A200536 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 3\,x - 12\,x^{2} - 6\,x^{3} + 4\,x^{4} + 9\,n x + 16\,n x^{2} - 6\,n x^{3} - 16\,n x^{4} + 9\,n^{2} x^{2} + 24\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n}(x) = \left(1 + 9\,x + 33\,x^{2} + 63\,x^{3} + 66\,x^{4} + 36\,x^{5} + 8\,x^{6} + 9\,n x + 61\,n x^{2} + 162\,n x^{3} + 210\,n x^{4} + 132\,n x^{5} + 32\,n x^{6} + 9\,n^{2} x^{2} + 51\,n^{2} x^{3} + 106\,n^{2} x^{4} + 96\,n^{2} x^{5} + 32\,n^{2} x^{6}\right)\,P_{n-1}(x)\)

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

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	12	18													31
2	1	24	117	192	100											434
3	1	36	297	1008	1650	1296	392									4680
4	1	48	558	2880	8025	12960	12152	6144	1296							44064
5	1	60	900	6240	24625	60588	96530	99840	64800	24000	3872					381456
6	1	72	1323	11520	58875	195048	440461	693504	763020	576000	284592	82944	10816			3118176
7	1	84	1827	19152	120225	501228	1467697	3112896	4852386	5569200	4655112	2757888	1097824	263424	28800	24447744

All 15 rows

Technical details

Node id: n-43519ccf905be15d · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-6a29f8d72993234b

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,  12,   18;
  2: 1,  24,  117,   192,    100;
  3: 1,  36,  297,  1008,   1650,    1296,     392;
  4: 1,  48,  558,  2880,   8025,   12960,   12152,     6144,     1296;
  5: 1,  60,  900,  6240,  24625,   60588,   96530,    99840,    64800,     24000,      3872;
  6: 1,  72, 1323, 11520,  58875,  195048,  440461,   693504,   763020,    576000,    284592,     82944,     10816;
  7: 1,  84, 1827, 19152, 120225,  501228, 1467697,  3112896,  4852386,   5569200,   4655112,   2757888,   1097824,    263424,     28800;
  8: 1,  96, 2412, 29568, 220150, 1106784, 3985660, 10646016, 21524049,  33268800,  39368560,  35417088,  23811424,  11590656,   3859200,   786432,    73984;
  9: 1, 108, 3078, 43200, 372150, 2190888, 9368016, 30198528, 75122721, 146256300, 224440722, 271786752, 258480768, 190850688, 107179200, 44236800, 12651264, 2239488, 184832;

%F Formula

The row polynomials satisfy (1+(9*n-3)*x+(9*n^2+16*n-12)*x^2+(24*n^2-6*n-6)*x^3+(16*n^2-16*n+4)*x^4)*P(n,x) = (1+(9*n+9)*x+(9*n^2+61*n+33)*x^2+(51*n^2+162*n+63)*x^3+(106*n^2+210*n+66)*x^4+(96*n^2+132*n+36)*x^5+(32*n^2+32*n+8)*x^6)*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 + (9*n + 9)*x + (9*n^2 + 61*n + 33)*x^2 + (51*n^2 + 162*n + 63)*x^3 + (106*n^2 + 210*n + 66)*x^4 + (96*n^2 + 132*n + 36)*x^5 + (32*n^2 + 32*n + 8)*x^6)*P[n - 1, x])/(1 + (9*n - 3)*x + (9*n^2 + 16*n - 12)*x^2 + (24*n^2 - 6*n - 6)*x^3 + (16*n^2 - 16*n + 4)*x^4)]];
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 + (9*n + 9)*x + (9*n^2 + 61*n + 33)*x^2 + (51*n^2 + 162*n + 63)*x^3 + (106*n^2 + 210*n + 66)*x^4 + (96*n^2 + 132*n + 36)*x^5 + (32*n^2 + 32*n + 8)*x^6)*v[i - 1])/(1 + (9*n - 3)*x + (9*n^2 + 16*n - 12)*x^2 + (24*n^2 - 6*n - 6)*x^3 + (16*n^2 - 16*n + 4)*x^4)); 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 of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.