A156289 → D+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 4 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	3														7
3	16	45	15													76
4	64	483	525	105												1177
5	256	4665	11655	6930	945											24451
6	1024	43263	218625	273735	103950	10395										650992
7	4096	394485	3798795	8768760	6621615	1756755	135135									21479641
8	16384	3570843	63533925	253017765	332432100	168378210	33108075	2027025								856084327
9	65536	32219505	1041505935	6893694990	14709579885	12455933490	4534860330	689188500	34459425							40391507596
10	262144	290303223	16889500425	181683774195	604691901450	798289336845	471841420050	129636356850	15713497800	654729075						2219691082057
11	1048576	2614039725	272263605075	4693809817620	23756266981755	46889121083805	41947924403385	18293130355500	3932302824450	389563799625	13749310575					140190747270091
12	4194304	23531600403	4374513862125	119793004895025	906676881264000	2604096658478700	3378961820359125	2180735592910875	730832163761700	126388245908925	10435726726425	316234143225				10062634378104832
13	16777216	211805375145	70156934608215	3034172219728650	33955182279243225	139328366344467300	254964169503143100	233272374565230000	113667426322324875	30201490085641875	4296040835711625	300422436063750	7905853580625			813097919201895601
14	67108864	1906332262383	1123993557803025	76485506116092255	1255715755776231750	7267001909572580175	18394351352643766500	23185223931009640500	15732389313446305500	5991394213930875750	1293560506374010875	154102479164258625	9249848689331250	213458046676875		73360814174726944327
15	268435456	17157325905765	17997241116467355	1922251688657380080	46046359081604540295	372369342814493599575	1285748840026410034275	2188367223301264005000	2008836926805267058500	1050157726888824303750	321154376499633426750	57476093129187232500	5820929780196155625	303323884327839375	6190283353629375	7338227598581930013676

Technical details

Node id: n-3b22cec39202b5b9 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e5341cf5487df687

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:      4,         3;
   3:     16,        45,          15;
   4:     64,       483,         525,          105;
   5:    256,      4665,       11655,         6930,          945;
   6:   1024,     43263,      218625,       273735,       103950,        10395;
   7:   4096,    394485,     3798795,      8768760,      6621615,      1756755,       135135;
   8:  16384,   3570843,    63533925,    253017765,    332432100,    168378210,     33108075,      2027025;
   9:  65536,  32219505,  1041505935,   6893694990,  14709579885,  12455933490,   4534860330,    689188500,    34459425;
  10: 262144, 290303223, 16889500425, 181683774195, 604691901450, 798289336845, 471841420050, 129636356850, 15713497800, 654729075;

%F Formula

The row polynomials satisfy P(n,x) = -(4+5*x+2*x^2-3*x^3)*P(n-2,x) - (5*x+2*x^2-3*x^3-2*x^4)*d/dx P(n-2,x) - (x^2-x^4)*d^2/dx^2 P(n-2,x) + (5+5*x-x^2)*P(n-1,x) + (5*x+2*x^2)*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) = 4 + 3*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(4 + 5*x + 2*x^2 - 3*x^3)*P[n - 2, x] - (5*x + 2*x^2 - 3*x^3 - 2*x^4)*D[P[n - 2, x], x] - (x^2 - x^4)*D[P[n - 2, x], {x, 2}] + (5 + 5*x - x^2)*P[n - 1, x] + (5*x + 2*x^2)*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] = 4 + 3*x; for(n = 3, N, my(i = n); v[i] = -(4 + 5*x + 2*x^2 - 3*x^3)*v[i - 2] - (5*x + 2*x^2 - 3*x^3 - 2*x^4)*deriv(v[i - 2], x) - (x^2 - x^4)*deriv(deriv(v[i - 2], x), x) + (5 + 5*x - x^2)*v[i - 1] + (5*x + 2*x^2)*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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=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.