A133214 → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(-2 + 16\,x - n + 24\,n x + 12\,n^{2} x + 2\,n^{3} x\right)\,P_{n}(x) = \left(-2 + 104\,x^{2} - n + 12\,n x + 164\,n x^{2} + 12\,n^{2} x + 80\,n^{2} x^{2} + 2\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x + 32\,x^{2} - 56\,x^{3} + 22\,n x^{2} - 44\,n x^{3} + 4\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1 + 16\,x + 12\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1	16	12								29
1	1	36	108	32							177
2	1	64	432	512	80						1089
3	1	100	1200	3200	2000	192					6693
4	1	144	2700	12800	18000	6912	448				41005
5	1	196	5292	39200	98000	84672	21952	1024			250337
6	1	256	9408	100352	392000	602112	351232	65536	2304		1523201
7	1	324	15552	225792	1270080	3048192	3161088	1327104	186624	5120	9239877

All 15 rows

Technical details

Node id: n-33a7ac25a5865eb2 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-b87d7959dbf825bd

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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,  16,    12;
  1: 1,  36,   108,     32;
  2: 1,  64,   432,    512,      80;
  3: 1, 100,  1200,   3200,    2000,      192;
  4: 1, 144,  2700,  12800,   18000,     6912,      448;
  5: 1, 196,  5292,  39200,   98000,    84672,    21952,      1024;
  6: 1, 256,  9408, 100352,  392000,   602112,   351232,     65536,     2304;
  7: 1, 324, 15552, 225792, 1270080,  3048192,  3161088,   1327104,   186624,     5120;
  8: 1, 400, 24300, 460800, 3528000, 12192768, 19756800,  14745600,  4665600,   512000,   11264;
  9: 1, 484, 36300, 871200, 8712000, 40981248, 95622912, 111513600, 62726400, 15488000, 1362944, 24576;

%F Formula

The row polynomials satisfy -(n+2-(2*n^3+12*n^2+24*n+16)*x)*P(n,x) = -(n+2-(2*n^3+12*n^2+12*n)*x-(12*n^3+80*n^2+164*n+104)*x^2)*P(n-1,x) - (2*x-(4*n^2+22*n+32)*x^2+(8*n^2+44*n+56)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 16*x + 12*x^2.

%t Mathematica

P[0, x_] := 1 + 16*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 2 - (2*n^3 + 12*n^2 + 12*n)*x - (12*n^3 + 80*n^2 + 164*n + 104)*x^2)*P[n - 1, x] - (2*x - (4*n^2 + 22*n + 32)*x^2 + (8*n^2 + 44*n + 56)*x^3)*D[P[n - 1, x], x])/(-n - 2 + (2*n^3 + 12*n^2 + 24*n + 16)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 16*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(n + 2 - (2*n^3 + 12*n^2 + 12*n)*x - (12*n^3 + 80*n^2 + 164*n + 104)*x^2)*v[i - 1] - (2*x - (4*n^2 + 22*n + 32)*x^2 + (8*n^2 + 44*n + 56)*x^3)*deriv(v[i - 1], x))/(-n - 2 + (2*n^3 + 12*n^2 + 24*n + 16)*x)); 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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Delannoy paths counted by number of weak peaks.?
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023