A038334 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(84 + 80\,x + 16\,x^{2} + 24\,n + 16\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 336 + 320\,x + 64\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	336	320	64								720
1	5184	8064	3840	512							17600
2	76032	165888	129024	40960	4096						416000
3	1078272	3041280	3317760	1720320	409600	32768					9600000
4	14929920	51757056	72990720	53084160	20643840	3932160	262144				217600000
5	203046912	836075520	1449197568	1362493440	743178240	231211008	36700160	2097152			4864000000
6	2723217408	12995002368	26754416640	30916214784	21799895040	9512681472	2466250752	335544320	16777216		107520000000
7	36118462464	196071653376	467820085248	642105999360	556491866112	313918488576	114152177664	25367150592	3019898880	134217728	2355200000000

All 15 rows

Technical details

Node id: n-68e39df9a140191c · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-7ad822d7676abaf6

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:           336,            320,              64;
  1:          5184,           8064,            3840,             512;
  2:         76032,         165888,          129024,           40960,            4096;
  3:       1078272,        3041280,         3317760,         1720320,          409600,           32768;
  4:      14929920,       51757056,        72990720,        53084160,        20643840,         3932160,          262144;
  5:     203046912,      836075520,      1449197568,      1362493440,       743178240,       231211008,        36700160,        2097152;
  6:    2723217408,    12995002368,     26754416640,     30916214784,     21799895040,      9512681472,      2466250752,      335544320,       16777216;
  7:   36118462464,   196071653376,    467820085248,    642105999360,    556491866112,    313918488576,    114152177664,    25367150592,     3019898880,     134217728;
  8:  474699792384,  2889476997120,   7842866135040,  12475202273280,  12842119987200,   8903869857792,   4185579847680,  1304596316160,   253671505920,   26843545600,   1073741824;
  9: 6191736422400, 41773581729792, 127136987873280, 230057406627840, 274454450012160, 226021311774720, 130590091247616, 52618718085120, 14350559477760, 2480343613440, 236223201280, 8589934592;

%F Formula

The row polynomials satisfy (2*n+5+2*x)*P(n,x) = (24*n+84+(16*n+80)*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 336 + 320*x + 64*x^2.

%t Mathematica

P[0, x_] := 336 + 320*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((24*n + 84 + (16*n + 80)*x + 16*x^2)*P[n - 1, x])/(2*n + 5 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 336 + 320*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24*n + 84 + (16*n + 80)*x + 16*x^2)*v[i - 1])/(2*n + 5 + 2*x)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.?
  - 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 [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Parent-row provers of polytool polytool ? 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
[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.