A257621 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	3														9
2	60	60	9													129
3	840	1260	474	27												2601
4	15120	30240	18600	3480	81											67521
5	332640	831600	714000	239400	24846	243										2142729
6	8648640	25945920	28652400	14061600	2881860	175380	729									80366529
7	259459200	908107200	1227108960	797504400	251270040	33454260	1232034	2187								3478138281
8	8821612800	35286451200	56423727360	45768602880	19675444320	4237410240	380317200	8637360	6561							170602209921
9	335221286400	1508495788800	2786591808000	2713424313600	1485537641280	450106826400	68885594400	4269862800	60500886	19683						9352593642249
10	14079294028800	70396470144000	147541474080000	167787075456000	111886435140480	44069715930240	9791970400800	1093061995200	47573499660	423624300	59049					566693494358529
11	647647525324800	3562061389286400	8352038350656000	10868712158304000	8557992393350400	4166746362979200	1229026815213120	205722553528800	17061958923240	527544739260	2965724394	177147				37607540018206761
12	32382376266240000	194294257597440000	503954250466867200	738740557691136000	671376666914553600	390300155083238400	144456848995979520	32793512927535360	4215843777665520	263315010198240	5832649375800	20761133640	531441			2712783638141894721
13	1748648318376960000	11366214069450240000	32313725628555571200	52697136193103001600	54323195092707340800	36768388614839481600	16433010277459776000	4752966259007366400	846999360441954720	84840701959213200	4031382244234800	64366754470200	145331124126	1594323		211339220410232328969
14	101421602465863680000	709951217261045760000	2195363198969740800000	3942813369424849920000	4547551050197137382400	3515803375979486592000	1845615950953014240000	652012306371474048000	150352072364689715520	21347480733267648960	1684538217621277200	61371868226104800	709487610413460	1017327434820	4782969	17683978245311359800129

Technical details

Node id: n-47ad3360622db9f1 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-b4bba0102ede147f

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:            6,             3;
  2:           60,            60,             9;
  3:          840,          1260,           474,            27;
  4:        15120,         30240,         18600,          3480,            81;
  5:       332640,        831600,        714000,        239400,         24846,          243;
  6:      8648640,      25945920,      28652400,      14061600,       2881860,       175380,         729;
  7:    259459200,     908107200,    1227108960,     797504400,     251270040,     33454260,     1232034,       2187;
  8:   8821612800,   35286451200,   56423727360,   45768602880,   19675444320,   4237410240,   380317200,    8637360,     6561;
  9: 335221286400, 1508495788800, 2786591808000, 2713424313600, 1485537641280, 450106826400, 68885594400, 4269862800, 60500886, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (4*n+2+(4*n-1)*x)*P(n-1,x) - (4*x+4*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[(4*n + 2 + (4*n - 1)*x)*P[n - 1, x] - (4*x + 4*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] = (4*n + 2 + (4*n - 1)*x)*v[i - 1] - (4*x + 4*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), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - 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), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257621 → 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