A166961 → θ+1

nonnegative coefficients

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

Recurrence

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

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

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

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	18	3													22
3	1	118	126	4												249
4	1	722	2775	616	5											4119
5	1	4350	49206	35748	2535	6										91846
6	1	26122	796815	1460200	337500	9522	7									2630167
7	1	156758	12383238	50447380	27964250	2653452	33964	8								93639051
8	1	940578	188805351	1594147464	1825924275	410027532	18569194	117408	9							4038531812
9	1	5643502	2853243270	47908755076	104086120005	46781799282	5038354832	120198640	397629	10						206794512247
10	1	33861050	42942558351	1398510007528	5462392670210	4461535536678	949699182173	54730675456	736954587	1327330	11					12370582773375
11	1	203166342	645103415190	40131644866740	272029732800600	379674671076024	143678998298444	16336015499928	543758121027	4346384850	4382730	12				853044478011888
12	1	1218998098	9682950967623	1140028676120200	13094135429445225	29955063481428384	18833188023144768	3766572124314656	248597149453902	5055042951710	24917527305	14346792	13			67052349028698677
13	1	7313988638	145286319948726	32191925558459044	616311775382889875	2244154552534955934	2238033721031170608	731796019833052416	84646693904840874	3446429655363920	44679779535622	139873499892	46631299	14		5950771231234336863
14	1	43883931882	2179569073804815	905829362781947496	28579067938139594280	162166108946036980794	248170133905779818019	126181814740500643200	23653460957499045306	1689392337800533260	44440682660831490	379796129696760	772803296590	150660258	15	591392809053240784166

Technical details

Node id: n-1a6a5162ada875b1 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-572d7783088b9b6d

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,      18,          3;
  3: 1,     118,        126,           4;
  4: 1,     722,       2775,         616,            5;
  5: 1,    4350,      49206,       35748,         2535,           6;
  6: 1,   26122,     796815,     1460200,       337500,        9522,          7;
  7: 1,  156758,   12383238,    50447380,     27964250,     2653452,      33964,         8;
  8: 1,  940578,  188805351,  1594147464,   1825924275,   410027532,   18569194,    117408,      9;
  9: 1, 5643502, 2853243270, 47908755076, 104086120005, 46781799282, 5038354832, 120198640, 397629, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(4*n^2-4*n-3)*x^2*P(n-2,x) - ((6*n+3)*x^2-(4*n+2)*x^3)*d/dx P(n-2,x) - (4*n+2)*x^3*d^2/dx^2 P(n-2,x) + (1+(4*n-2)*x)*P(n-1,x) + (5*x-2*x^2)*d/dx P(n-1,x) + 2*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[-(4*n^2 - 4*n - 3)*x^2*P[n - 2, x] - ((6*n + 3)*x^2 - (4*n + 2)*x^3)*D[P[n - 2, x], x] - (4*n + 2)*x^3*D[P[n - 2, x], {x, 2}] + (1 + (4*n - 2)*x)*P[n - 1, x] + (5*x - 2*x^2)*D[P[n - 1, x], x] + 2*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] = -(4*n^2 - 4*n - 3)*x^2*v[i - 2] - ((6*n + 3)*x^2 - (4*n + 2)*x^3)*deriv(v[i - 2], x) - (4*n + 2)*x^3*deriv(deriv(v[i - 2], x), x) + (1 + (4*n - 2)*x)*v[i - 1] + (5*x - 2*x^2)*deriv(v[i - 1], x) + 2*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,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 2.?
  - 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.