Real-rooted atlas

A132812 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-18\,x^{2} - 27\,x^{3} + 18\,n x^{2} + 27\,n x^{3}\right)\,P_{n-3}(x) + \left(4 + 12\,x - 8\,n - 24\,n x - 27\,n x^{2}\right)\,P_{n-2}(x) + \left(6 + 9\,x + 6\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 6\,x\)
\(P_{2}(x) = 15 + 45\,x + 27\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14610
215452787
356252324108740
42101260252018904056285
5792594016200194409720145853550
630032702793555155925127575459275103458115
71144012012050450410810801247400748440204120174963934600
8437585250962594592681080410216206875674840415768660525904933913881
916796022674601288872040098240742996808358714055724760204703203542940196830293244050
1064664696996906235515022447854049621572069470200861297236032837805098513415140733456495392542684463
1124961444118637629546748012087306003108164400522171619257713705284122407520181870920045467730054561276212576422101612780
129657700173838600137680171263103411801851947955036365523480484873646404363862817625838661420956987460020265616802072619906908733192530903461
1337442160730122120632772504032125373280106013731824238530896604373352707728407293862976305470397232152735198616482321679848769485088773778096223205221680415209270
14145422675305387617528748558475160170540075587291980275149081810377526834725867953450179040165315016347145520046494818358591354922152343096796953699626521528458665557174453514692052109915
Technical details
Node id
n-5ae74f80fa28d4d2 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-cb4056e025ca2479

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,       6;
  2:     15,      45,       27;
  3:     56,     252,      324,      108;
  4:    210,    1260,     2520,     1890,      405;
  5:    792,    5940,    16200,    19440,     9720,     1458;
  6:   3003,   27027,    93555,   155925,   127575,    45927,     5103;
  7:  11440,  120120,   504504,  1081080,  1247400,   748440,   204120,    17496;
  8:  43758,  525096,  2594592,  6810804, 10216206,  8756748,  4041576,   866052,   59049;
  9: 167960, 2267460, 12888720, 40098240, 74299680, 83587140, 55724760, 20470320, 3542940, 196830;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = ((18*n-18)*x^2+(27*n-27)*x^3)*P(n-3,x) - (8*n-4+(24*n-12)*x+27*n*x^2)*P(n-2,x) + (6*n+6+(9*n+9)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4 + 6*x, P(2,x) = 15 + 45*x + 27*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 6*x;
P[2, x_] := 15 + 45*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((18*n - 18)*x^2 + (27*n - 27)*x^3)*P[n - 3, x] - (8*n - 4 + (24*n - 12)*x + 27*n*x^2)*P[n - 2, x] + (6*n + 6 + (9*n + 9)*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 + 6*x; v[3] = 15 + 45*x + 27*x^2; for(n = 3, N, my(i = n + 1); v[i] = (((18*n - 18)*x^2 + (27*n - 27)*x^3)*v[i - 3] - (8*n - 4 + (24*n - 12)*x + 27*n*x^2)*v[i - 2] + (6*n + 6 + (9*n + 9)*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