A225473 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(3 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 4 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
1	4	4								8
2	12	44	32							88
3	36	356	704	384						1480
4	108	2636	10592	14208	6144					33688
5	324	18884	137600	340224	344064	122880				963976
6	972	133484	1664672	6754560	11980800	9707520	2949120			33191128
7	2916	938276	19379264	121294464	335708160	462888960	312606720	82575360		1335394120
8	8748	6579596	220678112	2051968128	8319166464	17360609280	19549716480	11312824320	2642411520	61463962648

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-7f0c1b7aed5a0d02 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-7edb340fc0df783f

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     4,         4;
   2:    12,        44,          32;
   3:    36,       356,         704,          384;
   4:   108,      2636,       10592,        14208,          6144;
   5:   324,     18884,      137600,       340224,        344064,         122880;
   6:   972,    133484,     1664672,      6754560,      11980800,        9707520,        2949120;
   7:  2916,    938276,    19379264,    121294464,     335708160,      462888960,      312606720,       82575360;
   8:  8748,   6579596,   220678112,   2051968128,    8319166464,    17360609280,    19549716480,    11312824320,     2642411520;
   9: 26244,  46092164,  2480096000,  33427659264,  190895652864,   565677342720,   944496967680,   898089615360,   454494781440,    95126814720;
  10: 78732, 322750124, 27649793312, 531176040960, 4161859952640, 16828491939840, 39077674352640, 54286693171200, 44646185041920, 20071757905920, 3805072588800;

%F Formula

The row polynomials satisfy P(n,x) = (3+4*x)*P(n-1,x) + (4*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 4 + 4*x.

%t Mathematica

P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[(3 + 4*x)*P[n - 1, x] + (4*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] = 4 + 4*x; for(n = 2, N, my(i = n); v[i] = (3 + 4*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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-d4e2c570702f66e5

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

obreschkoff_combination e-d4e2c570702f66e5

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= …
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check ma_wang_interval e-a7b6cc9ddce470c2 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
- premise (sequence): Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= …
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  
  no recorded property covers this requirement

Related sequences ?

No candidate equalities or related sequences.

References

[Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
[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
[Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.