Real-rooted atlas

A132812 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-8\,x^{2} - 8\,x^{3} + 8\,n x^{2} + 8\,n x^{3}\right)\,P_{n-3}(x) + \left(4 + 8\,x - 8\,n - 16\,n x - 12\,n x^{2}\right)\,P_{n-2}(x) + \left(6 + 6\,x + 6\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 4\,x\)
\(P_{2}(x) = 15 + 30\,x + 12\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
215301257
35616814432400
42108401120560802810
5792396072005760192019219824
63003180184158046200252006048448140497
7114408008022422432032024640098560179201024999968
84375835006411531522018016201801611531523548165068823047143966
91679601511640572832011880960146764801100736048921601198080138240512051206320
106466466466460277134006651216098017920914833925381376019219200384384036608011264368094122
112496144274575841313188803581424006139584006876334085066772482412748807096320011827200946176245762652720096
1296577001158924006119118721869730720365816880047888755204256778240255406694410081843202489344003514368023961605324819159794004
133744216048674808028123222409518629120209409840643141147609632777192448238379581441191897907239729930248364195841013841925963776114688138658606688
141454226752035917450127771371004745793780011600829240019632172560023558607072020193091776012291447168052145533440148987238402708858880285143040146227202457601005231020865
Technical details
Node id
n-74e46c56b4bf9150 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-665cb75b242a18c2

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:      4,       4;
  2:     15,      30,      12;
  3:     56,     168,     144,       32;
  4:    210,     840,    1120,      560,       80;
  5:    792,    3960,    7200,     5760,     1920,      192;
  6:   3003,   18018,   41580,    46200,    25200,     6048,     448;
  7:  11440,   80080,  224224,   320320,   246400,    98560,   17920,    1024;
  8:  43758,  350064, 1153152,  2018016,  2018016,  1153152,  354816,   50688,   2304;
  9: 167960, 1511640, 5728320, 11880960, 14676480, 11007360, 4892160, 1198080, 138240, 5120;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = ((8*n-8)*x^2+(8*n-8)*x^3)*P(n-3,x) - (8*n-4+(16*n-8)*x+12*n*x^2)*P(n-2,x) + (6*n+6+(6*n+6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4 + 4*x, P(2,x) = 15 + 30*x + 12*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[2, x_] := 15 + 30*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((8*n - 8)*x^2 + (8*n - 8)*x^3)*P[n - 3, x] - (8*n - 4 + (16*n - 8)*x + 12*n*x^2)*P[n - 2, x] + (6*n + 6 + (6*n + 6)*x)*P[n - 1, x])/(n + 2)]];
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] = 4 + 4*x; v[3] = 15 + 30*x + 12*x^2; for(n = 3, N, my(i = n + 1); v[i] = (((8*n - 8)*x^2 + (8*n - 8)*x^3)*v[i - 3] - (8*n - 4 + (16*n - 8)*x + 12*n*x^2)*v[i - 2] + (6*n + 6 + (6*n + 6)*x)*v[i - 1])/(n + 2)); 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