A038305 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(25 + 12\,x + 12\,n\right)\,P_{n}(x) = \left(481 + 600\,x + 144\,x^{2} + 156\,n + 144\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 481 + 600\,x + 144\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	481	600	144								1225
1	8281	17316	10800	1728							38125
2	134017	397488	415584	172800	20736						1140625
3	2084953	8041020	11924640	8311680	2592000	248832					33203125
4	31559905	150116616	289476720	286191360	149610240	37324800	2985984				947265625
5	468200473	2651032020	6304897872	8105348160	6010018560	2513452032	522547200	35831808			26611328125
6	6839588353	44947245408	127249536960	201756731904	194528355840	115392356352	40215232512	7166361600	429981696		738525390625
7	98703417241	738675542124	2427151252032	4580983330560	5447431761408	4201812486144	2077062414336	620463587328	96745881600	5159780352	20294189453125

All 15 rows

Technical details

Node id: n-d3760c0acdc98316 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-85b44f7305edc983

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:            481,             600,             144;
  1:           8281,           17316,           10800,             1728;
  2:         134017,          397488,          415584,           172800,            20736;
  3:        2084953,         8041020,        11924640,          8311680,          2592000,           248832;
  4:       31559905,       150116616,       289476720,        286191360,        149610240,         37324800,          2985984;
  5:      468200473,      2651032020,      6304897872,       8105348160,       6010018560,       2513452032,        522547200,         35831808;
  6:     6839588353,     44947245408,    127249536960,     201756731904,     194528355840,     115392356352,      40215232512,       7166361600,       429981696;
  7:    98703417241,    738675542124,   2427151252032,    4580983330560,    5447431761408,    4201812486144,    2077062414336,     620463587328,     96745881600,      5159780352;
  8:  1410398416609,  11844410068920,  44320532527440,   97086050081280,  137429499916800,  130738362273792,   84036249722880,   35606784245760,   9306953809920,   1289945088000,    61917364224;
  9: 19989481318105, 186172590992388, 781731064548720, 1950103431207360, 3203839652682240, 3628138797803520, 2876243970023424, 1584683566202880, 587511940055040, 136501989212160, 17027275161600, 743008370688;

%F Formula

The row polynomials satisfy (12*n+25+12*x)*P(n,x) = (156*n+481+(144*n+600)*x+144*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 481 + 600*x + 144*x^2.

%t Mathematica

P[0, x_] := 481 + 600*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((156*n + 481 + (144*n + 600)*x + 144*x^2)*P[n - 1, x])/(12*n + 25 + 12*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 481 + 600*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((156*n + 481 + (144*n + 600)*x + 144*x^2)*v[i - 1])/(12*n + 25 + 12*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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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

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.