A257608 → D

nonnegative coefficients

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

Recurrence

\(\left(-17 + 9\,n\right)\,P_{n}(x) = \left(80 - 416\,x + 3536\,x^{2} - 90\,n + 612\,n x - 7074\,n x^{2} - 162\,n^{2} x + 4131\,n^{2} x^{2} - 729\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(72\,x - 1296\,x^{2} + 1224\,x^{3} - 81\,n x + 2106\,n x^{2} - 2025\,n x^{3} - 729\,n^{2} x^{2} + 729\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-178 + 578\,x + 99\,n - 612\,n x + 162\,n^{2} x\right)\,P_{n-1}(x) + \left(-153\,x + 153\,x^{2} + 81\,n x - 81\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	20	2													22
3	219	438	3												660
4	2218	16644	6654	4											25520
5	22217	440444	660666	88868	5										1212200
6	222216	10012494	36997296	20024988	1111080	6									68368080
7	2222215	210681258	1591617705	2122156940	526703145	13333290	7								4466714560
8	22222214	4247387552	59102302542	157039613560	98503837570	12742162656	155555498	8							331813081600
9	222222213	83544806880	2005273944216	9435406924296	11794258655370	4010547888432	292406824080	1777777704	9						27623439043200
10	2222222212	1619795773818	64167971898528	496163478774792	1085071796294040	744245218162188	149725267763232	6479183095272	19999999908	10					2547494933984000
11	22222222211	31140564145310	1974070850391855	23833715650008360	84024541795292790	100829450154351348	41709002387514630	5264188934378280	140132538653895	222222222110	11				257806487319180800
12	222222222210	595715163203292	59104273200845070	1073990375155116540	5771826174584137140	11091239516978648280	8080556644417791996	2147980750310233080	177312819602535210	2978575816016460	2444444444310	12			28405587511895193600
13	2222222222209	11363032545304548	1736234769400911318	46199711084031706300	363505625763774692715	1053294423641887387752	1228843494248868619044	581609001222039508344	103949349939071339175	5787449231336371060	62496678999175014	26666666666508	13		3384999178500843904000
14	22222222222208	216382062805227974	50319028425021199104	1920052462129817039684	21456729171246885760640	89774286117210469408194	157291967263855183237632	119699048156280625877592	38622112508244394369152	4800131155324542599210	184503104225077730048	1298292376831367844	288888888888704	14	433800663952492764928000

Technical details

Node id: n-7a5ea6043e9c6d8c · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-63acddefa3bc865a

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:         1;
  2:        20,           2;
  3:       219,         438,             3;
  4:      2218,       16644,          6654,             4;
  5:     22217,      440444,        660666,         88868,              5;
  6:    222216,    10012494,      36997296,      20024988,        1111080,             6;
  7:   2222215,   210681258,    1591617705,    2122156940,      526703145,      13333290,            7;
  8:  22222214,  4247387552,   59102302542,  157039613560,    98503837570,   12742162656,    155555498,          8;
  9: 222222213, 83544806880, 2005273944216, 9435406924296, 11794258655370, 4010547888432, 292406824080, 1777777704, 9;

%F Formula

The row polynomials satisfy (9*n-17)*P(n,x) = -(90*n-80+(162*n^2-612*n+416)*x+(729*n^3-4131*n^2+7074*n-3536)*x^2)*P(n-2,x) - ((81*n-72)*x+(729*n^2-2106*n+1296)*x^2-(729*n^2-2025*n+1224)*x^3)*d/dx P(n-2,x) + (99*n-178+(162*n^2-612*n+578)*x)*P(n-1,x) + ((81*n-153)*x-(81*n-153)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(90*n - 80 + (162*n^2 - 612*n + 416)*x + (729*n^3 - 4131*n^2 + 7074*n - 3536)*x^2)*P[n - 2, x] - ((81*n - 72)*x + (729*n^2 - 2106*n + 1296)*x^2 - (729*n^2 - 2025*n + 1224)*x^3)*D[P[n - 2, x], x] + (99*n - 178 + (162*n^2 - 612*n + 578)*x)*P[n - 1, x] + ((81*n - 153)*x - (81*n - 153)*x^2)*D[P[n - 1, x], x])/(9*n - 17)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(90*n - 80 + (162*n^2 - 612*n + 416)*x + (729*n^3 - 4131*n^2 + 7074*n - 3536)*x^2)*v[i - 2] - ((81*n - 72)*x + (729*n^2 - 2106*n + 1296)*x^2 - (729*n^2 - 2025*n + 1224)*x^3)*deriv(v[i - 2], x) + (99*n - 178 + (162*n^2 - 612*n + 578)*x)*v[i - 1] + ((81*n - 153)*x - (81*n - 153)*x^2)*deriv(v[i - 1], x))/(9*n - 17)); 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) = 9*x + 1.?
  - 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) = 9*x + 1.?
    
    ↑ 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.