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

nonnegative coefficients

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

Recurrence

\(\left(2\,n - 8\,n x - 2\,n x^{2} - 4\,n^{2} + 24\,n^{2} x - 4\,n^{2} x^{2} - 16\,n^{3} x + 16\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-10\,x^{2} + 2\,x^{4} + 2\,n + 28\,n x + 82\,n x^{2} + 44\,n x^{3} + 12\,n x^{4} - 4\,n^{2} - 24\,n^{2} x + 12\,n^{2} x^{2} + 48\,n^{2} x^{3} + 16\,n^{2} x^{4} - 16\,n^{3} x - 80\,n^{3} x^{2} - 32\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-x^{2} - 5\,x^{3} + 5\,x^{4} + x^{5} + 2\,n x + 20\,n x^{2} + 56\,n x^{3} + 44\,n x^{4} + 6\,n x^{5} + 8\,n^{2} x^{2} + 56\,n^{2} x^{3} + 56\,n^{2} x^{4} + 8\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)

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

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	1	6	3																											10
2	1	14	39	28	5																									87
3	1	22	123	252	205	66	7																							676
4	1	30	255	924	1605	1386	595	120	9																					4925
5	1	38	435	2300	6445	10098	9023	4600	1305	190	11																			34446
6	1	46	663	4636	18245	43098	62923	57464	32841	11590	2431	276	13																	234227
7	1	54	939	8188	41805	134178	280567	389112	360729	223630	91971	24564	4069	378	15															1560200
8	1	62	1263	13212	83205	340170	939715	1793144	2391561	2241430	1476695	680340	216333	46242	6315	496	17													10230201
9	1	70	1635	19964	149805	747090	2585135	6363640	11308185	14625630	13821115	9545460	4800965	1743210	449415	79856	9265	630	19											66251090
10	1	78	2055	28700	250245	1476282	6164795	18723960	42060105	70597350	89071983	84716820	60753485	32766930	13210275	3936752	850833	129150	13015	780	21									424753615
11	1	86	2523	39676	394445	2690562	13194727	47878136	130920345	272981550	437286003	540560628	516792549	382174170	218200575	95756272	32044337	8071686	1498891	198380	17661	946	23							2700704172
12	1	94	3039	53148	593605	4600362	25952563	109815928	355410441	891292470	1747011783	2691578772	3270747597	3140175234	2382288795	1426067952	671330833	247085838	70442671	15334540	2493141	292314	23299	1128	25					17052595573
13	1	102	3603	69372	860205	7469874	47699743	231002616	865985049	2548316670	5943075435	11056876020	16482113349	19739070666	19017823095	14742501360	9184752945	4586970006	1828190659	577506540	143099229	27389538	3956943	416232	30025	1326	27			107035180630
14	1	110	4215	88604	1208005	11623194	82932395	452884600	1933457481	6548622470	17779704415	38978725140	69354227085	100498457010	118857723795	114855379440	90693989265	58468087710	30710398535	13097244940	4511400789	1245432650	272492155	46492776	6040025	575926	37935	1540	29	668407232235

Technical details

Node id: n-589abe4ee4d8f2c5 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-7a8d67db6efd091a

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: 1,  6,    3;
  2: 1, 14,   39,    28,      5;
  3: 1, 22,  123,   252,    205,     66,       7;
  4: 1, 30,  255,   924,   1605,   1386,     595,     120,        9;
  5: 1, 38,  435,  2300,   6445,  10098,    9023,    4600,     1305,      190,       11;
  6: 1, 46,  663,  4636,  18245,  43098,   62923,   57464,    32841,    11590,     2431,     276,      13;
  7: 1, 54,  939,  8188,  41805, 134178,  280567,  389112,   360729,   223630,    91971,   24564,    4069,     378,     15;
  8: 1, 62, 1263, 13212,  83205, 340170,  939715, 1793144,  2391561,  2241430,  1476695,  680340,  216333,   46242,   6315,   496,   17;
  9: 1, 70, 1635, 19964, 149805, 747090, 2585135, 6363640, 11308185, 14625630, 13821115, 9545460, 4800965, 1743210, 449415, 79856, 9265, 630, 19;

%F Formula

The row polynomials satisfy -(4*n^2-2*n+(16*n^3-24*n^2+8*n)*x-(16*n^3-4*n^2-2*n)*x^2)*P(n,x) = -(4*n^2-2*n+(16*n^3+24*n^2-28*n)*x+(80*n^3-12*n^2-82*n+10)*x^2+(32*n^3-48*n^2-44*n)*x^3-(16*n^2+12*n+2)*x^4)*P(n-1,x) + (2*n*x+(8*n^2+20*n-1)*x^2+(56*n^2+56*n-5)*x^3+(56*n^2+44*n+5)*x^4+(8*n^2+6*n+1)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 2*n + (16*n^3 + 24*n^2 - 28*n)*x + (80*n^3 - 12*n^2 - 82*n + 10)*x^2 + (32*n^3 - 48*n^2 - 44*n)*x^3 - (16*n^2 + 12*n + 2)*x^4)*P[n - 1, x] + (2*n*x + (8*n^2 + 20*n - 1)*x^2 + (56*n^2 + 56*n - 5)*x^3 + (56*n^2 + 44*n + 5)*x^4 + (8*n^2 + 6*n + 1)*x^5)*D[P[n - 1, x], x])/(-4*n^2 + 2*n - (16*n^3 - 24*n^2 + 8*n)*x + (16*n^3 - 4*n^2 - 2*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(4*n^2 - 2*n + (16*n^3 + 24*n^2 - 28*n)*x + (80*n^3 - 12*n^2 - 82*n + 10)*x^2 + (32*n^3 - 48*n^2 - 44*n)*x^3 - (16*n^2 + 12*n + 2)*x^4)*v[i - 1] + (2*n*x + (8*n^2 + 20*n - 1)*x^2 + (56*n^2 + 56*n - 5)*x^3 + (56*n^2 + 44*n + 5)*x^4 + (8*n^2 + 6*n + 1)*x^5)*deriv(v[i - 1], x))/(-4*n^2 + 2*n - (16*n^3 - 24*n^2 + 8*n)*x + (16*n^3 - 4*n^2 - 2*n)*x^2)); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.