A056242 → reverse → D

nonnegative coefficients

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

Recurrence

\(\left(-14 - 20\,x + 27\,n + 38\,n x - 16\,n^{2} - 22\,n^{2} x + 3\,n^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(-16 - 56\,x - 70\,x^{2} + 36\,n + 121\,n x + 116\,n x^{2} - 26\,n^{2} - 82\,n^{2} x - 66\,n^{2} x^{2} + 6\,n^{3} + 17\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-8\,x - 28\,x^{2} - 20\,x^{3} + 14\,n x + 38\,n x^{2} + 24\,n x^{3} - 6\,n^{2} x - 14\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	5	2													7
4	16	18	3												37
5	44	82	42	4											172
6	112	292	255	80	5										744
7	272	912	1131	620	135	6									3076
8	640	2624	4224	3332	1295	210	7								12332
9	1472	7136	14136	14596	8260	2436	308	8							48352
10	3328	18624	43824	56008	41805	18144	4242	432	9						186416
11	7424	47104	128400	195760	181825	104814	36414	6960	585	10					709296
12	16384	116224	360192	638240	710000	511830	237699	68112	10890	770	11				2670352
13	35840	281088	976128	1971584	2556900	2215830	1292830	498168	120384	16390	990	12			9966144
14	77824	668672	2572032	5833472	8644640	8758860	6164795	2997280	979407	203060	23881	1248	13		36925184
15	167936	1568768	6620928	16662016	27781600	32239200	26602135	15653560	6480045	1826110	329329	33852	1547	14	135967040

Technical details

Node id: n-7b839014ea990f5b · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-e20dbbeeb0459415

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
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:    0;
   2:    1;
   3:    5,     2;
   4:   16,    18,     3;
   5:   44,    82,    42,     4;
   6:  112,   292,   255,    80,     5;
   7:  272,   912,  1131,   620,   135,     6;
   8:  640,  2624,  4224,  3332,  1295,   210,    7;
   9: 1472,  7136, 14136, 14596,  8260,  2436,  308,   8;
  10: 3328, 18624, 43824, 56008, 41805, 18144, 4242, 432, 9;

%F Formula

The row polynomials satisfy (3*n^3-16*n^2+27*n-14+(4*n^3-22*n^2+38*n-20)*x)*P(n,x) = (6*n^3-26*n^2+36*n-16+(17*n^3-82*n^2+121*n-56)*x+(12*n^3-66*n^2+116*n-70)*x^2)*P(n-1,x) - ((6*n^2-14*n+8)*x+(14*n^2-38*n+28)*x^2+(8*n^2-24*n+20)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((6*n^3 - 26*n^2 + 36*n - 16 + (17*n^3 - 82*n^2 + 121*n - 56)*x + (12*n^3 - 66*n^2 + 116*n - 70)*x^2)*P[n - 1, x] - ((6*n^2 - 14*n + 8)*x + (14*n^2 - 38*n + 28)*x^2 + (8*n^2 - 24*n + 20)*x^3)*D[P[n - 1, x], x])/(3*n^3 - 16*n^2 + 27*n - 14 + (4*n^3 - 22*n^2 + 38*n - 20)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((6*n^3 - 26*n^2 + 36*n - 16 + (17*n^3 - 82*n^2 + 121*n - 56)*x + (12*n^3 - 66*n^2 + 116*n - 70)*x^2)*v[i - 1] - ((6*n^2 - 14*n + 8)*x + (14*n^2 - 38*n + 28)*x^2 + (8*n^2 - 24*n + 20)*x^3)*deriv(v[i - 1], x))/(3*n^3 - 16*n^2 + 27*n - 14 + (4*n^3 - 22*n^2 + 38*n - 20)*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 reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
    no recorded property covers this requirement
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    ↑ Affine-current basis map theorem (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.