A144440 → D+1

nonnegative coefficients

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

Recurrence

\(\left(8\right)\,P_{n}(x) = \left(96 - 60\,x - 126\,x^{2} - 252\,x^{3} - 162\,n - 81\,n x + 324\,n x^{3} + 54\,n^{2} + 27\,n^{2} x - 81\,n^{2} x^{3}\right)\,P_{n-4}(x) + \left(48\,x - 54\,x^{2} - 252\,x^{3} - 81\,n x + 324\,n x^{3} + 27\,n^{2} x - 81\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(-336 + 586\,x + 480\,x^{2} + 528\,x^{3} + 663\,n - 114\,n x - 90\,n x^{2} - 1116\,n x^{3} - 351\,n^{2} - 99\,n^{2} x - 27\,n^{2} x^{2} + 540\,n^{2} x^{3} + 54\,n^{3} + 27\,n^{3} x - 81\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(112 - 398\,x + 174\,x^{2} + 978\,x^{3} + 252\,x^{4} - 189\,n + 507\,n x + 135\,n x^{2} - 1440\,n x^{3} - 324\,n x^{4} + 63\,n^{2} - 234\,n^{2} x - 54\,n^{2} x^{2} + 621\,n^{2} x^{3} + 81\,n^{2} x^{4} + 27\,n^{3} x - 81\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(48\,x - 102\,x^{2} - 198\,x^{3} + 252\,x^{4} - 81\,n x + 81\,n x^{2} + 324\,n x^{3} - 324\,n x^{4} + 27\,n^{2} x - 27\,n^{2} x^{2} - 81\,n^{2} x^{3} + 81\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(48 - 454\,x - 344\,x^{2} + 240\,x^{3} - 45\,n + 582\,n x + 240\,n x^{2} - 144\,n x^{3} - 9\,n^{2} - 108\,n^{2} x - 45\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-16 + 234\,x - 240\,x^{2} - 36\,x^{3} + 144\,x^{4} + 27\,n - 54\,n x - 54\,n x^{2} - 9\,n^{2} + 27\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(72\,x^{2} - 144\,x^{3} + 72\,x^{4}\right)\,P_{n-2}''(x) + \left(16 - 164\,x + 48\,x^{2} + 18\,n + 48\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 12 + 13\,x + x^{2}\)

\(P_{3}(x) = 55 + 162\,x + 57\,x^{2} + x^{3}\)

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	12	13	1													26
3	55	162	57	1												275
4	230	1807	1476	233	1											3747
5	933	16882	31900	11703	937	1										62356
6	3748	141003	554229	436379	87363	3753	1									1226476
7	15011	1101298	8147421	12673795	5250479	633204	15017	1								27836226
8	60066	8256943	106829773	299801118	237608225	58834412	4518894	60073	1							715969505
9	240289	60351354	1295868192	6096201873	8511396480	3950191398	632326284	31977837	240297	1						20578794005
10	961184	434113435	14895253845	111136508097	254527316706	204958705851	60811151892	6625501581	225226773	961193	1					653615700558
11	3844767	3091178506	164844153307	1871238294351	6646228164813	8668837751133	4417813791045	889669163550	68326497309	1582114530	3844777	1				22731638798089
12	15379102	21870857871	1775549805969	29708733999640	156713108629743	313063704088245	256402352538618	88127648681967	12573997860780	697484749378	11096909496	15379113	1			859095578879923
13	61516445	154118715298	18753422029902	451324736615165	3419031925158295	10009238375668698	12463582605658608	6850359197105262	1663024783736007	173529462802835	7072779070522	77766796755	61516457	1		35056149296390250
14	246065820	1083290949259	195260723542489	6630213021557979	70253116869034444	291099082104181174	527068504208526138	438526924967381379	169730676394964997	30135624595724976	2355297059333917	71404669667563	544721297367	246065833	1	1536067733118293336

Technical details

Node id: n-b493569528b7b7bc · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-16611f922ff0a6c9

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:     12,       13,          1;
  3:     55,      162,         57,          1;
  4:    230,     1807,       1476,        233,          1;
  5:    933,    16882,      31900,      11703,        937,          1;
  6:   3748,   141003,     554229,     436379,      87363,       3753,         1;
  7:  15011,  1101298,    8147421,   12673795,    5250479,     633204,     15017,        1;
  8:  60066,  8256943,  106829773,  299801118,  237608225,   58834412,   4518894,    60073,      1;
  9: 240289, 60351354, 1295868192, 6096201873, 8511396480, 3950191398, 632326284, 31977837, 240297, 1;

%F Formula

The row polynomials satisfy 8*P(n,x) = (54*n^2-162*n+96+(27*n^2-81*n-60)*x-126*x^2-(81*n^2-324*n+252)*x^3)*P(n-4,x) + ((27*n^2-81*n+48)*x-54*x^2-(81*n^2-324*n+252)*x^3)*d/dx P(n-4,x) + (54*n^3-351*n^2+663*n-336+(27*n^3-99*n^2-114*n+586)*x-(27*n^2+90*n-480)*x^2-(81*n^3-540*n^2+1116*n-528)*x^3)*P(n-3,x) + (63*n^2-189*n+112+(27*n^3-234*n^2+507*n-398)*x-(54*n^2-135*n-174)*x^2-(81*n^3-621*n^2+1440*n-978)*x^3+(81*n^2-324*n+252)*x^4)*d/dx P(n-3,x) + ((27*n^2-81*n+48)*x-(27*n^2-81*n+102)*x^2-(81*n^2-324*n+198)*x^3+(81*n^2-324*n+252)*x^4)*d^2/dx^2 P(n-3,x) - (9*n^2+45*n-48+(108*n^2-582*n+454)*x+(45*n^2-240*n+344)*x^2+(144*n-240)*x^3)*P(n-2,x) - (9*n^2-27*n+16+(54*n-234)*x-(27*n^2-54*n-240)*x^2+36*x^3-144*x^4)*d/dx P(n-2,x) + (72*x^2-144*x^3+72*x^4)*d^2/dx^2 P(n-2,x) + (18*n+16+(48*n-164)*x+48*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 12 + 13*x + x^2, P(3,x) = 55 + 162*x + 57*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 12 + 13*x + x^2;
P[3, x_] := 55 + 162*x + 57*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((54*n^2 - 162*n + 96 + (27*n^2 - 81*n - 60)*x - 126*x^2 - (81*n^2 - 324*n + 252)*x^3)*P[n - 4, x] + ((27*n^2 - 81*n + 48)*x - 54*x^2 - (81*n^2 - 324*n + 252)*x^3)*D[P[n - 4, x], x] + (54*n^3 - 351*n^2 + 663*n - 336 + (27*n^3 - 99*n^2 - 114*n + 586)*x - (27*n^2 + 90*n - 480)*x^2 - (81*n^3 - 540*n^2 + 1116*n - 528)*x^3)*P[n - 3, x] + (63*n^2 - 189*n + 112 + (27*n^3 - 234*n^2 + 507*n - 398)*x - (54*n^2 - 135*n - 174)*x^2 - (81*n^3 - 621*n^2 + 1440*n - 978)*x^3 + (81*n^2 - 324*n + 252)*x^4)*D[P[n - 3, x], x] + ((27*n^2 - 81*n + 48)*x - (27*n^2 - 81*n + 102)*x^2 - (81*n^2 - 324*n + 198)*x^3 + (81*n^2 - 324*n + 252)*x^4)*D[P[n - 3, x], {x, 2}] - (9*n^2 + 45*n - 48 + (108*n^2 - 582*n + 454)*x + (45*n^2 - 240*n + 344)*x^2 + (144*n - 240)*x^3)*P[n - 2, x] - (9*n^2 - 27*n + 16 + (54*n - 234)*x - (27*n^2 - 54*n - 240)*x^2 + 36*x^3 - 144*x^4)*D[P[n - 2, x], x] + (72*x^2 - 144*x^3 + 72*x^4)*D[P[n - 2, x], {x, 2}] + (18*n + 16 + (48*n - 164)*x + 48*x^2)*P[n - 1, x])/(8)]];
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] = 12 + 13*x + x^2; v[4] = 55 + 162*x + 57*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((54*n^2 - 162*n + 96 + (27*n^2 - 81*n - 60)*x - 126*x^2 - (81*n^2 - 324*n + 252)*x^3)*v[i - 4] + ((27*n^2 - 81*n + 48)*x - 54*x^2 - (81*n^2 - 324*n + 252)*x^3)*deriv(v[i - 4], x) + (54*n^3 - 351*n^2 + 663*n - 336 + (27*n^3 - 99*n^2 - 114*n + 586)*x - (27*n^2 + 90*n - 480)*x^2 - (81*n^3 - 540*n^2 + 1116*n - 528)*x^3)*v[i - 3] + (63*n^2 - 189*n + 112 + (27*n^3 - 234*n^2 + 507*n - 398)*x - (54*n^2 - 135*n - 174)*x^2 - (81*n^3 - 621*n^2 + 1440*n - 978)*x^3 + (81*n^2 - 324*n + 252)*x^4)*deriv(v[i - 3], x) + ((27*n^2 - 81*n + 48)*x - (27*n^2 - 81*n + 102)*x^2 - (81*n^2 - 324*n + 198)*x^3 + (81*n^2 - 324*n + 252)*x^4)*deriv(deriv(v[i - 3], x), x) - (9*n^2 + 45*n - 48 + (108*n^2 - 582*n + 454)*x + (45*n^2 - 240*n + 344)*x^2 + (144*n - 240)*x^3)*v[i - 2] - (9*n^2 - 27*n + 16 + (54*n - 234)*x - (27*n^2 - 54*n - 240)*x^2 + 36*x^3 - 144*x^4)*deriv(v[i - 2], x) + (72*x^2 - 144*x^3 + 72*x^4)*deriv(deriv(v[i - 2], x), x) + (18*n + 16 + (48*n - 164)*x + 48*x^2)*v[i - 1])/(8)); 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 T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = …?
  - 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 T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = …?
    
    ↑ 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.