A080248 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\right)\,P_{n}(x) = \left(-12 + 4\,x + 4\,x^{2} - 2\,x^{3}\right)\,P_{n-2}(x) + \left(-12\,x + 10\,x^{2} - 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} + 2\,x^{3} - x^{4}\right)\,P_{n-2}''(x) + \left(16 - 4\,x + 2\,x^{2}\right)\,P_{n-1}(x) + \left(12\,x\right)\,P_{n-1}'(x) + \left(2\,x^{2}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	5	6	1													12
3	14	33	13	1												61
4	41	186	133	24	1											385
5	122	1077	1297	413	40	1										2950
6	365	6342	12613	6400	1078	62	1									26861
7	1094	37689	123505	95781	25048	2478	91	1								285687
8	3281	225042	1217845	1419944	545899	82588	5166	128	1							3499894
9	9842	1346973	12070849	21041053	11570128	2518571	238282	9966	174	1						48805839
10	29525	8071998	120049765	312483600	242463376	73378050	9852403	617584	18051	230	1					766964583
11	88574	48402465	1196505937	4652599061	5061955216	2088234952	383059521	33731555	1465750	31031	297	1				13466074360
12	265721	290326218	11940990997	69422359864	105626388517	58736732432	14374620544	1708708560	103437906	3232086	51051	376	1			262207114273
13	797162	1741691589	119265145441	1037548441293	2205713552536	1643150342277	528522769744	82626172432	6690543132	289139422	6696547	80899	468	1		5625555372943
14	2391485	10449352374	1191781804357	15524632167200	46109928617194	45866491935814	19209406276053	3881133898784	409059408106	23469022564	746936190	13153088	124124	574	1	132227115087908

Technical details

Node id: n-2b1aebafcbe458a9 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-3c7966ab378e7773

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:    2,       1;
  2:    5,       6,        1;
  3:   14,      33,       13,        1;
  4:   41,     186,      133,       24,        1;
  5:  122,    1077,     1297,      413,       40,       1;
  6:  365,    6342,    12613,     6400,     1078,      62,      1;
  7: 1094,   37689,   123505,    95781,    25048,    2478,     91,    1;
  8: 3281,  225042,  1217845,  1419944,   545899,   82588,   5166,  128,   1;
  9: 9842, 1346973, 12070849, 21041053, 11570128, 2518571, 238282, 9966, 174, 1;

%F Formula

The row polynomials satisfy 4*P(n,x) = -(12-4*x-4*x^2+2*x^3)*P(n-2,x) - (12*x-10*x^2+4*x^3)*d/dx P(n-2,x) - (2*x^2-2*x^3+x^4)*d^2/dx^2 P(n-2,x) + (16-4*x+2*x^2)*P(n-1,x) + 12*x*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) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(12 - 4*x - 4*x^2 + 2*x^3)*P[n - 2, x] - (12*x - 10*x^2 + 4*x^3)*D[P[n - 2, x], x] - (2*x^2 - 2*x^3 + x^4)*D[P[n - 2, x], {x, 2}] + (16 - 4*x + 2*x^2)*P[n - 1, x] + 12*x*D[P[n - 1, x], x] + 2*x^2*D[P[n - 1, x], {x, 2}])/(4)]];
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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(12 - 4*x - 4*x^2 + 2*x^3)*v[i - 2] - (12*x - 10*x^2 + 4*x^3)*deriv(v[i - 2], x) - (2*x^2 - 2*x^3 + x^4)*deriv(deriv(v[i - 2], x), x) + (16 - 4*x + 2*x^2)*v[i - 1] + 12*x*deriv(v[i - 1], x) + 2*x^2*deriv(deriv(v[i - 1], x), x))/(4)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Stirling-like number triangle defined by sequence A000217.?
  - 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.