A257607 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9 + 4\,x + n + n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	10	5														15
2	110	110	25													245
3	1320	1980	910	125												4335
4	17160	34320	23870	6710	625											82685
5	240240	600600	561000	240900	46510	3125										1692375
6	3603600	10810800	12698400	7378800	2197910	310310	15625									37015445
7	57657600	201801600	285885600	210210000	85405320	18798780	2018110	78125								861855135
8	980179200	3920716800	6508101600	5801796000	3002159160	908827920	153790670	12889910	390625							21288851885
9	17643225600	79394515200	151178227200	158616057600	100048188240	38834195400	9116111400	1218323700	81245710	1953125						556132043175
10	335221286400	1676106432000	3602158560000	4351995648000	3245499457200	1544160618000	468017206800	87548775600	9421968710	507005510	9765625					15320646723845
11	6704425728000	36874341504000	88304344128000	120811987296000	104151939091200	58862903299200	21993847075200	5350288944000	813453829320	71530841580	3139689310	48828125				443942250253935
12	140792940288000	844757641728000	2230897660992000	3410876589120000	3341138799398400	2191629721881600	977399642419200	294842515968000	58727500277160	7366000733520	535252473470	19326417110	244140625			13498983835837085
13	3097444686336000	20133390461184000	58134075487488000	98270120108160000	107892770636851200	80692509664166400	41937950941286400	15174033806131200	3770610163560240	624305512030200	65351035549800	3959357902500	118399908910	1220703125		429796641481257975
14	71241227785728000	498688594500096000	1563083223077376000	2895547609963008000	3524014443722572800	2962963962870912000	1761947791636032000	746039952193536000	223473524320846800	46446378804025200	6468313070419200	570320579228400	29017904315510	722606484710	6103515625	14300515089138096245

Technical details

Node id: n-fa0ad7db8c6303f8 · revision 0 · JSON
Definition source: derived:shift_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:          10,           5;
  2:         110,         110,           25;
  3:        1320,        1980,          910,          125;
  4:       17160,       34320,        23870,         6710,          625;
  5:      240240,      600600,       561000,       240900,        46510,        3125;
  6:     3603600,    10810800,     12698400,      7378800,      2197910,      310310,      15625;
  7:    57657600,   201801600,    285885600,    210210000,     85405320,    18798780,    2018110,      78125;
  8:   980179200,  3920716800,   6508101600,   5801796000,   3002159160,   908827920,  153790670,   12889910,   390625;
  9: 17643225600, 79394515200, 151178227200, 158616057600, 100048188240, 38834195400, 9116111400, 1218323700, 81245710, 1953125;

%F Formula

The row polynomials satisfy P(n,x) = (n+9+(n+4)*x)*P(n-1,x) - (x+x^2)*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[(n + 9 + (n + 4)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
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] = (n + 9 + (n + 4)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); 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]
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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
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) = 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/(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