A269945 → θ+1

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	0	2														2
3	0	2	3													5
4	0	2	15	4												21
5	0	2	63	56	5											126
6	0	2	255	588	150	6										1001
7	0	2	1023	5632	3135	330	7									10129
8	0	2	4095	52052	57200	12012	637	8								126006
9	0	2	16383	473928	980265	368940	36946	1120	9							1877593
10	0	2	65535	4287196	16276650	10399818	1760486	97104	1836	10						32888637
11	0	2	262143	38672144	265785395	279527430	75510617	6770080	226746	2850	11					666757418
12	0	2	1048575	348398820	4300906500	7307128224	3044497547	418031768	22128084	482790	4235	12				15442626557
13	0	2	4194303	3136987480	69250002525	187839293400	118126894620	23962982400	1886483115	63692750	954569	6072	13			404271491249
14	0	2	16777215	28238479724	1111921274750	4779082338030	4471714048620	1309188302880	147693274560	7255205100	165518925	1776060	8450	14		11855277004330
15	0	2	67108863	254168687136	17826038495655	120811363980450	166557301811355	69260757182400	10925206412580	751775251500	24532618110	395469360	3140865	11466	15	386411610169757

Technical details

Node id: n-37d876d4d4dcb1c8 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-64728cdfc8b98a6c

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 -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: 1;
   2: 0, 2;
   3: 0, 2,     3;
   4: 0, 2,    15,       4;
   5: 0, 2,    63,      56,        5;
   6: 0, 2,   255,     588,      150,        6;
   7: 0, 2,  1023,    5632,     3135,      330,       7;
   8: 0, 2,  4095,   52052,    57200,    12012,     637,     8;
   9: 0, 2, 16383,  473928,   980265,   368940,   36946,  1120,    9;
  10: 0, 2, 65535, 4287196, 16276650, 10399818, 1760486, 97104, 1836, 10;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[-(2*x + x^2)*P[n - 2, x] - x*D[P[n - 2, x], x] - (x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (1 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x] + x^2*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] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = -(2*x + x^2)*v[i - 2] - x*deriv(v[i - 2], x) - (x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (1 + 2*x)*v[i - 1] + x*deriv(v[i - 1], x) + 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, 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 Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(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.