basis-map coefficients of A297191, shifted → basis A008288 → D+1

nonnegative coefficients

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

Recurrence

\(\left(12 - 7\,x - 10\,x^{2} + 4\,x^{3} + x^{4} - 20\,n - 10\,n x + 12\,n x^{2} + 2\,n x^{3} + 8\,n^{2} + 8\,n^{2} x\right)\,P_{n}(x) = \left(9\,x^{3} - 2\,x^{4} - 6\,x^{5} - x^{6} + 4\,n x^{2} - 6\,n x^{3} - 12\,n x^{4} - 2\,n x^{5} - 8\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-2}(x) + \left(-27\,x - 78\,x^{2} - 25\,x^{3} + 23\,x^{4} + 10\,x^{5} + x^{6} - 12\,n - 50\,n x - 8\,n x^{2} + 48\,n x^{3} + 20\,n x^{4} + 2\,n x^{5} + 8\,n^{2} + 40\,n^{2} x + 40\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 4 + 5\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	4	5	1																											10
2	8	33	34	11	1																									87
3	12	93	230	227	96	17	1																							676
4	16	185	778	1515	1476	741	190	23	1																					4925
5	20	309	1870	5731	9704	9417	5314	1735	316	29	1																			34446
6	24	465	3698	15755	39564	61117	59270	36375	14080	3369	474	35	1																	234227
7	28	653	6454	35491	120176	262849	380554	369287	241348	105973	30878	5803	664	41	1															1560200
8	32	873	10330	69867	300116	862165	1703246	2349975	2283016	1566593	757890	256387	59580	9197	886	47	1													10230201
9	36	1125	15518	124835	652536	2340345	5937490	10847175	14419532	14027213	10006470	5227115	1988000	543969	104826	13711	1140	53	1											66251090
10	40	1409	22210	207371	1280284	5530157	17264150	39727255	68210960	88034265	85754538	63139875	35099780	14670085	4571390	1046831	172024	19505	1426	59	1									424753615
11	44	1725	30598	325475	2321024	11758193	43778330	122358407	260230100	424830885	535267182	522085995	394629288	230952025	104318210	36144143	9508220	1868429	267350	26739	1744	65	1							2700704172
12	48	2073	40874	488171	3952356	23011781	99796446	329647383	841653272	1677321777	2626099794	3243449859	3167472972	2447767773	1495757958	720970031	272848896	80462153	18275322	3141147	397748	35573	2094	71	1					17052595573
13	52	2453	53230	705507	6396936	42130473	208941538	798639815	2389705564	5657627517	10675749270	16135664811	19596991920	19159901841	15088949898	9565879695	4872417924	1986801765	644851774	165160307	32958808	5029881	570930	46167	2476	77	1			107035180630
14	56	2865	67858	988555	9927596	73022109	408121510	1775239447	6108588832	16818229897	37346834810	67267513635	98654936340	118112336757	115600766478	92537509935	60554801160	32342288865	14058719458	4951434427	1403650684	317255245	56403062	7735623	795376	58681	2890	83	1	668407232235

Technical details

Node id: n-a4933e776449d358 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-eedfb0e3e62b3e80

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:  4,    5,     1;
  2:  8,   33,    34,     11,      1;
  3: 12,   93,   230,    227,     96,      17,       1;
  4: 16,  185,   778,   1515,   1476,     741,     190,       23,        1;
  5: 20,  309,  1870,   5731,   9704,    9417,    5314,     1735,      316,       29,        1;
  6: 24,  465,  3698,  15755,  39564,   61117,   59270,    36375,    14080,     3369,      474,      35,       1;
  7: 28,  653,  6454,  35491, 120176,  262849,  380554,   369287,   241348,   105973,    30878,    5803,     664,     41,      1;
  8: 32,  873, 10330,  69867, 300116,  862165, 1703246,  2349975,  2283016,  1566593,   757890,  256387,   59580,   9197,    886,    47,    1;
  9: 36, 1125, 15518, 124835, 652536, 2340345, 5937490, 10847175, 14419532, 14027213, 10006470, 5227115, 1988000, 543969, 104826, 13711, 1140, 53, 1;

%F Formula

The row polynomials satisfy (8*n^2-20*n+12+(8*n^2-10*n-7)*x+(12*n-10)*x^2+(2*n+4)*x^3+x^4)*P(n,x) = -((8*n^2-4*n)*x^2+(8*n^2+6*n-9)*x^3+(12*n+2)*x^4+(2*n+6)*x^5+x^6)*P(n-2,x) + (8*n^2-12*n+(40*n^2-50*n-27)*x+(40*n^2-8*n-78)*x^2+(8*n^2+48*n-25)*x^3+(20*n+23)*x^4+(2*n+10)*x^5+x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 5*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-((8*n^2 - 4*n)*x^2 + (8*n^2 + 6*n - 9)*x^3 + (12*n + 2)*x^4 + (2*n + 6)*x^5 + x^6)*P[n - 2, x] + (8*n^2 - 12*n + (40*n^2 - 50*n - 27)*x + (40*n^2 - 8*n - 78)*x^2 + (8*n^2 + 48*n - 25)*x^3 + (20*n + 23)*x^4 + (2*n + 10)*x^5 + x^6)*P[n - 1, x])/(8*n^2 - 20*n + 12 + (8*n^2 - 10*n - 7)*x + (12*n - 10)*x^2 + (2*n + 4)*x^3 + x^4)]];
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] = 4 + 5*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (-((8*n^2 - 4*n)*x^2 + (8*n^2 + 6*n - 9)*x^3 + (12*n + 2)*x^4 + (2*n + 6)*x^5 + x^6)*v[i - 2] + (8*n^2 - 12*n + (40*n^2 - 50*n - 27)*x + (40*n^2 - 8*n - 78)*x^2 + (8*n^2 + 48*n - 25)*x^3 + (20*n + 23)*x^4 + (2*n + 10)*x^5 + x^6)*v[i - 1])/(8*n^2 - 20*n + 12 + (8*n^2 - 10*n - 7)*x + (12*n - 10)*x^2 + (2*n + 4)*x^3 + x^4)); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.?
  - Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
    no recorded property covers this requirement
  - Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
    no recorded property covers this requirement
  - Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
    no recorded property covers this requirement
  - Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
    no recorded property covers this requirement
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.?
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine-current basis map theorem (above)

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.