A257625 → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(27 - 90\,x + 135\,x^{2} - 54\,n + 216\,n x - 414\,n x^{2} - 72\,n^{2} x + 324\,n^{2} x^{2} - 72\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(18\,x - 72\,x^{2} + 54\,x^{3} - 36\,n x + 180\,n x^{2} - 144\,n x^{3} - 72\,n^{2} x^{2} + 72\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-30 + 54\,x + 24\,n - 72\,n x + 24\,n^{2} x\right)\,P_{n-1}(x) + \left(-18\,x + 18\,x^{2} + 12\,n x - 12\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	3														3
2	54	18													72
3	621	1242	81												1944
4	6156	37260	18468	324											62208
5	57591	817452	1226178	230364	1215										2332800
6	526338	15371694	51499476	30743388	2631690	4374									100776960
7	4765473	265313718	1704042603	2272056804	663284295	28592838	15309								4938071040
8	42987672	4351412664	48917923704	122691067416	81529872840	13054237992	300913704	52488							270888468480
9	387223659	69140080440	1281834038628	5465047463208	6831309329010	2563668077256	241990281540	3097789272	177147						16456474460160
10	3486134862	1076598019818	31585470240888	214211651515272	450866415714660	321317477272908	73699430562072	4306392079272	31375213758	590490					1097098297344000
11	31378933845	16546389671538	745654623574725	7678842805365624	25321409729953290	30385691675943948	13437974909389842	1988412329532600	74458753521921	313789338450	1948617				79649336387174400
12	282422627748	252149590737540	17073463411985724	258033270147953148	1267541428890274920	2370083950723627944	1774558000446384888	516066540295906296	51220390235957172	1260747953687700	3106648905228	6377292			6256093330774425600
13	2541843507807	3821218183692324	382370867976397734	8264481513305650020	58278488125311085545	160880511238296335496	187693929778012391412	93245581000497736872	18595083404937712545	1274569559921325780	21016700010307782	30502122093684	20726199		528639886450438963200
14	22876720710426	57699549281555910	8425284309516507948	255260153769269960196	2512034415958293386910	9840983411562595519698	16892453680021115227080	13121311215416794026264	4521661948724928096438	638150384423174900490	30892709134893862476	346197295689335460	297397369235538	66961566	47821577420439709286400

Technical details

Node id: n-e5b7396a6427b373 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-a7d11e539b3a8882

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:         3;
  2:        54,          18;
  3:       621,        1242,            81;
  4:      6156,       37260,         18468,           324;
  5:     57591,      817452,       1226178,        230364,          1215;
  6:    526338,    15371694,      51499476,      30743388,       2631690,          4374;
  7:   4765473,   265313718,    1704042603,    2272056804,     663284295,      28592838,        15309;
  8:  42987672,  4351412664,   48917923704,  122691067416,   81529872840,   13054237992,    300913704,      52488;
  9: 387223659, 69140080440, 1281834038628, 5465047463208, 6831309329010, 2563668077256, 241990281540, 3097789272, 177147;

%F Formula

The row polynomials satisfy (2*n-3)*P(n,x) = -(54*n-27+(72*n^2-216*n+90)*x+(72*n^3-324*n^2+414*n-135)*x^2)*P(n-2,x) - ((36*n-18)*x+(72*n^2-180*n+72)*x^2-(72*n^2-144*n+54)*x^3)*d/dx P(n-2,x) + (24*n-30+(24*n^2-72*n+54)*x)*P(n-1,x) + ((12*n-18)*x-(12*n-18)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(54*n - 27 + (72*n^2 - 216*n + 90)*x + (72*n^3 - 324*n^2 + 414*n - 135)*x^2)*P[n - 2, x] - ((36*n - 18)*x + (72*n^2 - 180*n + 72)*x^2 - (72*n^2 - 144*n + 54)*x^3)*D[P[n - 2, x], x] + (24*n - 30 + (24*n^2 - 72*n + 54)*x)*P[n - 1, x] + ((12*n - 18)*x - (12*n - 18)*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(54*n - 27 + (72*n^2 - 216*n + 90)*x + (72*n^3 - 324*n^2 + 414*n - 135)*x^2)*v[i - 2] - ((36*n - 18)*x + (72*n^2 - 180*n + 72)*x^2 - (72*n^2 - 144*n + 54)*x^3)*deriv(v[i - 2], x) + (24*n - 30 + (24*n^2 - 72*n + 54)*x)*v[i - 1] + ((12*n - 18)*x - (12*n - 18)*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - 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), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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.