A357367 → D

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	6	24													30
3	24	240	360												624
4	120	2160	7560	6720											16560
5	720	20160	120960	241920	151200										534960
6	5040	201600	1814400	6048000	8316000	3991680									20376720
7	40320	2177280	27216000	133056000	299376000	311351040	121080960								894297600
8	362880	25401600	419126400	2794176000	9081072000	15256200960	12713500800	4151347200							44441187840
9	3628800	319334400	6706022400	58118860800	254270016000	610248038400	813664051200	564583219200	158789030400						2466702201600
10	39916800	4311014400	112086374400	1220496076800	6865290432000	21968929382400	41496866611200	45731240755200	27152924198400	6704425728000					151256610489600
11	479001600	62270208000	1961511552000	26153487360000	183074411520000	746943599001600	1867358997504000	2896311914496000	2715292419840000	1407929402880000	309744468633600				10154832961996800
12	6227020800	958961203200	35961045120000	575376721920000	4890702136320000	24649138767052800	78055606095667200	159297155297280000	209077516327680000	170359457748480000	78365350564300800	15543540607795200			740850770499840000
13	87178291200	15692092416000	690452066304000	13041872363520000	132048957680640000	802857662698291200	3122224243826688000	8028576626982912000	13799116077626880000	15673070112860160000	11284610481259315200	4663062182338560000	841941782922240000		58361256231896217600
14	1307674368000	271996268544000	13871809695744000	305179813306368000	3624010283013120000	26092874037694464000	121766745509240832000	382695485886185472000	825187141442087424000	1222499468803092480000	1222499468803092480000	788057508815216640000	295521565805706240000	48962152914554880000	4937225747226829056000

Technical details

Node id: n-1ea5485793a82d6a · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-6832a290ef8b97c2

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 -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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:       0;
  1:       2;
  2:       6,        24;
  3:      24,       240,        360;
  4:     120,      2160,       7560,        6720;
  5:     720,     20160,     120960,      241920,       151200;
  6:    5040,    201600,    1814400,     6048000,      8316000,      3991680;
  7:   40320,   2177280,   27216000,   133056000,    299376000,    311351040,    121080960;
  8:  362880,  25401600,  419126400,  2794176000,   9081072000,  15256200960,  12713500800,   4151347200;
  9: 3628800, 319334400, 6706022400, 58118860800, 254270016000, 610248038400, 813664051200, 564583219200, 158789030400;

%F Formula

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

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - 1 + (2*n^2 + 2*n)*x)*P[n - 1, x] + (2*n*x + 2*n*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 - 1 + (2*n^2 + 2*n)*x)*v[i - 1] + (2*n*x + 2*n*x^2)*deriv(v[i - 1], x))/(n - 1)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.?
  - 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
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

References

[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.