A048854 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-6 - x + n x\right)\,P_{n}(x) = \left(36 - 6\,x - 24\,n - 10\,n x + n x^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-24\,x + 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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	12	24	3													39
4	120	360	90	4												574
5	1680	6720	2520	224	5											11149
6	30240	151200	75600	10080	450	6										267576
7	665280	3991680	2494800	443520	29700	792	7									7625779
8	17297280	121080960	90810720	20180160	1801800	72072	1274	8								251244274
9	518918400	4151347200	3632428800	968647680	108108000	5765760	152880	1920	9							9385370649
10	17643225600	158789030400	158789030400	49401031680	6616209600	441080640	15593760	293760	2754	10						391695498604
11	670442572800	6704425728000	7542478944000	2681770291200	419026608000	33522128640	1481407200	37209600	523260	3800	11					18053185416511
12	28158588057600	309744468633600	387180585792000	154872234316800	27655756128000	2581203905280	136882025280	4297708800	80582040	877800	5082	12				910334098032294
13	1295295050649600	15543540607795200	21372368335718400	9498830371430400	1908247172832000	203546365102080	12593146325760	474467051520	11120321520	161515200	1402632	6624	13			49834906800150949
14	64764752532480000	841941782922240000	1262912674383360000	617423974142976000	137817851371200000	16538142164544000	1169363587392000	51400597248000	1445641797600	26246220000	303903600	2152800	8450	14		2942621413895522464
15	3497296636753920000	48962152914554880000	79563498486151680000	42433865859280896000	10419029563662720000	1389203941821696000	110504859008544000	5551264502784000	182150866497600	3968428464000	57437780400	542505600	3194100	10584	15	186381289703015572299

Technical details

Node id: n-be026e31b07cd533 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-b8ec366cca2adfe1

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:          12,           24,            3;
   4:         120,          360,           90,           4;
   5:        1680,         6720,         2520,         224,          5;
   6:       30240,       151200,        75600,       10080,        450,         6;
   7:      665280,      3991680,      2494800,      443520,      29700,       792,        7;
   8:    17297280,    121080960,     90810720,    20180160,    1801800,     72072,     1274,      8;
   9:   518918400,   4151347200,   3632428800,   968647680,  108108000,   5765760,   152880,   1920,    9;
  10: 17643225600, 158789030400, 158789030400, 49401031680, 6616209600, 441080640, 15593760, 293760, 2754, 10;

%F Formula

The row polynomials satisfy -(6-(n-1)*x)*P(n,x) = -(24*n-36-(4*n^2-10*n-6)*x-n*x^2)*P(n-1,x) - (24*x-4*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[(-(24*n - 36 - (4*n^2 - 10*n - 6)*x - n*x^2)*P[n - 1, x] - (24*x - 4*n*x^2)*D[P[n - 1, x], x])/(-6 + (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] = (-(24*n - 36 - (4*n^2 - 10*n - 6)*x - n*x^2)*v[i - 1] - (24*x - 4*n*x^2)*deriv(v[i - 1], x))/(-6 + (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[4,1].?
  - 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

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.