A048854 → D

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	12	2													14
4	180	60	3												243
5	3360	1680	168	4											5212
6	75600	50400	7560	360	5										133925
7	1995840	1663200	332640	23760	660	6									4016106
8	60540480	60540480	15135120	1441440	60060	1092	7								137718679
9	2075673600	2421619200	726485760	86486400	4804800	131040	1680	8							5315202488
10	79394515200	105859353600	37050773760	5292967680	367567200	13366080	257040	2448	9						227978803017
11	3352212864000	5028319296000	2011327718400	335221286400	27935107200	1269777600	32558400	465120	3420	10					10756319076550
12	154872234316800	258120390528000	116154175737600	22124604902400	2151003254400	117327450240	3760495200	71628480	790020	4620	11				553543569107771
13	7771770303897600	14248245557145600	7124122778572800	1526597738265600	169621970918400	10794125422080	415158670080	9884730240	145363680	1275120	6072	12			30851577664267284
14	420970891461120000	841941782922240000	463067980607232000	110254281096960000	13781785137120000	1002311646336000	44975522592000	1285014931200	23621598000	276276000	1973400	7800	13		1851065317308386413
15	24481076457277440000	53042332324101120000	31825399394460672000	8335223650930176000	1157669951518080000	94718450578752000	4857356439936000	161911881331200	3571585617600	52216164000	497296800	2948400	9828	14	118941443121489543842

Technical details

Node id: n-2fd4025bef24ce96 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-f136838b00e1c323

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:           0;
   2:           1;
   3:          12,            2;
   4:         180,           60,           3;
   5:        3360,         1680,         168,          4;
   6:       75600,        50400,        7560,        360,         5;
   7:     1995840,      1663200,      332640,      23760,       660,        6;
   8:    60540480,     60540480,    15135120,    1441440,     60060,     1092,      7;
   9:  2075673600,   2421619200,   726485760,   86486400,   4804800,   131040,   1680,    8;
  10: 79394515200, 105859353600, 37050773760, 5292967680, 367567200, 13366080, 257040, 2448, 9;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (4*n^2-10*n+6+(n-1)*x)*P(n-1,x) + (4*n-4)*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 10*n + 6 + (n - 1)*x)*P[n - 1, x] + (4*n - 4)*x*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 10*n + 6 + (n - 1)*x)*v[i - 1] + (4*n - 4)*x*deriv(v[i - 1], x))/(n - 2)); 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

Related ?

Same fingerprint ?

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.