A225469 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 5\,x + 4\,n x\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	3														4
3	1	10	9													20
4	1	21	79	27												128
5	1	36	310	580	81											1008
6	1	55	850	3990	4141	243										9280
7	1	78	1895	16740	48031	29230	729									96704
8	1	105	3689	52745	299131	557571	205339	2187								1120768
9	1	136	6524	137592	1301286	5044536	6338620	1439560	6561							14274816
10	1	171	10740	313740	4465902	29768970	82006660	71164380	10083481	19683						197833728
11	1	210	16725	646680	12936882	132484716	647617090	1301264280	792891661	70604050	59049					2958521344
12	1	253	24915	1232055	32983962	481780530	3694765558	13605988990	20311855861	8792412321	494287399	177147				47415508992
13	1	300	35794	2203740	76105887	1504283352	16702839868	98585596824	278825646671	313470250236	97210822930	3460188940	531441			809838505984
14	1	351	49894	3742882	162051747	4167989397	63335623780	549562273260	2546294373623	5611157536985	4799264576470	1072779241170	24221854021	1594323		14670950907904
15	1	406	67795	6087900	322995673	10488007530	209215252675	2512966610440	17384475751643	64175928130314	111411257779185	73061747888220	11824793506891	169554572470	4782969	280760761434112

Technical details

Node id: n-d16ab74d3d5eb9b8 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-b86b6dd33b0582b0

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,   3;
   3: 1,  10,     9;
   4: 1,  21,    79,     27;
   5: 1,  36,   310,    580,      81;
   6: 1,  55,   850,   3990,    4141,      243;
   7: 1,  78,  1895,  16740,   48031,    29230,      729;
   8: 1, 105,  3689,  52745,  299131,   557571,   205339,     2187;
   9: 1, 136,  6524, 137592, 1301286,  5044536,  6338620,  1439560,     6561;
  10: 1, 171, 10740, 313740, 4465902, 29768970, 82006660, 71164380, 10083481, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n-5)*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n - 5)*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (4*n - 5)*x)*v[i - 1] - 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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A225469: Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0,…

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.