A257622 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	8	4														12
2	88	88	16													192
3	1232	1848	744	64												3888
4	20944	41888	26664	5720	256											95472
5	418880	1047200	917840	329560	42088	1024										2756592
6	9634240	28902720	32358480	16545760	3758568	302808	4096									91506672
7	250490240	876715840	1196321280	799013600	267714832	40916568	2152424	16384								3433341168
8	7264216960	29056867840	46803137920	38710376320	17335369744	4053124768	432842344	15198040	65536							143671199472
9	232454942720	1046047242240	1943303281920	1920007689600	1082204544960	346269036960	58750414800	4495601880	106910568	262144						6633639927792
10	8135922995200	40679614976000	85645117958400	98502781977600	67130873705280	27487267093440	6539061642960	826693818720	46132035048	750471128	1048576					334994217722352
11	309165073817600	1700407905996800	4001711838924800	5254643980185600	4207972265777280	2109086364665280	653039307634560	118678781205600	11392868134032	469575532888	5261686504	4194304				18366573227755248
12	12675768026521600	76054608159129600	197873783166259200	292201674372608000	269161227152737920	160035552125729280	61419804098231040	14781322469168640	2092539257568144	154681343202848	4753633880424	36865359960	16777216			1086455750687173872
13	557733793166950400	3625269655585177600	10333224262205644800	16954767230694092800	17671411174438461440	12177545371922810880	5597803709697239040	1691206106951961600	322695912065567040	36110210955538720	2077408335962960	47941857763800	258191737448	67108864		68969893036136017392
14	26213488278846668800	183494417951926681600	568672980973838540800	1026610452467984384000	1194961476948945305600	937495198309122252800	503307577995058636800	183916451523181977600	44628540330451197120	6853426548356548160	613079316999989840	27677494376211680	482258686749928	1807879033048	268435456	4676795252205922613232

Technical details

Node id: n-4c6e90a40d8a7363 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-31113a40d89e3fd5

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:            8,             4;
  2:           88,            88,            16;
  3:         1232,          1848,           744,            64;
  4:        20944,         41888,         26664,          5720,           256;
  5:       418880,       1047200,        917840,        329560,         42088,         1024;
  6:      9634240,      28902720,      32358480,      16545760,       3758568,       302808,        4096;
  7:    250490240,     876715840,    1196321280,     799013600,     267714832,     40916568,     2152424,      16384;
  8:   7264216960,   29056867840,   46803137920,   38710376320,   17335369744,   4053124768,   432842344,   15198040,     65536;
  9: 232454942720, 1046047242240, 1943303281920, 1920007689600, 1082204544960, 346269036960, 58750414800, 4495601880, 106910568, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (3*n+5+(3*n+1)*x)*P(n-1,x) - (3*x+3*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[(3*n + 5 + (3*n + 1)*x)*P[n - 1, x] - (3*x + 3*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] = (3*n + 5 + (3*n + 1)*x)*v[i - 1] - (3*x + 3*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) = 3*x + 4.?
  - 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) = 3*x + 4.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257622 → 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