A281260 → basis A007318

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	5	8	3													16
3	14	30	20	4												68
4	42	112	105	40	5											304
5	132	420	504	280	70	6										1412
6	429	1584	2310	1680	630	112	7									6752
7	1430	6006	10296	9240	4620	1260	168	8								33028
8	4862	22880	45045	48048	30030	11088	2310	240	9							164512
9	16796	87516	194480	240240	180180	84084	24024	3960	330	10						831620
10	58786	335920	831402	1166880	1021020	576576	210210	48048	6435	440	11					4255728
11	208012	1293292	3527160	5542680	5542680	3675672	1633632	480480	90090	10010	572	12				22004292
12	742900	4992288	14872858	25865840	29099070	22170720	11639628	4200768	1021020	160160	15015	728	13			114781008
13	2674440	19315400	62403600	118982864	148728580	128035908	77597520	33256080	9976824	2042040	272272	21840	910	14		603308292
14	9694845	74884320	260757900	540831200	743642900	713897184	490804314	243877920	87297210	22170720	3879876	445536	30940	1120	15	3192216000

Other recurrences

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

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

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

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

Technical details

Node id: n-03d40c9aa990124e · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-0d103fa5e9d4e10d)
Fingerprint: f6-2d598a2367210863

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:     2,     2;
  2:     5,     8,      3;
  3:    14,    30,     20,      4;
  4:    42,   112,    105,     40,      5;
  5:   132,   420,    504,    280,     70,     6;
  6:   429,  1584,   2310,   1680,    630,   112,     7;
  7:  1430,  6006,  10296,   9240,   4620,  1260,   168,    8;
  8:  4862, 22880,  45045,  48048,  30030, 11088,  2310,  240,   9;
  9: 16796, 87516, 194480, 240240, 180180, 84084, 24024, 3960, 330, 10;

%F Formula

The row polynomials satisfy (n^2+2*n)*P(n,x) = (4*n^2+2*n+(3*n^2+2*n+1)*x)*P(n-1,x) - ((2*n+1)*x+(2*n+1)*x^2)*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 + (3*n^2 + 2*n + 1)*x)*P[n - 1, x] - ((2*n + 1)*x + (2*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*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] = ((4*n^2 + 2*n + (3*n^2 + 2*n + 1)*x)*v[i - 1] - ((2*n + 1)*x + (2*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <…?
  - 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
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
Ore-algebra identity of two definitions polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <…?
  - ↑ Row-hypergeometric rows (multiplier sequences) (above)
  - ↑ Liu–Wang theorem for three-term recurrences (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.