A286724 → θ+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	2														4
3	8	16	3													27
4	48	144	54	4												250
5	384	1536	864	128	5											2917
6	3840	19200	14400	3200	250	6										40896
7	46080	276480	259200	76800	9000	432	7									667999
8	645120	4515840	5080320	1881600	294000	21168	686	8								12438742
9	10321920	82575360	108380160	48168960	9408000	903168	43904	1024	9							259802505
10	185794560	1672151040	2508226560	1300561920	304819200	36578304	2370816	82944	1458	10						6010586812
11	3715891200	37158912000	62705664000	37158912000	10160640000	1463132160	118540800	5529600	145800	2000	11					152487369571
12	81749606400	899245670400	1686085632000	1124057088000	351267840000	59012997120	5737374720	334540800	11761200	242000	2662	12				4207502755314
13	1961990553600	23543886643200	48559266201600	35969826816000	12645642240000	2427963310080	275393986560	19269550080	846806400	23232000	383328	3456	13			125404109726317
14	51011754393600	663152807116800	1492093816012800	1215780146380800	474914119680000	102581449850880	13297595351040	1085517987840	57244112640	1963104000	43188288	584064	4394	14		4013976457767160
15	1428329123020800	19996607722291200	48741731323084800	43325983398297600	18616633491456000	4467992037949440	651582172200960	60789007319040	3739948692480	153907353600	4232452224	76317696	861224	5488	15	137293546441302567

Technical details

Node id: n-c047d0120ec62185 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-baf7e4e24f192751

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:         1;
   2:         2,          2;
   3:         8,         16,          3;
   4:        48,        144,         54,          4;
   5:       384,       1536,        864,        128,         5;
   6:      3840,      19200,      14400,       3200,       250,        6;
   7:     46080,     276480,     259200,      76800,      9000,      432,       7;
   8:    645120,    4515840,    5080320,    1881600,    294000,    21168,     686,     8;
   9:  10321920,   82575360,  108380160,   48168960,   9408000,   903168,   43904,  1024,    9;
  10: 185794560, 1672151040, 2508226560, 1300561920, 304819200, 36578304, 2370816, 82944, 1458, 10;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 4 - (2*n^2 - 4*n - 2)*x - n*x^2)*P[n - 1, x] - (4*x - 2*n*x^2)*D[P[n - 1, x], x])/(-2 + (n - 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(4*n - 4 - (2*n^2 - 4*n - 2)*x - n*x^2)*v[i - 1] - (4*x - 2*n*x^2)*deriv(v[i - 1], x))/(-2 + (n - 1)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from 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 Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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 ?

A079621 → (θ+1)²

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.