A290319 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(1 + 3\,x + 4\,n\right)\,P_{n-1}(x)\)

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

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	3														8
2	45	42	9													96
3	585	681	243	27												1536
4	9945	13332	6174	1188	81											30720
5	208845	309807	169650	43470	5265	243										737280
6	5221125	8371710	5170671	1595700	262035	21870	729									20643840
7	151412625	258442965	175064589	61787313	12386115	1420335	86751	2187								660602880
8	4996616625	8982855720	6552460332	2564175096	594103734	84029400	7123788	332424	6561							23781703680
9	184874815125	347355511515	269389599444	114531859548	29674363446	4891399002	515668356	33671052	1240029	19683						951268147200
10	7579867420125	14796200417490	12087040111749	5503975039800	1560244479930	289570449420	35816599602	2927518200	151854345	4527090	59049					41855798476800
11	341094033905625	688568621047425	588305406281175	283939997126247	86722926716250	17711403663690	2480458330350	239188117806	15616000125	659282085	16238475	177147				2009078326886400
12	16713607661375625	34763144533040700	30892670770919850	15677976078029628	5101243400474991	1128027559669560	174676669178220	19161592763544	1482748359543	79152822540	2773531530	57395628	531441			104472072998092800
13	885821206052908125	1892587483235283975	1741600984457874150	923610744448329834	317399828459263407	75089190863911653	12641946145454340	1539594424002492	136070441346411	8643344673249	384455638710	11362562874	200353257	1594323		5850436087893196800
14	50491808745015763125	110534950162569910950	104949018563804678475	57870615386928422988	20862622455523003701	5232283364620754442	945858502882632339	125682720604505064	12374798428752903	900881970414426	47844005426217	1801032999948	45507824271	691936182	4782969	351026165273591808000

Technical details

Node id: n-a0debbe5cee8bfdf · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-bae587d88b0fc2e7

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:            1;
  1:            5,            3;
  2:           45,           42,            9;
  3:          585,          681,          243,           27;
  4:         9945,        13332,         6174,         1188,          81;
  5:       208845,       309807,       169650,        43470,        5265,        243;
  6:      5221125,      8371710,      5170671,      1595700,      262035,      21870,       729;
  7:    151412625,    258442965,    175064589,     61787313,    12386115,    1420335,     86751,     2187;
  8:   4996616625,   8982855720,   6552460332,   2564175096,   594103734,   84029400,   7123788,   332424,    6561;
  9: 184874815125, 347355511515, 269389599444, 114531859548, 29674363446, 4891399002, 515668356, 33671052, 1240029, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (4*n+1+3*x)*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 + 1 + 3*x)*P[n - 1, 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 + 1 + 3*x)*v[i - 1]); 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 Geometric basis map (elementary), Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary), Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary), Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A …?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1…?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1…?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
Geometric basis map (elementary) polytool ? step ›
- from basis_map_A007318 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Geometric basis map (elementary) polytool ? step ›
    
    from basis_map_A007318 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A …?
    
    Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1…?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1…?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1…?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from basis_map_A007318 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A …?
    
    ↑ Geometric basis map (elementary) (above)
    
    ↑ Geometric basis map (elementary) (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Geometric basis map (elementary) (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Geometric basis map (elementary) (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A …?
  - ↑ Geometric basis map (elementary) (above)
  - ↑ Geometric basis map (elementary) (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.