A038330 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(160 + 168\,x + 36\,x^{2} + 48\,n + 36\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 640 + 672\,x + 144\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	640	672	144								1456
1	13312	23040	12096	1728							50176
2	262144	638976	552960	193536	20736						1668352
3	4980736	15728640	19169280	11059200	2903040	248832					54089728
4	92274688	358612992	566231040	460062720	199065600	41803776	2985984				1721036800
5	1677721600	7751073792	15061745664	15854469120	9661317120	3344302080	585252864	35831808			53971714048
6	30064771072	161061273600	372051542016	481975861248	380507258880	185497288704	53508833280	8026324992	429981696		1673123135488
7	532575944704	3246995275776	8697308774400	13393855512576	13013348253696	8218956791808	3338951196672	825564856320	108355387392	5159780352	51381071773696

All 15 rows

Technical details

Node id: n-8c5ad656fba85f85 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-87f28ed98a2aba0a

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:             640,              672,              144;
  1:           13312,            23040,            12096,             1728;
  2:          262144,           638976,           552960,           193536,             20736;
  3:         4980736,         15728640,         19169280,         11059200,           2903040,            248832;
  4:        92274688,        358612992,        566231040,        460062720,         199065600,          41803776,          2985984;
  5:      1677721600,       7751073792,      15061745664,      15854469120,        9661317120,        3344302080,        585252864,         35831808;
  6:     30064771072,     161061273600,     372051542016,     481975861248,      380507258880,      185497288704,      53508833280,       8026324992,       429981696;
  7:    532575944704,    3246995275776,    8697308774400,   13393855512576,    13013348253696,     8218956791808,    3338951196672,     825564856320,    108355387392,      5159780352;
  8:   9345848836096,   63909113364480,  194819716546560,  347892350976000,   401815665377280,   312320358088704,  164379135836160,   57239163371520,  12383472844800,   1444738498560,    61917364224;
  9: 162727720910848, 1233652046364672, 4218001482055680, 8572067528048640, 11480447582208000, 10607933565960192, 6871047877951488, 3099720847196160, 944446195630080, 181624268390400, 19070548180992, 743008370688;

%F Formula

The row polynomials satisfy (3*n+7+3*x)*P(n,x) = (48*n+160+(36*n+168)*x+36*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 640 + 672*x + 144*x^2.

%t Mathematica

P[0, x_] := 640 + 672*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((48*n + 160 + (36*n + 168)*x + 36*x^2)*P[n - 1, x])/(3*n + 7 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 640 + 672*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((48*n + 160 + (36*n + 168)*x + 36*x^2)*v[i - 1])/(3*n + 7 + 3*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 basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^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.