A159854 → basis A008277 → reverse

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 1 + 3\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	3	1															5
1	1	7	10	2														20
2	1	12	38	32	4													87
3	1	18	98	184	100	8												409
4	1	25	206	674	836	308	16											2066
5	1	33	381	1910	4206	3652	940	32										11155
6	1	42	645	4577	15666	24682	15548	2852	64									64077
7	1	52	1023	9737	47705	118678	138958	65044	8620	128								389946
8	1	63	1543	18944	125601	452613	851026	759834	268796	25988	256							2504665
9	1	75	2236	34374	296097	1457421	4019317	5865990	4067966	1101172	78220	512						16923381
10	1	88	3136	58970	639837	4122294	15678685	34001209	39263906	21441002	4482908	235172	1024					119928232
11	1	102	4280	96602	1288507	10520664	52779331	159430689	277272369	257024438	111687918	18166804	706540	2048				888980293
12	1	117	5708	152242	2447731	24694241	157985971	634444668	1552717881	2197931021	1653834546	576606394	73373756	2121668	4096			6876320041
13	1	133	7463	232154	4426877	54067013	429622622	2214304378	7262719893	14619674069	17039351693	10499613670	2956405726	295616692	6369100	8192		55382419676
14	1	150	9591	344099	7677033	111616414	1078426778	6940153220	29405763673	79984153106	133996744245	129775075521	65954087746	15077645322	1188835868	19115492	16384	463539664643

Technical details

Node id: n-ea8536338ecfd7fc · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-c748c34fca57e6cb

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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,  3,    1;
  1: 1,  7,   10,     2;
  2: 1, 12,   38,    32,      4;
  3: 1, 18,   98,   184,    100,       8;
  4: 1, 25,  206,   674,    836,     308,      16;
  5: 1, 33,  381,  1910,   4206,    3652,     940,      32;
  6: 1, 42,  645,  4577,  15666,   24682,   15548,    2852,      64;
  7: 1, 52, 1023,  9737,  47705,  118678,  138958,   65044,    8620,     128;
  8: 1, 63, 1543, 18944, 125601,  452613,  851026,  759834,  268796,   25988,   256;
  9: 1, 75, 2236, 34374, 296097, 1457421, 4019317, 5865990, 4067966, 1101172, 78220, 512;

%F Formula

The row polynomials satisfy P(n,x) = (1+(n+3)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 3*x + x^2.

%t Mathematica

P[0, x_] := 1 + 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (n + 3)*x)*P[n - 1, 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 + 3*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (n + 3)*x)*v[i - 1] - 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 (−∞, 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 basis_map_A008277 of Riordan array (1-x,x/(1-x)).?
  - Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    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
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Related ?

Same fingerprint ?

A159854 → basis A008277

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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.