A272867 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 - x^{2} + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(1 + 8\,x + 18\,x^{2} + 8\,x^{3} + x^{4} + 4\,n x + 18\,n x^{2} + 12\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	8	3													12
2	1	16	54	32	5											108
3	1	24	153	352	255	72	7									864
4	1	32	300	1216	2270	1824	700	128	9							6480
5	1	40	495	2880	8850	14544	12390	5760	1485	200	11					46656
6	1	48	738	5600	24075	61344	92652	81792	43335	14000	2706	288	13			326592
7	1	56	1029	9632	53305	185304	412629	587136	530523	308840	117271	28896	4459	392	15	2239488

All 15 rows

Technical details

Node id: n-6b0327e8f50ac23f · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-2383397e4235b687

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 1,  8,    3;
  2: 1, 16,   54,    32,      5;
  3: 1, 24,  153,   352,    255,     72,       7;
  4: 1, 32,  300,  1216,   2270,   1824,     700,     128,        9;
  5: 1, 40,  495,  2880,   8850,  14544,   12390,    5760,     1485,      200,       11;
  6: 1, 48,  738,  5600,  24075,  61344,   92652,   81792,    43335,    14000,     2706,      288,      13;
  7: 1, 56, 1029,  9632,  53305, 185304,  412629,  587136,   530523,   308840,   117271,    28896,    4459,     392,     15;
  8: 1, 64, 1368, 15232, 103180, 455616, 1352008, 2720512,  3703158,  3400640,  2124584,   911232,  268268,   53312,   6840,   512,   17;
  9: 1, 72, 1755, 22656, 181620, 973728, 3622668, 9508608, 17683758, 23259760, 21613482, 14262912, 6727812, 2272032, 544860, 90624, 9945, 648, 19;

%F Formula

The row polynomials satisfy (1+4*n*x+(2*n-1)*x^2)*P(n,x) = (1+(4*n+8)*x+(18*n+18)*x^2+(12*n+8)*x^3+(2*n+1)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (4*n + 8)*x + (18*n + 18)*x^2 + (12*n + 8)*x^3 + (2*n + 1)*x^4)*P[n - 1, x])/(1 + 4*n*x + (2*n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (4*n + 8)*x + (18*n + 18)*x^2 + (12*n + 8)*x^3 + (2*n + 1)*x^4)*v[i - 1])/(1 + 4*n*x + (2*n - 1)*x^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

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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
  - ↑ Parent-row provers of polytool (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.