A088729 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-2}(x) + \left(-3\,x^{2}\right)\,P_{n-2}'(x) + \left(-2\,x^{2}\right)\,P_{n-2}''(x) + \left(3 + 2\,x\right)\,P_{n-1}(x) + \left(2 + 3\,x\right)\,P_{n-1}'(x) + \left(2\,x\right)\,P_{n-1}''(x)\)

\(P_{1}(x) = 13 + 18\,x + 3\,x^{2}\)

\(P_{2}(x) = 75 + 158\,x + 54\,x^{2} + 4\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	13	18	3															34
2	75	158	54	4														291
3	541	1530	795	120	5													2991
4	4683	16622	11610	2660	225	6												35806
5	47293	201978	177303	54600	7000	378	7											488559
6	545835	2724878	2881494	1115604	192150	15708	588	8										7476265
7	7087261	40492890	50101635	23378040	5062155	555660	31458	864	9									126709972
8	102247563	657944942	932458770	509327620	132937875	18341442	1393560	57840	1215	10								2354710837
9	1622632573	11611834938	18546943503	11609221800	3547799750	590007726	56331429	3136320	99495	1650	11							47588009195
10	28091567595	221291822798	393306236334	277569511604	97228909800	18933361656	2178661716	152385816	6482025	162250	2178	12						1038759103784
11	526858348381	4530383962650	8867945070075	6966567567960	2752063619305	614338845120	82774627936	6976857888	372733218	12505350	253253	2808	13					24348294393957
12	10641342970443	99179581033262	212005297797930	183488773393380	80706106005525	20320348766718	3138727912320	309754016000	19938628710	839959120	22792770	381108	3549	14				609810733660849
13	230283190977853	2312402554523898	5359614972812103	5067134403294600	2455969995160500	688602014694594	119976253287891	13573358472960	1022889620610	51911359500	1768344578	39606840	556010	4410	15			16248633352716362
14	5315654681981355	57211901491595918	142913557846397574	146542805981754004	77603568666255450	23980290495423444	4653059466539484	593806192832424	51297688277550	3047553723500	125042721804	3515080296	66082380	789880	5400	16		458869118689460479
15	130370767029135901	1497211181084718810	4009810380526544115	4432379279563085880	2546246562242196455	859789792907971380	183884261272900734	26135234974978272	2546773253637327	173432587485300	8329368347014	281984858064	6652195914	106457400	1096500	6528	17	13688556283555615611

Technical details

Node id: n-98e09500466a3809 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c9e06b6396582449

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:          13,           18,            3;
   2:          75,          158,           54,            4;
   3:         541,         1530,          795,          120,           5;
   4:        4683,        16622,        11610,         2660,         225,           6;
   5:       47293,       201978,       177303,        54600,        7000,         378,          7;
   6:      545835,      2724878,      2881494,      1115604,      192150,       15708,        588,         8;
   7:     7087261,     40492890,     50101635,     23378040,     5062155,      555660,      31458,       864,       9;
   8:   102247563,    657944942,    932458770,    509327620,   132937875,    18341442,    1393560,     57840,    1215,     10;
   9:  1622632573,  11611834938,  18546943503,  11609221800,  3547799750,   590007726,   56331429,   3136320,   99495,   1650,   11;
  10: 28091567595, 221291822798, 393306236334, 277569511604, 97228909800, 18933361656, 2178661716, 152385816, 6482025, 162250, 2178, 12;

%F Formula

The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 3*x^2*d/dx P(n-2,x) - 2*x^2*d^2/dx^2 P(n-2,x) + (3+2*x)*P(n-1,x) + (2+3*x)*d/dx P(n-1,x) + 2*x*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 13 + 18*x + 3*x^2, P(2,x) = 75 + 158*x + 54*x^2 + 4*x^3.

%t Mathematica

P[1, x_] := 13 + 18*x + 3*x^2;
P[2, x_] := 75 + 158*x + 54*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 3*x^2*D[P[n - 2, x], x] - 2*x^2*D[P[n - 2, x], {x, 2}] + (3 + 2*x)*P[n - 1, x] + (2 + 3*x)*D[P[n - 1, x], x] + 2*x*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 13 + 18*x + 3*x^2; v[2] = 75 + 158*x + 54*x^2 + 4*x^3; for(n = 3, N, my(i = n); v[i] = -x^2*v[i - 2] - 3*x^2*deriv(v[i - 2], x) - 2*x^2*deriv(deriv(v[i - 2], x), x) + (3 + 2*x)*v[i - 1] + (2 + 3*x)*deriv(v[i - 1], x) + 2*x*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.?
  - 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.