A308737 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,n x - n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 16\,x - 6\,x^{2} - 8\,x^{3} - 2\,n x + 21\,n x^{2} - n x^{4} - 3\,n^{2} x^{2} + 9\,n^{2} x^{3} - n^{3} x^{3} + n^{3} x^{4}\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2} - 10\,x^{3} - 4\,x^{4} + 3\,n x^{2} - n x^{3} - 5\,n x^{4} - n x^{5} + n^{2} x^{3} - n^{2} x^{5}\right)\,P_{n-1}'(x)\)

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

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	6														7
2	1	16	21													38
3	1	34	93	60												188
4	1	68	288	376	155											888
5	1	134	774	1528	1265	378										4080
6	1	264	1935	5120	6375	3816	889									18400
7	1	522	4635	15380	25575	22986	10717	2040								81856
8	1	1036	10794	43064	89600	107436	75166	28624	4599							360320
9	1	2062	24636	114800	286790	429996	401212	229216	73665	10230						1572608
10	1	4112	55377	295104	860370	1548288	1806042	1375872	663309	184240	22517					6815232
11	1	8210	122985	737580	2458050	5161212	7225050	6880560	4423005	1842850	450461	49140				29359104
12	1	16404	270468	1802680	6759225	16220952	26492928	30276576	24328755	13515700	4955676	1081224	106483			125827072
13	1	32790	589986	4325992	18023775	48662262	90833820	121109472	116782479	81098050	39647586	12975480	2555761	229362		536866816
14	1	65560	1278147	10224448	46860385	140578152	295209915	449839104	506065131	421718440	257715601	112457280	33225907	5963608	491505	2281693184

Technical details

Node id: n-bac037884d2391c5 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-dae00e5755e6d397

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: 1,    6;
  2: 1,   16,    21;
  3: 1,   34,    93,     60;
  4: 1,   68,   288,    376,    155;
  5: 1,  134,   774,   1528,   1265,    378;
  6: 1,  264,  1935,   5120,   6375,   3816,    889;
  7: 1,  522,  4635,  15380,  25575,  22986,  10717,   2040;
  8: 1, 1036, 10794,  43064,  89600, 107436,  75166,  28624,  4599;
  9: 1, 2062, 24636, 114800, 286790, 429996, 401212, 229216, 73665, 10230;

%F Formula

The row polynomials satisfy (2+2*n*x+(n^2-n)*x^2)*P(n,x) = (2-(2*n-16)*x-(3*n^2-21*n+6)*x^2-(n^3-9*n^2+8)*x^3+(n^3-n)*x^4)*P(n-1,x) + (2*x+(3*n-4)*x^2+(n^2-n-10)*x^3-(5*n+4)*x^4-(n^2+n)*x^5)*d/dx 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[((2 - (2*n - 16)*x - (3*n^2 - 21*n + 6)*x^2 - (n^3 - 9*n^2 + 8)*x^3 + (n^3 - n)*x^4)*P[n - 1, x] + (2*x + (3*n - 4)*x^2 + (n^2 - n - 10)*x^3 - (5*n + 4)*x^4 - (n^2 + n)*x^5)*D[P[n - 1, x], x])/(2 + 2*n*x + (n^2 - n)*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] = ((2 - (2*n - 16)*x - (3*n^2 - 21*n + 6)*x^2 - (n^3 - 9*n^2 + 8)*x^3 + (n^3 - n)*x^4)*v[i - 1] + (2*x + (3*n - 4)*x^2 + (n^2 - n - 10)*x^3 - (5*n + 4)*x^4 - (n^2 + n)*x^5)*deriv(v[i - 1], x))/(2 + 2*n*x + (n^2 - n)*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 of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is t…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? 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
[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.