A038331 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(34 + 40\,x + 17\,n + 20\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 680 + 800\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	680	800															1480
1	17340	40800	24000														82140
2	393040	1387200	1632000	640000													4052240
3	8352100	39304000	69360000	54400000	16000000												187416100
4	170382840	1002252000	2358240000	2774400000	1632000000	384000000											8321274840
5	3379259660	23853597600	70157640000	110051200000	97104000000	45696000000	8960000000										359201697260
6	65654187680	540681545600	1908287808000	3741740800000	4402048000000	3107328000000	1218560000000	204800000000									15189100341280
7	1255636339380	11817753782400	48661339104000	114497268480000	168378336000000	158473728000000	93219840000000	31334400000000	4608000000000								632246301705780
8	23717575299400	251127267876000	1181775378240000	3244089273600000	5724863424000000	6735133440000000	5282457600000000	2663424000000000	783360000000000	102400000000000							25992347959015400
9	443518658098780	5217866565868000	27623999466360000	86663527737600000	178424910048000000	251893990656000000	246954892800000000	166020096000000000	73244160000000000	19148800000000000	2252800000000000						1057888561931926780
10	8225255113831920	106444477943707200	626143987904160000	2209919957308800000	5199811664256000000	8564395682304000000	10075759626240000000	8467024896000000000	4980602880000000000	1953177600000000000	459571200000000000	49152000000000000					42700229227070499120
11	151481781679737860	2138566329596299200	13837782132681936000	54265812285027200000	143644797225072000000	270390206541312000000	371123812899840000000	374242500403200000000	275178309120000000000	143884083200000000000	50782617600000000000	10862592000000000000	1064960000000000000				1711567521518409173060
12	2773281849213662360	42414898870326600800	299399286143481888000	1291526332383647360000	3798606859951904000000	8044108644604032000000	12618209638594560000000	14844952515993600000000	13098487514112000000000	8561102950400000000000	4028754329600000000000	1292648448000000000000	253460480000000000000	22937600000000000000			68199382780502765511160
13	50513347967820278700	831984554764098708000	6362234830548990120000	29939928614348188800000	96864474928773552000000	227916411597114240000000	402205432230201600000000	540780413082624000000000	556685719349760000000000	436616250470400000000000	256833088512000000000000	109875118080000000000000	32316211200000000000000	5849088000000000000000	491520000000000000000		2703618388798502489906700
14	915975376483141053760	16164271349702489184000	133117528762255793280000	678638381925225612800000	2395194289147855104000000	6199326395441507328000000	12155541951846092800000000	18386534044809216000000000	21631216523304960000000000	19793270021324800000000000	13971720015052800000000000	7471508029440000000000000	2930003148800000000000000	795475968000000000000000	133693440000000000000000	10485760000000000000000	106702805744580898268317760

Technical details

Node id: n-e6b78a920c1c667f · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-8c1ae502a0a48e51

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:             680,              800;
  1:           17340,            40800,             24000;
  2:          393040,          1387200,           1632000,            640000;
  3:         8352100,         39304000,          69360000,          54400000,           16000000;
  4:       170382840,       1002252000,        2358240000,        2774400000,         1632000000,          384000000;
  5:      3379259660,      23853597600,       70157640000,      110051200000,        97104000000,        45696000000,         8960000000;
  6:     65654187680,     540681545600,     1908287808000,     3741740800000,      4402048000000,      3107328000000,      1218560000000,       204800000000;
  7:   1255636339380,   11817753782400,    48661339104000,   114497268480000,    168378336000000,    158473728000000,     93219840000000,     31334400000000,     4608000000000;
  8:  23717575299400,  251127267876000,  1181775378240000,  3244089273600000,   5724863424000000,   6735133440000000,   5282457600000000,   2663424000000000,   783360000000000,   102400000000000;
  9: 443518658098780, 5217866565868000, 27623999466360000, 86663527737600000, 178424910048000000, 251893990656000000, 246954892800000000, 166020096000000000, 73244160000000000, 19148800000000000, 2252800000000000;

%F Formula

The row polynomials satisfy (n+1)*P(n,x) = (17*n+34+(20*n+40)*x)*P(n-1,x) for n >= 1, with P(0,x) = 680 + 800*x.

%t Mathematica

P[0, x_] := 680 + 800*x;
P[n_, x_] := P[n, x] = Expand[Together[((17*n + 34 + (20*n + 40)*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] = 680 + 800*x; for(n = 1, N, my(i = n + 1); v[i] = ((17*n + 34 + (20*n + 40)*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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
    
    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.