A048854 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3\,x + n\right)\,P_{n}(x) = \left(-6 + 15\,x - 3\,x^{2} - 2\,n - 11\,n x + 4\,n^{2}\right)\,P_{n-1}(x) + \left(-12\,x - 12\,x^{2} + 4\,n x\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	1														4
3	24	14	1													39
4	300	240	33	1												574
5	5040	5040	1008	60	1											11149
6	105840	126000	32760	2880	95	1										267576
7	2661120	3659040	1164240	134640	6600	138	1									7625779
8	77837760	121080960	45405360	6486480	420420	13104	189	1								251244274
9	2594592000	4497292800	1937295360	328648320	26426400	1092000	23520	248	1							9385370649
10	97037740800	185253868800	89980450560	17643225600	1690809120	86879520	2484720	39168	315	1						391695498604
11	4022655436800	8380532160000	4525487366400	1005663859200	111740428800	6856799040	244188000	5116320	61560	390	1					18053185416511
12	183030822374400	412992624844800	245214371001600	60842663481600	7682154480000	547528101120	23315070240	608842080	9743580	92400	473	1				910334098032294
13	9067065354547200	22020015861043200	14248245557145600	3901305331123200	551271405484800	44718519605760	2214179573760	69193111680	1380954960	17426640	133584	564	1			49834906800150949
14	485735643993600000	1262912674383360000	884038872068352000	264610274632704000	41345355411360000	3758668673760000	212027463648000	7710089587200	184248464400	2900898000	29601000	187200	663	1		2942621413895522464
15	27978373094031360000	77523408781378560000	58346565556511232000	18943690115750400000	3241475864250624000	326252440882368000	20643764869728000	855819944179200	23810570784000	449059010400	5718913200	48157200	255528	770	1	186381289703015572299

Technical details

Node id: n-92baaccdeb9edfec · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-aa579d4e84ab5b11

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:           3,            1;
   3:          24,           14,           1;
   4:         300,          240,          33,           1;
   5:        5040,         5040,        1008,          60,          1;
   6:      105840,       126000,       32760,        2880,         95,        1;
   7:     2661120,      3659040,     1164240,      134640,       6600,      138,       1;
   8:    77837760,    121080960,    45405360,     6486480,     420420,    13104,     189,     1;
   9:  2594592000,   4497292800,  1937295360,   328648320,   26426400,  1092000,   23520,   248,   1;
  10: 97037740800, 185253868800, 89980450560, 17643225600, 1690809120, 86879520, 2484720, 39168, 315, 1;

%F Formula

The row polynomials satisfy (n-3*x)*P(n,x) = (4*n^2-2*n-6-(11*n-15)*x-3*x^2)*P(n-1,x) + ((4*n-12)*x-12*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^2 - 2*n - 6 - (11*n - 15)*x - 3*x^2)*P[n - 1, x] + ((4*n - 12)*x - 12*x^2)*D[P[n - 1, x], x])/(n - 3*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^2 - 2*n - 6 - (11*n - 15)*x - 3*x^2)*v[i - 1] + ((4*n - 12)*x - 12*x^2)*deriv(v[i - 1], x))/(n - 3*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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] 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.