A166962 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5\,x^{2} + 12\,n x^{2} - 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2} + 3\,x^{3} - 12\,n x^{2} + 9\,n x^{3}\right)\,P_{n-2}'(x) + \left(-3\,x^{3} - 9\,n x^{3}\right)\,P_{n-2}''(x) + \left(1 - 4\,x + 6\,n x\right)\,P_{n-1}(x) + \left(7\,x - 3\,x^{2}\right)\,P_{n-1}'(x) + \left(3\,x^{2}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 1 + 2\,x\)

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	2														3
2	1	24	3													28
3	1	206	207	4												418
4	1	1668	6510	1264	5											9448
5	1	13370	161730	111320	6645	6										293072
6	1	106992	3657045	6609200	1405975	32472	7									11811692
7	1	855974	79365753	327756780	174003375	14927502	152467	8								597061860
8	1	6847836	1691076072	14803407264	16636267050	3521080692	142186044	698400	9							36801563368
9	1	54782738	35738576100	634976884384	1377425503530	597549512412	59990018592	1260406752	3144825	10						2706998829344
10	1	438261960	752564450775	26447406940960	104613358733570	83811322859328	17041506744318	907431800832	10633288797	13979800	11					233584677060352
11	1	3506095742	15822260468595	1082981759330900	7527172008558450	10431071573620884	3830998423701750	412896097147728	12601097738685	86617701310	61514871	12				23313633405878928
12	1	28048766004	332430503292498	43909968097396880	523109360535389875	1200100212093547944	740744876900929012	142722255702832128	8873852411989647	164265153463300	687932871098	268433424	13			2659957937648911824
13	1	224390128106	6982471056208662	1770139184698631784	35538957318209232675	130902901177042298874	129321640738224622492	41203862902118627360	4565617395348648879	174172111142907790	2041501797140718	5362826398104	1163217757	14		343486320388018063216
14	1	1795121024928	146644345832491785	71122106075826730704	2378625119090672105505	13760787478887890909784	21017779502514275175261	10503395409333954638016	1907996483486231657595	129934722592879050160	3186813302168566235	24458961866506992	41227607926739	5010792024	15	49772998781619337575744

Technical details

Node id: n-8c87e3909ebe3753 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-49673c01b4703ab3

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,        2;
  2: 1,       24,           3;
  3: 1,      206,         207,            4;
  4: 1,     1668,        6510,         1264,             5;
  5: 1,    13370,      161730,       111320,          6645,            6;
  6: 1,   106992,     3657045,      6609200,       1405975,        32472,           7;
  7: 1,   855974,    79365753,    327756780,     174003375,     14927502,      152467,          8;
  8: 1,  6847836,  1691076072,  14803407264,   16636267050,   3521080692,   142186044,     698400,       9;
  9: 1, 54782738, 35738576100, 634976884384, 1377425503530, 597549512412, 59990018592, 1260406752, 3144825, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-12*n-5)*x^2*P(n-2,x) - ((12*n+4)*x^2-(9*n+3)*x^3)*d/dx P(n-2,x) - (9*n+3)*x^3*d^2/dx^2 P(n-2,x) + (1+(6*n-4)*x)*P(n-1,x) + (7*x-3*x^2)*d/dx P(n-1,x) + 3*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 12*n - 5)*x^2*P[n - 2, x] - ((12*n + 4)*x^2 - (9*n + 3)*x^3)*D[P[n - 2, x], x] - (9*n + 3)*x^3*D[P[n - 2, x], {x, 2}] + (1 + (6*n - 4)*x)*P[n - 1, x] + (7*x - 3*x^2)*D[P[n - 1, x], x] + 3*x^2*D[P[n - 1, x], {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; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 12*n - 5)*x^2*v[i - 2] - ((12*n + 4)*x^2 - (9*n + 3)*x^3)*deriv(v[i - 2], x) - (9*n + 3)*x^3*deriv(deriv(v[i - 2], x), x) + (1 + (6*n - 4)*x)*v[i - 1] + (7*x - 3*x^2)*deriv(v[i - 1], x) + 3*x^2*deriv(deriv(v[i - 1], x), 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n.?
  - 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.