A200536 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 - x + 4\,x^{2} + 7\,n + 8\,n x\right)\,P_{n}(x) = \left(18 + 84\,x + 146\,x^{2} + 112\,x^{3} + 32\,x^{4} + 42\,n + 146\,n x + 168\,n x^{2} + 64\,n x^{3}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	20	30	8													58
2	204	752	964	480	64											2464
3	1728	10296	24720	30200	19296	5760	512									92512
4	13392	110592	391968	774784	926736	677632	288256	61440	4096							3248896
5	98496	1041120	4890240	13408320	23682880	28013024	22293120	11627520	3706880	614400	32768					109408768
6	699840	9020160	52809408	185448960	434260800	712608000	837344320	706263552	421002240	170639360	43499520	5898240	262144			3579756544
7	4852224	73809792	517757184	2218043520	6476004864	13613314176	21208748288	24815796096	21837831680	14309349376	6817800192	2259582976	479854592	55050240	2097152	114689892352

All 15 rows

Technical details

Node id: n-5fedd6b839deb652 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a1bc1a919c7f051b

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:        20,         30,           8;
  2:       204,        752,         964,          480,           64;
  3:      1728,      10296,       24720,        30200,        19296,          5760,           512;
  4:     13392,     110592,      391968,       774784,       926736,        677632,        288256,          61440,           4096;
  5:     98496,    1041120,     4890240,     13408320,     23682880,      28013024,      22293120,       11627520,        3706880,         614400,          32768;
  6:    699840,    9020160,    52809408,    185448960,    434260800,     712608000,     837344320,      706263552,      421002240,      170639360,       43499520,        5898240,        262144;
  7:   4852224,   73809792,   517757184,   2218043520,   6476004864,   13613314176,   21208748288,    24815796096,    21837831680,    14309349376,     6817800192,     2259582976,     479854592,      55050240,       2097152;
  8:  33032448,  579280896,  4736517120,  23955886080,  83826925056,  215009206272,  417706791936,   626123509760,   730506461440,   663951826944,   466921570304,   250110672896,   99241820160,   27818721280,    5062524928,    503316480,    16777216;
  9: 221709312, 4403953152, 41130436608, 239889475584, 978675664896, 2964140282880, 6902446731264, 12625313605632, 18368827610112, 21387462516224, 19939656812544, 14813410590720, 8674147958784, 3928985174016, 1335616339968, 324362305536, 51640270848, 4529848320, 134217728;

%F Formula

The row polynomials satisfy (7*n-4+(8*n-1)*x+4*x^2)*P(n,x) = (42*n+18+(146*n+84)*x+(168*n+146)*x^2+(64*n+112)*x^3+32*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[((42*n + 18 + (146*n + 84)*x + (168*n + 146)*x^2 + (64*n + 112)*x^3 + 32*x^4)*P[n - 1, x])/(7*n - 4 + (8*n - 1)*x + 4*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] = ((42*n + 18 + (146*n + 84)*x + (168*n + 146)*x^2 + (64*n + 112)*x^3 + 32*x^4)*v[i - 1])/(7*n - 4 + (8*n - 1)*x + 4*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_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.