A038333 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(38 + 42\,x + 19\,n + 21\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 798 + 882\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	798	882															1680
1	22743	50274	27783														100800
2	576156	1910412	2111508	777924													5376000
3	13683705	60496380	100296630	73902780	20420505												268800000
4	311988474	1724146830	3811271940	4212458460	2327937570	514596726											12902400000
5	6915744507	45862305678	126724792005	186752325060	154807848405	68441364558	12607619787										602112000000
6	150170452152	1161845077176	3852433676952	7096588352280	7843597652520	5201543706408	1916358207624	302582874888									27525120000000
7	3209893414749	28382215456728	109794359793132	242703321647976	335313799645230	296487991265256	163848626751852	51741671605848	7148520419229								1238630400000000
8	67764416533590	674077617097290	2980132622956440	7685605185519240	12741924386518740	14083179585099660	10377079694283960	4915458802555560	1358218879653510	166798809782010							55050240000000000
9	1416276305552031	15653580219259290	77855964774736995	229470211967645880	443843699463736110	588676906657165788	542202414026336910	342443629911370680	141933872923791795	34860951244440090	3853052505964431						2422210560000000000
10	29355545242351188	356901628999111812	1972351107626670540	6539901041077907580	14456623353961690440	22369722452972299944	24724430079600963096	19519286904948128760	10786974342208176420	3974148441866170260	878495971359890268	88269930136639692					105696460800000000000
11	604234972905061953	8014063851161874324	48717072358378762338	179483950794027019140	446348246053567192335	789331635126308298024	1017822371610239647452	964252773104437560744	666095665631354893935	327204888380314684740	108494252462946448098	21802672743750003924	2008140910608552993				4580179968000000000000
12	12363577137903575346	177645082034088214182	1178067386120795525628	4774273091121118709124	13192070383360985906790	26245276867949750909298	38677250121189106603176	42748539607630065192984	35436289411588080357342	21759125077290926535210	9619823718381251731356	2899755474918750521892	534165482221875096138	45414879055301121534			197300060160000000000000
13	251687106021608498115	3894526798439626233990	27979100420368893733665	123697075542683530190940	375974005925788098343515	831100434151742112127770	1377877035567361922738145	1740476255453509797142920	1683223747050433816973745	1240270129405582812506970	685412439934664185859115	275476770117281299579740	76118581216617201199665	12943240530760819637190	1021834778744275234515		8455716864000000000000000
14	5100858682037932228464	84566867623260455366640	654280502137857207310320	3133659247081316098170480	10390554345585416536038960	25265453198212960208684208	46541624312497558279155120	66138097707233372291430960	73100002729047411480002640	62840353223216195833686480	41673076348027582500234192	20936234528913378768060240	7713349563283876388232720	1967372560675644584852880	310637772738259671292560	22889099043871765253136	360777252864000000000000000

Technical details

Node id: n-e89c31f5f9f52ed6 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-7745de9e47db299c

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:              798,               882;
  1:            22743,             50274,             27783;
  2:           576156,           1910412,           2111508,             777924;
  3:         13683705,          60496380,         100296630,           73902780,           20420505;
  4:        311988474,        1724146830,        3811271940,         4212458460,         2327937570,          514596726;
  5:       6915744507,       45862305678,      126724792005,       186752325060,       154807848405,        68441364558,        12607619787;
  6:     150170452152,     1161845077176,     3852433676952,      7096588352280,      7843597652520,      5201543706408,      1916358207624,       302582874888;
  7:    3209893414749,    28382215456728,   109794359793132,    242703321647976,    335313799645230,    296487991265256,    163848626751852,     51741671605848,      7148520419229;
  8:   67764416533590,   674077617097290,  2980132622956440,   7685605185519240,  12741924386518740,  14083179585099660,  10377079694283960,   4915458802555560,   1358218879653510,   166798809782010;
  9: 1416276305552031, 15653580219259290, 77855964774736995, 229470211967645880, 443843699463736110, 588676906657165788, 542202414026336910, 342443629911370680, 141933872923791795, 34860951244440090, 3853052505964431;

%F Formula

The row polynomials satisfy (n+1)*P(n,x) = (19*n+38+(21*n+42)*x)*P(n-1,x) for n >= 1, with P(0,x) = 798 + 882*x.

%t Mathematica

P[0, x_] := 798 + 882*x;
P[n_, x_] := P[n, x] = Expand[Together[((19*n + 38 + (21*n + 42)*x)*P[n - 1, 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] = 798 + 882*x; for(n = 1, N, my(i = n + 1); v[i] = ((19*n + 38 + (21*n + 42)*x)*v[i - 1])/(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
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 basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.?
  - Geometric basis map (elementary) 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

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.