Real-rooted atlas

A114655 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
122
26410
32228858
4901689616370
5394968832288322514
618065500632033608006417850
78558310924476032240120002112128130890
841586175504303744276640141120394245376256983650
9206098990736200345622086401436288555520121856133125127536418
10103771855971721295164816770432132814086596352202137635942432256102458648810
1152934463165582082493160122736480114710400697643522763264069196801021440768002048462306266
122729773817925151251948336087337272094147680067864473633080678410771200022556160281600018022440963683602130
131420787461016258904324218822460788904007427676960619693747235944212481446939648396126720706252807569408417792819229620138994
147453870385768527596200898041764156323600056783140128538526651523626981529617436976128592676198413883217922137907201991475295846416384240059315610
1539376030383278093053212375034744828004658427242309656188844973139411234523256768019329408384078729530880229861632004672059392628805632514375682179072327681958940281322
Technical details
Node id
n-36ea7ad71f537a76 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
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:       2;
   2:       6,       4;
   3:      22,      28,        8;
   4:      90,     168,       96,       16;
   5:     394,     968,      832,      288,       32;
   6:    1806,    5500,     6320,     3360,      800,      64;
   7:    8558,   31092,    44760,    32240,    12000,    2112,     128;
   8:   41586,  175504,   303744,   276640,   141120,   39424,    5376,    256;
   9:  206098,  990736,  2003456,  2208640,  1436288,  555520,  121856,  13312,   512;
  10: 1037718, 5597172, 12951648, 16770432, 13281408, 6596352, 2021376, 359424, 32256, 1024;
%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) = 2.
%t Mathematica
P[1, x_] := 2;
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] = 2; 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.