Real-rooted atlas

A046716 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + 8\,x + 2\,x^{2} - 4\,n - 4\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(-1 - x + 2\,n + n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
3117119
45350141118
53454461973211021
629474983316189799112088
73141166641564062381750244271183727
84004371029172109943962480719920933791238613389242
959279211795396423258646167045007182619184877226373816088173156249
10998165153487291175321809024633040662516462378731578318898634233260212569111804349548
11188274165974617287471312608486313450935474884834753938746232371129335213211258298230126851112103150009179819
1239310397557174388094150347744402156410451759505317552462471168256635920618305264561555922092725658149732503642041097620611537920654526
13899919305929442027042657098596932266111318275890245211738677289770730927444329432473367977111029339870556228125414854336635528262976703487119481320151952155415381
14224109221773471207887248925312981411480705354456244294226194492030170749463217089568991201678438479573196427376736983117928836477614355265906778447466306153138391912587142154266711911990785926432
156031209392347553539976066276309958287838159846015841413499986755178415388131486131445653263327252086698503980506613896345370614859131216542781687732703637901574458631532011997716002052975553380133953818348340155176137201611236999
Technical details
Node id
n-28dca68df9044857 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-490c0ba8997aaced

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,         1;
   3:       11,         7,         1;
   4:       53,        50,        14,         1;
   5:      345,       446,       197,        32,         1;
   6:     2947,      4983,      3161,       897,        99,        1;
   7:    31411,     66641,     56406,     23817,      5024,      427,        1;
   8:   400437,   1029172,   1099439,    624807,    199209,    33791,     2386,       1;
   9:  5927921,  17953964,  23258646,  16704500,   7182619,  1848772,   263738,   16088,      1;
  10: 99816515, 348729117, 532180902, 463304066, 251646237, 87315783, 18898634, 2332602, 125691, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(4*n-8+(4*n-8)*x+(n-2)*x^2)*P(n-2,x) + (2*n-1+(n-1)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[-(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*P[n - 2, x] + (2*n - 1 + (n - 1)*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] = 3 + x; for(n = 3, N, my(i = n); v[i] = -(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*v[i - 2] + (2*n - 1 + (n - 1)*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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023