A144442 → D+1

nonnegative coefficients

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

Recurrence

\(\left(40\right)\,P_{n}(x) = \left(1780 - 610\,x - 1750\,x^{2} - 7200\,x^{3} - 2450\,n - 1225\,n x + 8000\,n x^{3} + 750\,n^{2} + 375\,n^{2} x - 1875\,n^{2} x^{3}\right)\,P_{n-4}(x) + \left(890\,x - 750\,x^{2} - 7200\,x^{3} - 1225\,n x + 8000\,n x^{3} + 375\,n^{2} x - 1875\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(-6586 + 8638\,x + 6760\,x^{2} + 18160\,x^{3} + 10845\,n - 1130\,n x - 1150\,n x^{2} - 29600\,n x^{3} - 5225\,n^{2} - 1525\,n^{2} x - 375\,n^{2} x^{2} + 13250\,n^{2} x^{3} + 750\,n^{3} + 375\,n^{3} x - 1875\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(1958 - 6812\,x + 2020\,x^{2} + 28110\,x^{3} + 7200\,x^{4} - 2695\,n + 7995\,n x + 2075\,n x^{2} - 37600\,n x^{3} - 8000\,n x^{4} + 825\,n^{2} - 3400\,n^{2} x - 750\,n^{2} x^{2} + 15125\,n^{2} x^{3} + 1875\,n^{2} x^{4} + 375\,n^{3} x - 1875\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(890\,x - 1640\,x^{2} - 6450\,x^{3} + 7200\,x^{4} - 1225\,n x + 1225\,n x^{2} + 8000\,n x^{3} - 8000\,n x^{4} + 375\,n^{2} x - 375\,n^{2} x^{2} - 1875\,n^{2} x^{3} + 1875\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(902 - 8990\,x - 4900\,x^{2} + 3600\,x^{3} - 655\,n + 8750\,n x + 3500\,n x^{2} - 2000\,n x^{3} - 75\,n^{2} - 1500\,n^{2} x - 625\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-178 + 3250\,x - 3160\,x^{2} - 500\,x^{3} + 2000\,x^{4} + 245\,n - 750\,n x - 850\,n x^{2} - 75\,n^{2} + 375\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(1000\,x^{2} - 2000\,x^{3} + 1000\,x^{4}\right)\,P_{n-2}''(x) + \left(60 - 1420\,x + 400\,x^{2} + 150\,n + 400\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 18 + 19\,x + x^{2}\)

\(P_{3}(x) = 119 + 354\,x + 121\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	18	19	1													38
3	119	354	121	1												595
4	730	6091	4868	733	1											12423
5	4401	87890	166980	59345	4405	1										323022
6	26432	1136911	4602975	3570205	687395	26437	1									10050356
7	158623	13792314	107186901	167283425	67540915	7768578	158629	1								363889386
8	951774	161021259	2225998211	6363029136	4988532775	1195529614	86697154	951781	1							15022711705
9	5710685	1836588906	42796338388	207624701263	290496602256	132177898930	20368373536	961124583	5710693	1						696273049241
10	34264156	20651718415	780471060695	6069056802675	14076568278840	11243200891757	3250129451190	339190594895	10617104525	34264165	1					35789954431314
11	205584987	230206990946	13719303063675	163840821981085	594475528191565	776508620383559	389715398335097	76104653260400	5569773780385	117056552840	205584997	1				2020281773709537
12	1233509978	2552561285659	234969229577151	4172161278249450	22649539788200255	45666822375365441	37166634770236198	12513721632528427	1724646552822850	90654494108180	1289232497588	1233509989	1			124222994381891167
13	7401059929	28212215559714	3950101043663226	101715432140233371	798202843662824875	2373780820326991038	2958880564298885568	1606823247359975592	380528120441041947	38220853603401675	1467126025219934	14191219969029	7401059941	1		8263611527139885840
14	44406359640	311212630268271	65519901670867283	2399466054723680493	26493857631177872386	112140234279631208520	204477359486904764552	169278171969466259781	64445713100934879271	11124082500968812762	834088651600358255	23655115196422781	156161394629623	44406359653	1	591282616155112743272

Technical details

Node id: n-eca254ae7d32dfb0 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e76b964912e6b0dd

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:       2,          1;
  2:      18,         19,           1;
  3:     119,        354,         121,            1;
  4:     730,       6091,        4868,          733,            1;
  5:    4401,      87890,      166980,        59345,         4405,            1;
  6:   26432,    1136911,     4602975,      3570205,       687395,        26437,           1;
  7:  158623,   13792314,   107186901,    167283425,     67540915,      7768578,      158629,         1;
  8:  951774,  161021259,  2225998211,   6363029136,   4988532775,   1195529614,    86697154,    951781,       1;
  9: 5710685, 1836588906, 42796338388, 207624701263, 290496602256, 132177898930, 20368373536, 961124583, 5710693, 1;

%F Formula

The row polynomials satisfy 40*P(n,x) = (750*n^2-2450*n+1780+(375*n^2-1225*n-610)*x-1750*x^2-(1875*n^2-8000*n+7200)*x^3)*P(n-4,x) + ((375*n^2-1225*n+890)*x-750*x^2-(1875*n^2-8000*n+7200)*x^3)*d/dx P(n-4,x) + (750*n^3-5225*n^2+10845*n-6586+(375*n^3-1525*n^2-1130*n+8638)*x-(375*n^2+1150*n-6760)*x^2-(1875*n^3-13250*n^2+29600*n-18160)*x^3)*P(n-3,x) + (825*n^2-2695*n+1958+(375*n^3-3400*n^2+7995*n-6812)*x-(750*n^2-2075*n-2020)*x^2-(1875*n^3-15125*n^2+37600*n-28110)*x^3+(1875*n^2-8000*n+7200)*x^4)*d/dx P(n-3,x) + ((375*n^2-1225*n+890)*x-(375*n^2-1225*n+1640)*x^2-(1875*n^2-8000*n+6450)*x^3+(1875*n^2-8000*n+7200)*x^4)*d^2/dx^2 P(n-3,x) - (75*n^2+655*n-902+(1500*n^2-8750*n+8990)*x+(625*n^2-3500*n+4900)*x^2+(2000*n-3600)*x^3)*P(n-2,x) - (75*n^2-245*n+178+(750*n-3250)*x-(375*n^2-850*n-3160)*x^2+500*x^3-2000*x^4)*d/dx P(n-2,x) + (1000*x^2-2000*x^3+1000*x^4)*d^2/dx^2 P(n-2,x) + (150*n+60+(400*n-1420)*x+400*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 18 + 19*x + x^2, P(3,x) = 119 + 354*x + 121*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 18 + 19*x + x^2;
P[3, x_] := 119 + 354*x + 121*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((750*n^2 - 2450*n + 1780 + (375*n^2 - 1225*n - 610)*x - 1750*x^2 - (1875*n^2 - 8000*n + 7200)*x^3)*P[n - 4, x] + ((375*n^2 - 1225*n + 890)*x - 750*x^2 - (1875*n^2 - 8000*n + 7200)*x^3)*D[P[n - 4, x], x] + (750*n^3 - 5225*n^2 + 10845*n - 6586 + (375*n^3 - 1525*n^2 - 1130*n + 8638)*x - (375*n^2 + 1150*n - 6760)*x^2 - (1875*n^3 - 13250*n^2 + 29600*n - 18160)*x^3)*P[n - 3, x] + (825*n^2 - 2695*n + 1958 + (375*n^3 - 3400*n^2 + 7995*n - 6812)*x - (750*n^2 - 2075*n - 2020)*x^2 - (1875*n^3 - 15125*n^2 + 37600*n - 28110)*x^3 + (1875*n^2 - 8000*n + 7200)*x^4)*D[P[n - 3, x], x] + ((375*n^2 - 1225*n + 890)*x - (375*n^2 - 1225*n + 1640)*x^2 - (1875*n^2 - 8000*n + 6450)*x^3 + (1875*n^2 - 8000*n + 7200)*x^4)*D[P[n - 3, x], {x, 2}] - (75*n^2 + 655*n - 902 + (1500*n^2 - 8750*n + 8990)*x + (625*n^2 - 3500*n + 4900)*x^2 + (2000*n - 3600)*x^3)*P[n - 2, x] - (75*n^2 - 245*n + 178 + (750*n - 3250)*x - (375*n^2 - 850*n - 3160)*x^2 + 500*x^3 - 2000*x^4)*D[P[n - 2, x], x] + (1000*x^2 - 2000*x^3 + 1000*x^4)*D[P[n - 2, x], {x, 2}] + (150*n + 60 + (400*n - 1420)*x + 400*x^2)*P[n - 1, x])/(40)]];
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] = 2 + x; v[3] = 18 + 19*x + x^2; v[4] = 119 + 354*x + 121*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((750*n^2 - 2450*n + 1780 + (375*n^2 - 1225*n - 610)*x - 1750*x^2 - (1875*n^2 - 8000*n + 7200)*x^3)*v[i - 4] + ((375*n^2 - 1225*n + 890)*x - 750*x^2 - (1875*n^2 - 8000*n + 7200)*x^3)*deriv(v[i - 4], x) + (750*n^3 - 5225*n^2 + 10845*n - 6586 + (375*n^3 - 1525*n^2 - 1130*n + 8638)*x - (375*n^2 + 1150*n - 6760)*x^2 - (1875*n^3 - 13250*n^2 + 29600*n - 18160)*x^3)*v[i - 3] + (825*n^2 - 2695*n + 1958 + (375*n^3 - 3400*n^2 + 7995*n - 6812)*x - (750*n^2 - 2075*n - 2020)*x^2 - (1875*n^3 - 15125*n^2 + 37600*n - 28110)*x^3 + (1875*n^2 - 8000*n + 7200)*x^4)*deriv(v[i - 3], x) + ((375*n^2 - 1225*n + 890)*x - (375*n^2 - 1225*n + 1640)*x^2 - (1875*n^2 - 8000*n + 6450)*x^3 + (1875*n^2 - 8000*n + 7200)*x^4)*deriv(deriv(v[i - 3], x), x) - (75*n^2 + 655*n - 902 + (1500*n^2 - 8750*n + 8990)*x + (625*n^2 - 3500*n + 4900)*x^2 + (2000*n - 3600)*x^3)*v[i - 2] - (75*n^2 - 245*n + 178 + (750*n - 3250)*x - (375*n^2 - 850*n - 3160)*x^2 + 500*x^3 - 2000*x^4)*deriv(v[i - 2], x) + (1000*x^2 - 2000*x^3 + 1000*x^4)*deriv(deriv(v[i - 2], x), x) + (150*n + 60 + (400*n - 1420)*x + 400*x^2)*v[i - 1])/(40)); 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) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (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.