A375853 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-2\,x + 4\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	2															2
2	16	8														24
3	96	96	20													212
4	512	768	336	40												1656
5	2560	5120	3456	896	70											12102
6	12288	30720	28160	11520	2016	112										84816
7	57344	172032	199680	112640	31680	4032	168									577576
8	262144	917504	1290240	931840	366080	76032	7392	240								3851472
9	1179648	4718592	7798784	6881280	3494400	1025024	164736	12672	330							25275466
10	5242880	23592960	44826624	46792704	29245440	11182080	2562560	329472	20592	440						163795752
11	23068672	115343360	247726080	298844160	222265344	105283584	31682560	5857280	617760	32032	572					1050721404
12	100663296	553648128	1326448640	1816657920	1568931840	889061376	333398016	81469440	12446720	1098240	48048	728				6683872392
13	436207616	2617245696	6920601600	10611589120	10445783040	6903300096	3111714816	952565760	193489920	24893440	1867008	69888	910			42219328910
14	1879048192	12213813248	35332816896	59978547200	66322432000	50139758592	26462650368	9779675136	2500485120	429977600	47297536	3055104	99008	1120		265089657120
15	8053063680	56371445760	177100292096	329772957696	404855193600	344876646400	208915660800	90729086976	28116566016	6112296960	902952960	85995520	4837248	137088	1360	1655897134160

Technical details

Node id: n-8c995547482f9480 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-4748f217d058d591

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:       2;
   2:      16,        8;
   3:      96,       96,       20;
   4:     512,      768,      336,       40;
   5:    2560,     5120,     3456,      896,       70;
   6:   12288,    30720,    28160,    11520,     2016,      112;
   7:   57344,   172032,   199680,   112640,    31680,     4032,     168;
   8:  262144,   917504,  1290240,   931840,   366080,    76032,    7392,    240;
   9: 1179648,  4718592,  7798784,  6881280,  3494400,  1025024,  164736,  12672,   330;
  10: 5242880, 23592960, 44826624, 46792704, 29245440, 11182080, 2562560, 329472, 20592, 440;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (4*n+(3*n-2)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.

%t Mathematica

P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + (3*n - 2)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((4*n + (3*n - 2)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A375853 → x ↦ x/(x+1)

References

[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