A272867 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 3\,x^{2} + 6\,n + 6\,n x\right)\,P_{n}(x) = \left(12 + 72\,x + 144\,x^{2} + 108\,x^{3} + 27\,x^{4} + 36\,n + 144\,n x + 162\,n x^{2} + 54\,n x^{3}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	24	36	9													69
2	252	1080	1404	648	81											3465
3	2160	15552	41796	52488	32076	8748	729									153549
4	16848	171072	707616	1539648	1907064	1364688	542376	104976	6561							6360849
5	124416	1632960	9156240	28693440	55287360	68024448	53800200	26768880	7873200	1180980	59049					252601173
6	886464	14276736	101196864	415704960	1097524080	1953813312	2395342368	2029396032	1172713140	446410440	104162436	12754584	531441			9744712857
7	6158592	117573120	1008189504	5132066688	17274430656	40553908416	68284263600	83545359552	74421580464	47855199168	21740188428	6696156600	1294590276	133923132	4782969	368068371165

All 15 rows

Technical details

Node id: n-eb325714e1fa06da · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8b65e409dcd1ef0f

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:        24,         36,           9;
  2:       252,       1080,        1404,          648,            81;
  3:      2160,      15552,       41796,        52488,         32076,          8748,            729;
  4:     16848,     171072,      707616,      1539648,       1907064,       1364688,         542376,         104976,           6561;
  5:    124416,    1632960,     9156240,     28693440,      55287360,      68024448,       53800200,       26768880,        7873200,        1180980,          59049;
  6:    886464,   14276736,   101196864,    415704960,    1097524080,    1953813312,     2395342368,     2029396032,     1172713140,      446410440,      104162436,       12754584,         531441;
  7:   6158592,  117573120,  1008189504,   5132066688,   17274430656,   40553908416,    68284263600,    83545359552,    74421580464,    47855199168,    21740188428,     6696156600,     1294590276,      133923132,       4782969;
  8:  41990400,  927148032,  9331946496,  56717273088,  232477330176,  679972382208,  1465019861760,  2366978692608,  2892518571744,  2676830053248,  1864924254144,   964246550400,   360521071344,    93210499872,   15382028304,    1377495072,     43046721;
  9: 282175488, 7074542592, 81765386496, 577935710208, 2794988519424, 9805316032512, 25818951336192, 52095026962944, 81529341661440, 99559833823872, 94941092847456, 70401018139776, 40189796220672, 17356437907200, 5510669035536, 1227348109152, 176663742984, 13947137604, 387420489;

%F Formula

The row polynomials satisfy (6*n-4+6*n*x+3*x^2)*P(n,x) = (36*n+12+(144*n+72)*x+(162*n+144)*x^2+(54*n+108)*x^3+27*x^4)*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[((36*n + 12 + (144*n + 72)*x + (162*n + 144)*x^2 + (54*n + 108)*x^3 + 27*x^4)*P[n - 1, x])/(6*n - 4 + 6*n*x + 3*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] = ((36*n + 12 + (144*n + 72)*x + (162*n + 144)*x^2 + (54*n + 108)*x^3 + 27*x^4)*v[i - 1])/(6*n - 4 + 6*n*x + 3*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 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 basis_map_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.