A110608 → D

nonnegative coefficients

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

Recurrence

\(\left(6 + 16\,n + 14\,n^{2} + 4\,n^{3}\right)\,P_{n}(x) = \left(48 + 12\,x + 96\,n + 45\,n x + 65\,n^{2} + 43\,n^{2} x + 15\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(1 + 18\,x - 19\,x^{2} + 2\,n + 23\,n x - 25\,n x^{2} + n^{2} + 7\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 8 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	8	2															10
1	45	36	3														84
2	224	336	96	4													660
3	1050	2400	1350	200	5												5005
4	4752	14850	13200	3960	360	6											37128
5	21021	84084	105105	50960	9555	588	7										271320
6	91520	448448	733824	509600	156800	20160	896	8									1961256
7	393822	2291328	4678128	4318272	1927800	411264	38556	1296	9								14060475
8	1679600	11337300	27907200	32558400	19535040	6104700	957600	68400	1800	10							100150050
9	7113106	54716200	158286150	225118080	172356030	72997848	16897650	2032800	114345	2420	11						709634640
10	29953728	258885792	862952640	1456232580	1370571840	746200224	235642176	42078960	4007520	182160	3168	12					5006710800
11	125550100	1205280960	4557468630	8936213000	10053239625	6772708800	2765522760	677270880	96203250	7436000	278850	4056	13				35197176924
12	524190240	5536759410	23449804560	52544932440	69138069000	56001835890	28445376960	9050801760	1770809040	204954750	13117104	412776	5096	14			246681069040
13	2181340125	25149568500	118063252125	298265058000	451125900225	429643714500	263645006625	104802984000	26746594875	4279455180	411486075	22167600	593775	6300	15		1724337127920
14	9051563520	113144544000	583587648000	1643771875200	2817894643200	3099684107520	2246147904000	1082964168000	346548533760	72567855360	9675714048	785366400	36108800	833280	7680	16	12025860872784

Technical details

Node id: n-34093db33eff2903 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-a93f0b751fdd5dc4

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:       8,        2;
  1:      45,       36,         3;
  2:     224,      336,        96,         4;
  3:    1050,     2400,      1350,       200,         5;
  4:    4752,    14850,     13200,      3960,       360,        6;
  5:   21021,    84084,    105105,     50960,      9555,      588,        7;
  6:   91520,   448448,    733824,    509600,    156800,    20160,      896,       8;
  7:  393822,  2291328,   4678128,   4318272,   1927800,   411264,    38556,    1296,      9;
  8: 1679600, 11337300,  27907200,  32558400,  19535040,  6104700,   957600,   68400,   1800,   10;
  9: 7113106, 54716200, 158286150, 225118080, 172356030, 72997848, 16897650, 2032800, 114345, 2420, 11;

%F Formula

The row polynomials satisfy (4*n^3+14*n^2+16*n+6)*P(n,x) = (15*n^3+65*n^2+96*n+48+(12*n^3+43*n^2+45*n+12)*x)*P(n-1,x) + (n^2+2*n+1+(7*n^2+23*n+18)*x-(8*n^2+25*n+19)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 8 + 2*x.

%t Mathematica

P[0, x_] := 8 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((15*n^3 + 65*n^2 + 96*n + 48 + (12*n^3 + 43*n^2 + 45*n + 12)*x)*P[n - 1, x] + (n^2 + 2*n + 1 + (7*n^2 + 23*n + 18)*x - (8*n^2 + 25*n + 19)*x^2)*D[P[n - 1, x], x])/(4*n^3 + 14*n^2 + 16*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 8 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((15*n^3 + 65*n^2 + 96*n + 48 + (12*n^3 + 43*n^2 + 45*n + 12)*x)*v[i - 1] + (n^2 + 2*n + 1 + (7*n^2 + 23*n + 18)*x - (8*n^2 + 25*n + 19)*x^2)*deriv(v[i - 1], x))/(4*n^3 + 14*n^2 + 16*n + 6)); 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 Number triangle T(n,k) = binomial(n,k)*binomial(2n,n-k).?
  - 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.