A257607 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

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

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	110	110													245
3	125	910	1980	1320												4335
4	625	6710	23870	34320	17160											82685
5	3125	46510	240900	561000	600600	240240										1692375
6	15625	310310	2197910	7378800	12698400	10810800	3603600									37015445
7	78125	2018110	18798780	85405320	210210000	285885600	201801600	57657600								861855135
8	390625	12889910	153790670	908827920	3002159160	5801796000	6508101600	3920716800	980179200							21288851885
9	1953125	81245710	1218323700	9116111400	38834195400	100048188240	158616057600	151178227200	79394515200	17643225600						556132043175
10	9765625	507005510	9421968710	87548775600	468017206800	1544160618000	3245499457200	4351995648000	3602158560000	1676106432000	335221286400					15320646723845
11	48828125	3139689310	71530841580	813453829320	5350288944000	21993847075200	58862903299200	104151939091200	120811987296000	88304344128000	36874341504000	6704425728000				443942250253935
12	244140625	19326417110	535252473470	7366000733520	58727500277160	294842515968000	977399642419200	2191629721881600	3341138799398400	3410876589120000	2230897660992000	844757641728000	140792940288000			13498983835837085
13	1220703125	118399908910	3959357902500	65351035549800	624305512030200	3770610163560240	15174033806131200	41937950941286400	80692509664166400	107892770636851200	98270120108160000	58134075487488000	20133390461184000	3097444686336000		429796641481257975
14	6103515625	722606484710	29017904315510	570320579228400	6468313070419200	46446378804025200	223473524320846800	746039952193536000	1761947791636032000	2962963962870912000	3524014443722572800	2895547609963008000	1563083223077376000	498688594500096000	71241227785728000	14300515089138096245

Technical details

Node id: n-f296d09d2ec50000 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-ec2ecf19f26d1b35

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,      110,        110;
  3:     125,      910,       1980,       1320;
  4:     625,     6710,      23870,      34320,       17160;
  5:    3125,    46510,     240900,     561000,      600600,       240240;
  6:   15625,   310310,    2197910,    7378800,    12698400,     10810800,      3603600;
  7:   78125,  2018110,   18798780,   85405320,   210210000,    285885600,    201801600,     57657600;
  8:  390625, 12889910,  153790670,  908827920,  3002159160,   5801796000,   6508101600,   3920716800,   980179200;
  9: 1953125, 81245710, 1218323700, 9116111400, 38834195400, 100048188240, 158616057600, 151178227200, 79394515200, 17643225600;

%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) + (x+3*x^2+3*x^3+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] + (x + 3*x^2 + 3*x^3 + 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] + (x + 3*x^2 + 3*x^3 + 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) = 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) = x + 5.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257607 → 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