A386789 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(2\,n - n^{2}\right)\,P_{n-2}(x) + \left(3 + 6\,x - 9\,n - 18\,n x + 6\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	6														8
2	9	45	54													108
3	44	312	720	540												1616
4	225	2070	7020	10395	5670											25380
5	1182	13374	59724	131544	142884	61236										409944
6	6321	84903	469980	1372140	2228310	1908522	673596									6743772
7	34232	532320	3514752	12772080	27585360	35411904	25019280	7505784								112375712
8	187137	3306636	25356240	110207790	296943570	507891384	538540002	323686935	84440070							1890559764
9	1030490	20392710	178085160	900705960	2907564660	6212426220	8785937160	7931111760	4146945660	956987460						32041187240
10	5707449	125046801	1225023426	7066232316	26577216666	68105607570	120422798496	145079298936	113977206486	52730009046	10909657044					546223804236
11	31760676	763199928	8288120880	53691690780	230538639240	688898141568	1461905643744	2203159130640	2310863719320	1606694946480	666480866688	124965162504				9356281022448
12	177435297	4639968450	55319836572	397616764545	1918890513540	6550444476576	16218810808200	29348189632476	38520180746190	35765492342580	22299338997936	8383079651310	1437099368796			160899280542468
13	994551222	28116779166	365093556660	2883031295760	15447080852280	59300580250296	167812033834368	354437040822480	558746477552340	649330199193060	540717849684936	305549435027088	105018800027400	16581915793800		2776218649220856
14	5590402785	169900981515	2386613843640	20538018507240	120961903516740	515794442892300	1642133700104040	3965405776829640	7298533288293390	10189319663527290	10621533001467960	8016961478783400	4141925680779900	1311155770266900	191876454185400	48038701284382140

Technical details

Node id: n-5bf7e7b307eaaec8 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-bbe4c1ef283db3b0

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,        6;
  2:       9,       45,        54;
  3:      44,      312,       720,       540;
  4:     225,     2070,      7020,     10395,       5670;
  5:    1182,    13374,     59724,    131544,     142884,      61236;
  6:    6321,    84903,    469980,   1372140,    2228310,    1908522,     673596;
  7:   34232,   532320,   3514752,  12772080,   27585360,   35411904,   25019280,    7505784;
  8:  187137,  3306636,  25356240, 110207790,  296943570,  507891384,  538540002,  323686935,   84440070;
  9: 1030490, 20392710, 178085160, 900705960, 2907564660, 6212426220, 8785937160, 7931111760, 4146945660, 956987460;

%F Formula

The row polynomials satisfy (n^2-n)*P(n,x) = -(n^2-2*n)*P(n-2,x) + (6*n^2-9*n+3+(12*n^2-18*n+6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 2*n)*P[n - 2, x] + (6*n^2 - 9*n + 3 + (12*n^2 - 18*n + 6)*x)*P[n - 1, x])/(n^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; v[2] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 2*n)*v[i - 2] + (6*n^2 - 9*n + 3 + (12*n^2 - 18*n + 6)*x)*v[i - 1])/(n^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 (−∞, 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)
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 Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).?
  - 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

Related ?

Same fingerprint ?

A357367 → basis A013610

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.