A124848 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(2 + 3\,x - 2\,n - 3\,n x\right)\,P_{n-2}(x) + \left(2 + 9\,x + 3\,n + 3\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 12\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	12														17
2	19	84	90													193
3	63	396	810	540												1809
4	192	1548	4590	5940	2835											15105
5	552	5400	20790	39420	36855	13608										116625
6	1520	17424	82080	203580	280665	204120	61236									850625
7	4048	53088	294840	899640	1630125	1755432	1041012	262440								5940625
8	10496	154752	987840	3568320	7983360	11335464	9981468	4986360	1082565							40090625
9	26624	435456	3136320	13063680	34700400	60991056	70972524	52750440	22733865	4330260						263140625
10	66304	1190400	9538560	44945280	137985120	288544032	416404800	409668840	263063295	99595980	16888014					1687890625
11	162560	3176448	28005120	147121920	511932960	1239416640	2131257744	2603929680	2216010555	1251445140	422200350	64481508				10619140625
12	393216	8303616	79833600	462274560	1796256000	4936111488	9839890368	14341821120	15173231040	11366932500	5725036746	1741000716	241805655			65712890625
13	940032	21325824	221944320	1403550720	6019453440	18494251776	41892282432	70854600960	89506474200	83426789160	55781110242	25341232644	7012363995	892820880		400869140625
14	2224128	53932032	603832320	4138767360	19407548160	65881880064	167008697856	321247555200	471288177360	524879475120	436926698208	263664886212	109054350405	27677447280	3252418920	2415087890625

Technical details

Node id: n-35eb54760ab979ef · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-0aae4bfdf29522d2

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:     5,     12;
  2:    19,     84,      90;
  3:    63,    396,     810,      540;
  4:   192,   1548,    4590,     5940,     2835;
  5:   552,   5400,   20790,    39420,    36855,    13608;
  6:  1520,  17424,   82080,   203580,   280665,   204120,    61236;
  7:  4048,  53088,  294840,   899640,  1630125,  1755432,  1041012,   262440;
  8: 10496, 154752,  987840,  3568320,  7983360, 11335464,  9981468,  4986360,  1082565;
  9: 26624, 435456, 3136320, 13063680, 34700400, 60991056, 70972524, 52750440, 22733865, 4330260;

%F Formula

The row polynomials satisfy n*P(n,x) = -(2*n-2+(3*n-3)*x)*P(n-2,x) + (3*n+2+(3*n+9)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 12*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 2 + (3*n - 3)*x)*P[n - 2, x] + (3*n + 2 + (3*n + 9)*x)*P[n - 1, x])/(n)]];
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] = 5 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 2 + (3*n - 3)*x)*v[i - 2] + (3*n + 2 + (3*n + 9)*x)*v[i - 1])/(n)); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) = (k+1)*(k+2)*(k+3)*binomial(n,k)/6 (0 <= k <= n).?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

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.