A201637 → reverse (shift +1) → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,x + n\right)\,P_{n}(x) = \left(2 + 6\,x - 19\,x^{2} - 3\,x^{3} + 2\,x^{4} - 5\,n - 13\,n x + 14\,n x^{2} + 8\,n x^{3} - 2\,n x^{4} + 4\,n^{2} + 9\,n^{2} x - 3\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 4\,x^{2} + 3\,x^{3} - 7\,x^{4} + 2\,x^{5} + 3\,n x - 2\,n x^{2} - 4\,n x^{3} + 3\,n x^{4} - n^{2} x + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(2 + 2\,x - 9\,x^{2} + 2\,x^{3} - 5\,n - 3\,n x + 5\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 4\,x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

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

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

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	1														2
2	2	4	1													7
3	6	22	11	1												40
4	24	140	124	26	1											315
5	120	1008	1428	536	57	1										3150
6	720	8136	16908	10208	2022	120	1									38115
7	5040	73008	208404	186620	60170	7050	247	1								540540
8	40320	722592	2697048	3361384	1623020	315500	23408	502	1							8783775
9	362880	7830720	36812808	60786552	41267564	12140500	1533320	75292	1013	1						160810650
10	3628800	92306880	530899776	1117806528	1015691824	423947384	82048120	7076944	237082	2036	1					3273645375
11	39916800	1176629760	8090800128	21076137024	24655639392	13893000912	3848972344	516320952	31499246	735758	4083	1				73329656400
12	479001600	16134439680	130192624512	409587462720	598101955344	437428825728	164884124272	31960822992	3085370160	136582636	2260452	8178	1			1791993478275
13	6227020800	236899192320	2209124396160	8230248084096	14632407028944	13458997327248	6632228966992	1762307408176	248351780208	17742410536	580955932	6895968	16369	1		47435121483750
14	87178291200	3709163404800	39463704610560	171309216241152	363331453755168	409694615927424	255593743927376	89378567261248	17396920761136	1834837034696	99104180116	2435925664	20933582	32752	1	1351900962286875

Technical details

Node id: n-b043c5c4d656c481 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-1cd7af8da2954497

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,       1;
  2:      2,       4,        1;
  3:      6,      22,       11,        1;
  4:     24,     140,      124,       26,        1;
  5:    120,    1008,     1428,      536,       57,        1;
  6:    720,    8136,    16908,    10208,     2022,      120,       1;
  7:   5040,   73008,   208404,   186620,    60170,     7050,     247,     1;
  8:  40320,  722592,  2697048,  3361384,  1623020,   315500,   23408,   502,    1;
  9: 362880, 7830720, 36812808, 60786552, 41267564, 12140500, 1533320, 75292, 1013, 1;

%F Formula

The row polynomials satisfy (n-2+2*x)*P(n,x) = -(n^3-4*n^2+5*n-2+(2*n^3-9*n^2+13*n-6)*x+(n^3-14*n+19)*x^2+(3*n^2-8*n+3)*x^3+(2*n-2)*x^4)*P(n-2,x) - ((n^2-3*n+2)*x+(2*n-4)*x^2-(n^2-4*n+3)*x^3-(3*n-7)*x^4-2*x^5)*d/dx P(n-2,x) + (2*n^2-5*n+2+(2*n^2-3*n+2)*x+(5*n-9)*x^2+2*x^3)*P(n-1,x) + ((n-2)*x-(n-4)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n^2 + 5*n - 2 + (2*n^3 - 9*n^2 + 13*n - 6)*x + (n^3 - 14*n + 19)*x^2 + (3*n^2 - 8*n + 3)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] - ((n^2 - 3*n + 2)*x + (2*n - 4)*x^2 - (n^2 - 4*n + 3)*x^3 - (3*n - 7)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (2*n^2 - 5*n + 2 + (2*n^2 - 3*n + 2)*x + (5*n - 9)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 2)*x - (n - 4)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 2 + 2*x)]];
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 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 4*n^2 + 5*n - 2 + (2*n^3 - 9*n^2 + 13*n - 6)*x + (n^3 - 14*n + 19)*x^2 + (3*n^2 - 8*n + 3)*x^3 + (2*n - 2)*x^4)*v[i - 2] - ((n^2 - 3*n + 2)*x + (2*n - 4)*x^2 - (n^2 - 4*n + 3)*x^3 - (3*n - 7)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (2*n^2 - 5*n + 2 + (2*n^2 - 3*n + 2)*x + (5*n - 9)*x^2 + 2*x^3)*v[i - 1] + ((n - 2)*x - (n - 4)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 2 + 2*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 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 Row reversed version of triangle A201637 (second-order Eulerian triangle).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.?
    
    ↑ Liu–Wang theorem (above)

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.