A257622 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + 3\,n\right)\,P_{n}(x) = \left(-28 + 40\,x - 10\,x^{2} - 84\,n + 96\,n x - 9\,n x^{2} - 72\,n^{2} x + 54\,n^{2} x^{2} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-12\,x + 18\,x^{2} - 6\,x^{3} - 36\,n x + 45\,n x^{2} - 9\,n x^{3} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-10 + 8\,x + 33\,n - 24\,n x + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 6\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	4														4
2	56	32													88
3	552	1104	192												1848
4	4696	22080	14088	1024											41888
5	36968	342896	514344	147872	5120										1047200
6	278232	4611936	13372944	9223872	1391160	24576									28902720
7	2037736	56692176	284534736	379379648	141730440	12226416	114688								876715840
8	14673752	656582144	5314678584	12140148736	8857797640	1969746432	102716264	524288							29056867840
9	104551272	7299509040	90758032344	328880903904	411101129880	181516064688	25548281640	836410176	2359296						1046047242240
10	739985368	78849961632	1453586009472	7924330077888	15621842935440	11886495116832	3391700688768	315399846528	6659868312	10485760					40679614976000
11	5215549160	834378709136	22208316511680	175242148228608	514734092915280	617680911498336	306673759400064	59222177364480	3754704191112	52155491600	46137344				1700407905996800
12	36664033368	8698444434240	327506724626664	3632984854760448	15256264897534320	27177960105926784	21358770856548048	7265969709520896	982520173879992	43492222171200	403304367048	201326592			76054608159129600
13	257319322216	89697582811632	4701208086292872	71664702294069280	417023502969767760	1055603069256418848	1231536914132488656	667237604751628416	161245580161655880	15670693620976240	493336705463976	3087831866592	872415232		3625269655585177600
14	1804120936664	917561373893600	66093920967852912	1359756669950385344	10700453770320772040	37235257223425411392	61576845706624432800	49647009631233881856	19260816786577389672	3399391674875963360	242344376882127344	5505368243361600	23453572176632	3758096384	183494417951926681600

Technical details

Node id: n-d5ced0bffca38ecf · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-afbfd84852f44762

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:         4;
  2:        56,         32;
  3:       552,       1104,         192;
  4:      4696,      22080,       14088,         1024;
  5:     36968,     342896,      514344,       147872,         5120;
  6:    278232,    4611936,    13372944,      9223872,      1391160,        24576;
  7:   2037736,   56692176,   284534736,    379379648,    141730440,     12226416,      114688;
  8:  14673752,  656582144,  5314678584,  12140148736,   8857797640,   1969746432,   102716264,    524288;
  9: 104551272, 7299509040, 90758032344, 328880903904, 411101129880, 181516064688, 25548281640, 836410176, 2359296;

%F Formula

The row polynomials satisfy (3*n-2)*P(n,x) = -(84*n+28+(72*n^2-96*n-40)*x+(27*n^3-54*n^2+9*n+10)*x^2)*P(n-2,x) - ((36*n+12)*x+(27*n^2-45*n-18)*x^2-(27*n^2-9*n-6)*x^3)*d/dx P(n-2,x) + (33*n-10+(18*n^2-24*n+8)*x)*P(n-1,x) + ((9*n-6)*x-(9*n-6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(84*n + 28 + (72*n^2 - 96*n - 40)*x + (27*n^3 - 54*n^2 + 9*n + 10)*x^2)*P[n - 2, x] - ((36*n + 12)*x + (27*n^2 - 45*n - 18)*x^2 - (27*n^2 - 9*n - 6)*x^3)*D[P[n - 2, x], x] + (33*n - 10 + (18*n^2 - 24*n + 8)*x)*P[n - 1, x] + ((9*n - 6)*x - (9*n - 6)*x^2)*D[P[n - 1, x], x])/(3*n - 2)]];
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] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(84*n + 28 + (72*n^2 - 96*n - 40)*x + (27*n^3 - 54*n^2 + 9*n + 10)*x^2)*v[i - 2] - ((36*n + 12)*x + (27*n^2 - 45*n - 18)*x^2 - (27*n^2 - 9*n - 6)*x^3)*deriv(v[i - 2], x) + (33*n - 10 + (18*n^2 - 24*n + 8)*x)*v[i - 1] + ((9*n - 6)*x - (9*n - 6)*x^2)*deriv(v[i - 1], x))/(3*n - 2)); 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 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.?
    
    ↑ 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.