A109983 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(4 + 6\,x + 2\,x^{2} - n - n x\right)\,P_{n-1}(x) + \left(4 + 10\,x + 8\,x^{2} + 2\,x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	5	2													10
2	13	44	55	30	6											148
3	63	321	675	749	462	150	20									2440
4	321	2184	6456	10824	11251	7420	3030	700	70							42256
5	1683	14325	54570	122480	179305	178835	123020	57610	17570	3150	252					752800
6	8989	91860	428331	1204790	2276205	3042360	2949115	2088450	1072050	388920	94626	13860	924			13660480
7	48639	580097	3200015	10820145	25049563	41995765	52570035	49906925	36100190	19796490	8099826	2397318	485100	60060	3432	251113600

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(n\right)\,P_{n}(x) = \left(1 + 2\,x + x^{2} - n - 2\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(-3 - 5\,x - 2\,x^{2} + 6\,n + 10\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 5\,x + 2\,x^{2}\)

Technical details

Node id: n-d9640fb05f96c50b · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-fbebeadd27b9d988)
Fingerprint: f6-a7f447fa676c998a

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:       3,        5,         2;
  2:      13,       44,        55,        30,          6;
  3:      63,      321,       675,       749,        462,        150,         20;
  4:     321,     2184,      6456,     10824,      11251,       7420,       3030,         700,          70;
  5:    1683,    14325,     54570,    122480,     179305,     178835,     123020,       57610,       17570,        3150,         252;
  6:    8989,    91860,    428331,   1204790,    2276205,    3042360,    2949115,     2088450,     1072050,      388920,       94626,      13860,        924;
  7:   48639,   580097,   3200015,  10820145,   25049563,   41995765,   52570035,    49906925,    36100190,    19796490,     8099826,    2397318,     485100,      60060,      3432;
  8:  265729,  3622928,  23077480,  91154672,  249867856,  503941760,  773456320,   921404240,   860905315,   632901720,   364827876,  163139592,   55470030,   13861848,   2400684,   257400,    12870;
  9: 1462563, 22437477, 162073818, 732361272, 2319658128, 5468415120, 9942660000, 14254786800, 16335019305, 15071939255, 11222589402, 6726888828, 3221438682, 1215692478, 353533752, 76439220, 11570130, 1093950, 48620;

%F Formula

The row polynomials satisfy n*P(n,x) = -(n-4+(n-6)*x-2*x^2)*P(n-1,x) + (4+10*x+8*x^2+2*x^3)*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[(-(n - 4 + (n - 6)*x - 2*x^2)*P[n - 1, x] + (4 + 10*x + 8*x^2 + 2*x^3)*D[P[n - 1, x], 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; for(n = 1, N, my(i = n + 1); v[i] = (-(n - 4 + (n - 6)*x - 2*x^2)*v[i - 1] + (4 + 10*x + 8*x^2 + 2*x^3)*deriv(v[i - 1], x))/(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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - 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
Ore-algebra identity of two definitions polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.?
  - ↑ Liu–Wang theorem for three-term recurrences (above)
  - ↑ Row-hypergeometric rows (multiplier sequences) (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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.