A049374 → θ+1

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	2														8
2	42	36	3													81
3	336	552	108	4												1000
4	3024	8400	2880	240	5											14549
5	30240	132048	70200	9840	450	6										242784
6	332640	2173248	1672272	349440	26250	756	7									4554613
7	3991680	37610496	40049856	11664576	1276800	59472	1176	8								94654064
8	51891840	684972288	977562432	380007936	56715120	3792096	119952	1728	9							2155063401
9	726485760	13118257152	24488956800	12323646720	2403324000	215949888	9702000	221760	2430	10						53286546520
10	10897286400	263818114560	631901305728	402361344000	99520898400	11521984320	688531536	22176000	383130	3300	11					1420732027385
11	174356582400	5561974978560	16825161120768	13315736738304	4085704022400	591826435200	45114765696	1917869184	46403280	627000	4356	12				40601839547160
12	2964061900800	122712162693120	462622312728576	448537123786752	167815719751680	29759555105280	2810858162112	151288929792	4802589792	90433200	981552	5616	13			1237377977068285
13	53353114214400	2828307865743360	13137493000439808	15418554835267584	6937668755297280	1481039733229056	169640672737536	11230722607104	448750253952	11033502480	166222056	1480752	7098	14		40037748651002480
14	1013709170073600	67986095006269440	385229758811369472	541747521733017600	289842275000678400	73490950768435200	10040272652906496	800334075340800	39110827916160	1204755552000	23606174592	290908800	2164890	8820	15	1370191256700816285

Technical details

Node id: n-8752bd5043cce6c9 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-827cc28cb33438df

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:         6,           2;
  2:        42,          36,           3;
  3:       336,         552,         108,           4;
  4:      3024,        8400,        2880,         240,          5;
  5:     30240,      132048,       70200,        9840,        450,         6;
  6:    332640,     2173248,     1672272,      349440,      26250,       756,       7;
  7:   3991680,    37610496,    40049856,    11664576,    1276800,     59472,    1176,      8;
  8:  51891840,   684972288,   977562432,   380007936,   56715120,   3792096,  119952,   1728,    9;
  9: 726485760, 13118257152, 24488956800, 12323646720, 2403324000, 215949888, 9702000, 221760, 2430, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+3*n-4+(2*n-2)*x+x^2)*P(n-2,x) - ((5*n-5)*x+5*x^2)*d/dx P(n-2,x) + (2*n+4+2*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 3*n - 4 + (2*n - 2)*x + x^2)*P[n - 2, x] - ((5*n - 5)*x + 5*x^2)*D[P[n - 2, x], x] + (2*n + 4 + 2*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];
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] = 6 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 3*n - 4 + (2*n - 2)*x + x^2)*v[i - 2] - ((5*n - 5)*x + 5*x^2)*deriv(v[i - 2], x) + (2*n + 4 + 2*x)*v[i - 1] + 5*x*deriv(v[i - 1], x)); 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 A triangle of numbers related to triangle A030527.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.