    3160,    1680,     400,      32;
   7:   4279,   15546,   22380,   16120,    6000,    1056,      64;
   8:  20793,   87752,  151872,  138320,   70560,   19712,    2688,    128;
   9: 103049,  495368, 1001728, 1104320,  718144,  277760,   60928,   6656,   256;
  10: 518859, 2798586, 6475824, 8385216, 6640704, 3298176, 1010688, 179712, 16128, 512;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (7*n-5+(6*n-6)*x)*P(n-1,x) - (2+6*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((7*n - 5 + (6*n - 6)*x)*P[n - 1, x] - (2 + 6*x + 4*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((7*n - 5 + (6*n - 6)*x)*v[i - 1] - (2 + 6*x + 4*x^2)*deriv(v[i - 1], x))/(n + 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

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. [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