Real-rooted atlas

A132812 → basis A007318 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(2\,n + n^{2}\right)\,P_{n}(x) = \left(-n - 4\,n x - 4\,n x^{2} - n^{2} - 4\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(3 + 2\,x + 9\,n + 6\,n x + 6\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
233421287
318033619232740
498524202080720806285
55418165001896010080240019253550
629953108822156660112840420007392448458115
7166344702016121497611065605644801576962150410243934600
89274414458312901555299388806463296249984054835259904230433913881
9518859027985860647582408385216066407040329817601010688017971201612805120293244050
1029113953174107010453712380675050640630907200383703936151979520380582405617920422400112642542684463
111637864281075509072311690016052402363205648860800407186841619848407046462720001353369601689600010813442457622101612780
12923511849660568287621074223456395127876004827990024040280093568233637381129405107712257482368045906432049201152271564853248192530903461
13521770926640379693172140628629232290942652000397481980896376968656064253888707072122058832896414873338889718252544149653504013940326467092481146881680415209270
1429532022785245856978990928127605860210034938204031732242141603372985455840258924425760014497056288005904714816001723962240003504044544047160422403857817601634304024576014692052109915
Technical details
Node id
n-31a1b9b1779ea532 · revision 0 · JSON
Definition source
derived:basis_map_A013609
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 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:       6,        4;
  2:      33,       42,       12;
  3:     180,      336,      192,       32;
  4:     985,     2420,     2080,      720,       80;
  5:    5418,    16500,    18960,    10080,     2400,      192;
  6:   29953,   108822,   156660,   112840,    42000,     7392,      448;
  7:  166344,   702016,  1214976,  1106560,   564480,   157696,    21504,    1024;
  8:  927441,  4458312,  9015552,  9938880,  6463296,  2499840,   548352,   59904,   2304;
  9: 5188590, 27985860, 64758240, 83852160, 66407040, 32981760, 10106880, 1797120, 161280, 5120;
%F Formula
The row polynomials satisfy (n^2+2*n)*P(n,x) = -(n^2+n+(4*n^2+4*n)*x+(4*n^2+4*n)*x^2)*P(n-2,x) + (6*n^2+9*n+3+(4*n^2+6*n+2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + n + (4*n^2 + 4*n)*x + (4*n^2 + 4*n)*x^2)*P[n - 2, x] + (6*n^2 + 9*n + 3 + (4*n^2 + 6*n + 2)*x)*P[n - 1, x])/(n^2 + 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + n + (4*n^2 + 4*n)*x + (4*n^2 + 4*n)*x^2)*v[i - 2] + (6*n^2 + 9*n + 3 + (4*n^2 + 6*n + 2)*x)*v[i - 1])/(n^2 + 2*n)); 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

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