A257615 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(5 + 20\,x + 25\,x^{2} + 10\,x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 6\,x^{3} + 2\,x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	10														15
2	25	120	120													265
3	125	1090	2520	1680												5415
4	625	8880	35760	53760	26880											125905
5	3125	68410	428400	1092000	1209600	483840										3285375
6	15625	510120	4676520	18009600	33196800	29030400	9676800									95115865
7	78125	3727090	48210120	263576880	719712000	1032998400	745113600	212889600								3026305815
8	390625	26870880	478616160	3574287360	13573486080	28449792000	33326899200	20437401600	5109350400							104977094305
9	1953125	192002410	4629996000	46017787200	233643916800	671069629440	1135553126400	1121502412800	597793996800	132843110400						3943247931375
10	9765625	1363548120	43973992920	571015603200	3773655513600	14271634944000	32725795737600	46290714624000	39469731840000	18598035456000	3719607091200					159465538116265
11	48828125	9642493090	412128513720	6896807536080	58193771328000	282000323404800	841771226419200	1599491084083200	1939841519616000	1453967843328000	613735170048000	111588212736000				6907907778334215
12	244140625	67985732880	3824866540560	81634682088960	866867947841280	5277492734976000	19950117317222400	48909297578803200	79124457929932800	83877139906560000	56054478864384000	21424936845312000	3570822807552000			319141139711086705
13	1220703125	478341536410	35239627659600	951529634546400	12575438235360000	94766014085783040	444701849092300800	1368179234976153600	2835436758419865600	3986410124029132800	3749921888993280000	2260107660754944000	789151840468992000	121407975456768000		15663646033337025375
14	6103515625	3360597786120	322896747373320	10960180767244800	178705171212422400	1647848099523225600	9455251716284774400	35778846144577536000	92380473566244864000	165408788571586560000	205367530697647718400	173520563510181888000	95208848517758976000	30594809815105536000	4370687116443648000	813923639370783068665

Technical details

Node id: n-49f8fa7d7f77a430 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-77f8f710632b75b3

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 0)
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:       1;
  1:       5,        10;
  2:      25,       120,        120;
  3:     125,      1090,       2520,        1680;
  4:     625,      8880,      35760,       53760,        26880;
  5:    3125,     68410,     428400,     1092000,      1209600,       483840;
  6:   15625,    510120,    4676520,    18009600,     33196800,     29030400,       9676800;
  7:   78125,   3727090,   48210120,   263576880,    719712000,   1032998400,     745113600,     212889600;
  8:  390625,  26870880,  478616160,  3574287360,  13573486080,  28449792000,   33326899200,   20437401600,   5109350400;
  9: 1953125, 192002410, 4629996000, 46017787200, 233643916800, 671069629440, 1135553126400, 1121502412800, 597793996800, 132843110400;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (5+20*x+25*x^2+10*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5 + 20*x + 25*x^2 + 10*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((5 + 20*x + 25*x^2 + 10*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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 [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from 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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.?
  - 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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257615 → 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