A008517 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-2\,n + 6\,n x - 6\,n x^{2} - 4\,n^{2} x + 10\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-n x + 3\,n x^{2} - 2\,n x^{3} - 2\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 2\,x + 3\,n - 6\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	8	12													20
3	22	116	72												210
4	52	656	1332	480											2520
5	114	2904	13200	14832	3600										34650
6	240	11220	96360	232560	169920	30240									540540
7	494	39900	587400	2576080	3926520	2046816	282240								9459450
8	1004	134520	3187500	23062000	62200320	66157776	26137440	2903040							183783600
9	2026	437696	15978480	179060000	776786920	1433429424	1135308384	354712320	32659200						3928374450
10	4072	1390076	75731712	1257478880	8241921520	24016170528	32495147664	20059849728	5115208320	399168000					91662070500
11	8166	4341252	344629128	8205931104	77744803920	336248392704	708263031168	737598321792	367094716416	78277190400	5269017600				2319050383650
12	16356	13399392	1521777828	50679271104	671617101120	4126320278304	12727320496128	20398923787392	16947716956416	6979294368000	1268696943360	74724249600			63246828645000
13	32738	41015976	6569489424	300064208912	5423382562080	45808992896352	198099370462784	459954987716736	582555376258848	396895780558080	138059944197120	21731910681600	1133317785600		1849969737866250
14	65504	124815780	27877580760	1719304025360	41542221639840	470348308987872	2760559114708640	8893979780557440	16174540986215040	16664244862714560	9515255785415424	2842899258516480	392552900812800	18307441152000	57775977967207500

Technical details

Node id: n-9414b3f39c23072d · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-69d5ef1a37e94893

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 -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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    0;
  1:    2;
  2:    8,     12;
  3:   22,    116,       72;
  4:   52,    656,     1332,       480;
  5:  114,   2904,    13200,     14832,      3600;
  6:  240,  11220,    96360,    232560,    169920,      30240;
  7:  494,  39900,   587400,   2576080,   3926520,    2046816,     282240;
  8: 1004, 134520,  3187500,  23062000,  62200320,   66157776,   26137440,   2903040;
  9: 2026, 437696, 15978480, 179060000, 776786920, 1433429424, 1135308384, 354712320, 32659200;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = -(2*n+(4*n^2-6*n)*x+(4*n^3-10*n^2+6*n)*x^2)*P(n-2,x) - (n*x+(2*n^2-3*n)*x^2-(2*n^2-2*n)*x^3)*d/dx P(n-2,x) + (3*n-2+(4*n^2-6*n+2)*x)*P(n-1,x) + ((n-1)*x-(n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + (4*n^2 - 6*n)*x + (4*n^3 - 10*n^2 + 6*n)*x^2)*P[n - 2, x] - (n*x + (2*n^2 - 3*n)*x^2 - (2*n^2 - 2*n)*x^3)*D[P[n - 2, x], x] + (3*n - 2 + (4*n^2 - 6*n + 2)*x)*P[n - 1, x] + ((n - 1)*x - (n - 1)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + (4*n^2 - 6*n)*x + (4*n^3 - 10*n^2 + 6*n)*x^2)*v[i - 2] - (n*x + (2*n^2 - 3*n)*x^2 - (2*n^2 - 2*n)*x^3)*deriv(v[i - 2], x) + (3*n - 2 + (4*n^2 - 6*n + 2)*x)*v[i - 1] + ((n - 1)*x - (n - 1)*x^2)*deriv(v[i - 1], x))/(n - 1)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Second-order Eulerian triangle T(n,k), 1 <= k <= n.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Second-order Eulerian triangle T(n,k), 1 <= k <= n.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A201637 → D

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.