A359363 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 8\,n^{2} + 5\,n^{3} + n^{4}\right)\,P_{n}(x) = \left(-2\,n - 2\,n x - n^{2} + 8\,n^{2} x + 2\,n^{3} + 20\,n^{3} x + n^{4} + 10\,n^{4} x\right)\,P_{n-1}(x) + \left(6\,n x + 9\,n^{2} x - 21\,n^{2} x^{2} + 3\,n^{3} x - 15\,n^{3} x^{2}\right)\,P_{n-1}'(x) + \left(2\,x^{2} + 2\,x^{3} + 10\,n x^{2} + 10\,n x^{3} + 6\,n^{2} x^{2} + 6\,n^{2} x^{3}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	2														2
2	0	2	3													5
3	0	2	12	4												18
4	0	2	30	40	5											77
5	0	2	60	200	100	6										368
6	0	2	105	700	875	210	7									1899
7	0	2	168	1960	4900	2940	392	8								10370
8	0	2	252	4704	20580	24696	8232	672	9							59147
9	0	2	360	10080	70560	148176	98784	20160	1080	10						349212
10	0	2	495	19800	207900	698544	814968	332640	44550	1650	11					2120560
11	0	2	660	36300	544500	2744280	5122656	3659040	980100	90750	2420	12				13180720
12	0	2	858	62920	1297725	9343620	26162136	29899584	14015430	2595450	173030	3432	13			83554200
13	0	2	1092	104104	2862860	28342314	113369256	194347296	145760472	47237190	6298292	312312	4732	14		538639936
14	0	2	1365	165620	5920915	78156078	429858429	1052714520	1184303835	614083470	143286143	14210196	538265	6370	15	3523245223

Technical details

Node id: n-bbf9ecd549bbacc7 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-637b1e74dc634e59

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: 0, 2;
  2: 0, 2,   3;
  3: 0, 2,  12,     4;
  4: 0, 2,  30,    40,     5;
  5: 0, 2,  60,   200,   100,      6;
  6: 0, 2, 105,   700,   875,    210,     7;
  7: 0, 2, 168,  1960,  4900,   2940,   392,     8;
  8: 0, 2, 252,  4704, 20580,  24696,  8232,   672,    9;
  9: 0, 2, 360, 10080, 70560, 148176, 98784, 20160, 1080, 10;

%F Formula

The row polynomials satisfy (n^4+5*n^3+8*n^2+4*n)*P(n,x) = (n^4+2*n^3-n^2-2*n+(10*n^4+20*n^3+8*n^2-2*n)*x)*P(n-1,x) + ((3*n^3+9*n^2+6*n)*x-(15*n^3+21*n^2)*x^2)*d/dx P(n-1,x) + ((6*n^2+10*n+2)*x^2+(6*n^2+10*n+2)*x^3)*d^2/dx^2 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[((n^4 + 2*n^3 - n^2 - 2*n + (10*n^4 + 20*n^3 + 8*n^2 - 2*n)*x)*P[n - 1, x] + ((3*n^3 + 9*n^2 + 6*n)*x - (15*n^3 + 21*n^2)*x^2)*D[P[n - 1, x], x] + ((6*n^2 + 10*n + 2)*x^2 + (6*n^2 + 10*n + 2)*x^3)*D[P[n - 1, x], {x, 2}])/(n^4 + 5*n^3 + 8*n^2 + 4*n)]];
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] = ((n^4 + 2*n^3 - n^2 - 2*n + (10*n^4 + 20*n^3 + 8*n^2 - 2*n)*x)*v[i - 1] + ((3*n^3 + 9*n^2 + 6*n)*x - (15*n^3 + 21*n^2)*x^2)*deriv(v[i - 1], x) + ((6*n^2 + 10*n + 2)*x^2 + (6*n^2 + 10*n + 2)*x^3)*deriv(deriv(v[i - 1], x), x))/(n^4 + 5*n^3 + 8*n^2 + 4*n)); 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. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 …?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.