A038333 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(19 + 7\,x + 7\,n\right)\,P_{n}(x) = \left(312 + 266\,x + 49\,x^{2} + 84\,n + 49\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 312 + 266\,x + 49\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	312	266	49								627
1	4752	6552	2793	343							14440
2	69120	133056	91728	26068	2401						322373
3	974592	2419200	2328480	1070160	228095	16807					7037334
4	13436928	40932864	50803200	32598720	11236680	1915998	117649				151042039
5	182145024	658409472	1002855168	829785600	399334320	110119464	15647317	823543			3199119908
6	2436562944	10200121344	18435465216	18719963136	11616998400	4472544384	1027781664	125178536	5764801		67040380425
7	32248627200	153503465472	321303822336	387144769536	294839419392	146374179840	46961716032	9250034976	985780971	40353607	1392652169362

All 15 rows

Technical details

Node id: n-f94383b851354fab · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a9de0b27ed80361e

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:           312,            266,             49;
  1:          4752,           6552,           2793,             343;
  2:         69120,         133056,          91728,           26068,            2401;
  3:        974592,        2419200,        2328480,         1070160,          228095,           16807;
  4:      13436928,       40932864,       50803200,        32598720,        11236680,         1915998,         117649;
  5:     182145024,      658409472,     1002855168,       829785600,       399334320,       110119464,       15647317,         823543;
  6:    2436562944,    10200121344,    18435465216,     18719963136,     11616998400,      4472544384,     1027781664,      125178536,       5764801;
  7:   32248627200,   153503465472,   321303822336,    387144769536,    294839419392,    146374179840,    46961716032,     9250034976,     985780971,     40353607;
  8:  423101988864,  2257403904000,  5372621291520,   7497089187840,   6775033466880,   4127751871488,  1707698764800,   469617160320,   80937806040,   7667185330,   282475249;
  9: 5510645415936, 32578853142528, 86910050304000, 137897279815680, 144318966865920, 104335515389952, 52972815684096, 18784686412800, 4520065168080, 692467896120, 59037327041, 1977326743;

%F Formula

The row polynomials satisfy (7*n+19+7*x)*P(n,x) = (84*n+312+(49*n+266)*x+49*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 312 + 266*x + 49*x^2.

%t Mathematica

P[0, x_] := 312 + 266*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((84*n + 312 + (49*n + 266)*x + 49*x^2)*P[n - 1, x])/(7*n + 19 + 7*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 312 + 266*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((84*n + 312 + (49*n + 266)*x + 49*x^2)*v[i - 1])/(7*n + 19 + 7*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)*7^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
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? 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
[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.