A286785 → D

nonnegative coefficients

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

Recurrence

\(\left(56 + 86\,n + 34\,n^{2} + 4\,n^{3}\right)\,P_{n}(x) = \left(392 + 112\,x + 387\,n + 186\,n x + 130\,n^{2} + 86\,n^{2} x + 15\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(2 + 68\,x - 70\,x^{2} + 3\,n + 45\,n x - 48\,n x^{2} + n^{2} + 7\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 14 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	14	4															18
1	72	54	6														132
2	330	440	132	8													910
3	1430	2860	1560	260	10												6120
4	6006	16380	13650	4200	450	12											40698
5	24752	86632	99960	47600	9520	714	14										269192
6	100776	434112	651168	434112	135660	19152	1064	16									1776060
7	406980	2093040	3907008	3418632	1508220	335160	35280	1512	18								11705850
8	1634380	9806280	22064130	24227280	14132580	4462920	743820	60720	2070	20							77134200
9	6537520	44945450	118973250	158631000	116886000	49092120	11688600	1518000	99000	2750	22						508373712
10	26075790	202470840	618660900	976833000	879149700	468879840	149189040	27799200	2895750	154440	3564	24					3352112088
11	103791870	899529540	3124681560	5728582860	6137767350	4017447720	1630268640	407567160	61135074	5225220	232232	4524	26				22116233776
12	412506150	3951374700	15410361330	32288376120	40360470150	31586454900	15793227450	5053832784	1020485466	125985860	8998990	338520	5642	28			146012418090
13	1637618400	17194993200	74511637200	176118415200	252691639200	231634002600	138980401560	54981257760	14254400160	2375733360	245765520	14894880	480480	6930	30		964641246480
14	6495886320	74238700800	354321072000	934586016000	1518702276000	1603749603456	1130849079360	538499561600	173089144800	37137900800	5199306112	457416960	23823800	666400	8400	32	6377350462840

Technical details

Node id: n-24fe2fab32454333 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-0fc3a9782387e944

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:      14,        4;
  1:      72,       54,         6;
  2:     330,      440,       132,         8;
  3:    1430,     2860,      1560,       260,        10;
  4:    6006,    16380,     13650,      4200,       450,       12;
  5:   24752,    86632,     99960,     47600,      9520,      714,       14;
  6:  100776,   434112,    651168,    434112,    135660,    19152,     1064,      16;
  7:  406980,  2093040,   3907008,   3418632,   1508220,   335160,    35280,    1512,    18;
  8: 1634380,  9806280,  22064130,  24227280,  14132580,  4462920,   743820,   60720,  2070,   20;
  9: 6537520, 44945450, 118973250, 158631000, 116886000, 49092120, 11688600, 1518000, 99000, 2750, 22;

%F Formula

The row polynomials satisfy (4*n^3+34*n^2+86*n+56)*P(n,x) = (15*n^3+130*n^2+387*n+392+(12*n^3+86*n^2+186*n+112)*x)*P(n-1,x) + (n^2+3*n+2+(7*n^2+45*n+68)*x-(8*n^2+48*n+70)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 4*x.

%t Mathematica

P[0, x_] := 14 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((15*n^3 + 130*n^2 + 387*n + 392 + (12*n^3 + 86*n^2 + 186*n + 112)*x)*P[n - 1, x] + (n^2 + 3*n + 2 + (7*n^2 + 45*n + 68)*x - (8*n^2 + 48*n + 70)*x^2)*D[P[n - 1, x], x])/(4*n^3 + 34*n^2 + 86*n + 56)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 14 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = ((15*n^3 + 130*n^2 + 387*n + 392 + (12*n^3 + 86*n^2 + 186*n + 112)*x)*v[i - 1] + (n^2 + 3*n + 2 + (7*n^2 + 45*n + 68)*x - (8*n^2 + 48*n + 70)*x^2)*deriv(v[i - 1], x))/(4*n^3 + 34*n^2 + 86*n + 56)); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.?
  - 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

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.