A143491 → θ+1

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 6 + 10\,x + 3\,x^{2}\)

\(P_{1}(x) = 24 + 52\,x + 27\,x^{2} + 4\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	6	10	3															19
1	24	52	27	4														107
2	120	308	213	56	5													702
3	720	2088	1740	620	100	6												5274
4	5040	16056	15312	6660	1475	162	7											44712
5	40320	138528	146580	73696	20125	3066	245	8										422568
6	362880	1327392	1527012	858704	273245	51744	5782	352	9									4407120
7	3628800	13999680	17261208	10623056	3805830	845334	118188	10128	486	10								50292720
8	39916800	161254080	210872808	139868560	55142950	13865670	2286291	246480	16740	650	11							623471040
9	479001600	2014882560	2772354816	1959586464	836551100	232559580	43612107	5570664	478170	26400	847	12						8344624320
10	6227020800	27151476480	39062936448	29171097120	13324647380	4027135860	838276901	122261040	12483207	874500	40051	1080	13					119938250880
11	87178291200	392574712320	587608324992	460479274944	223008934720	72369478896	16434201784	2669685304	312308568	26113230	1522664	58812	1352	14				1842662908800
12	1307674368000	6062977267200	9402986943360	7690666890816	3920733114480	1353152905104	330944085472	58827224456	7688024487	738707970	51564513	2543268	83993	1666	15			30136443724800
13	20922789888000	99622985011200	159542256994560	135587986177536	72345063445200	26355326219040	6873783756840	1319457403264	189189019305	20361576950	1637610975	96944484	4099095	117110	2025	16		522780938265600
14	355687428096000	1735436324966400	2861652846424320	2517718774344192	1399351061290320	534854621857920	147602204455160	30286528720448	4700602906857	556356829600	50237121220	3434540928	174707806	6405280	159900	2432	17	9587900602828800

Technical details

Node id: n-c71dcd281e71f7f9 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-13dd874d79e2c744

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:         6,         10,          3;
  1:        24,         52,         27,          4;
  2:       120,        308,        213,         56,         5;
  3:       720,       2088,       1740,        620,       100,         6;
  4:      5040,      16056,      15312,       6660,      1475,       162,        7;
  5:     40320,     138528,     146580,      73696,     20125,      3066,      245,       8;
  6:    362880,    1327392,    1527012,     858704,    273245,     51744,     5782,     352,      9;
  7:   3628800,   13999680,   17261208,   10623056,   3805830,    845334,   118188,   10128,    486,    10;
  8:  39916800,  161254080,  210872808,  139868560,  55142950,  13865670,  2286291,  246480,  16740,   650,  11;
  9: 479001600, 2014882560, 2772354816, 1959586464, 836551100, 232559580, 43612107, 5570664, 478170, 26400, 847, 12;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+4*n+4+(2*n+4)*x+x^2)*P(n-2,x) + (2*n+5+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 6 + 10*x + 3*x^2, P(1,x) = 24 + 52*x + 27*x^2 + 4*x^3.

%t Mathematica

P[0, x_] := 6 + 10*x + 3*x^2;
P[1, x_] := 24 + 52*x + 27*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 4*n + 4 + (2*n + 4)*x + x^2)*P[n - 2, x] + (2*n + 5 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 6 + 10*x + 3*x^2; v[2] = 24 + 52*x + 27*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 4*n + 4 + (2*n + 4)*x + x^2)*v[i - 2] + (2*n + 5 + 2*x)*v[i - 1]); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Unsigned 2-Stirling numbers of the first kind.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - 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

Related ?

Same fingerprint ?

A136124 → θ+1

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
[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.