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	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	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
9	26244	46092164	2480096000	33427659264	190895652864	565677342720	944496967680	898089615360	454494781440	95126814720							3184735048456
10	78732	322750124	27649793312	531176040960	4161859952640	16828491939840	39077674352640	54286693171200	44646185041920	20071757905920	3805072588800						183436883616088
11	236196	2259565796	306729727424	8299438134144	87574155755520	470292513669120	1458981014077440	2777062370181120	3299790657945600	2390061219840000	966488437555200	167423193907200					11626281990594760
12	708588	15817905356	3392103528032	128172328741248	1796699969501184	12568210929500160	50679507708149760	126940401869783040	204358668873891840	212004851259801600	137161451608473600	50394381366067200	8036313307545600				804072067143597208
13	2125764	110728171844	37439682051200	1963290173455104	36188056680382464	325002850768527360	1669983770428047360	5354178673791467520	11214646270419271680	15625101278592368640	14378136469556428800	8403639794977996800	2828782284256051200	417888291992371200			60255548582048716936
14	6377292	775105705964	412722327937952	29898628786440960	718985719702548480	8198826701283778560	52889630220001935360	212739084459520819200	563846337026001469440	1013106215600196157440	1243263919334621184000	1027609075024448716800	547642606656002457600	170080534840895078400	23401744351572787200		4863527272060473394648
15	19131876	5425765450916	4546146452965184	453432099731869824	14139106734931476480	202952728523577876480	1624791856770862940160	8075821264405199585280	26542272498614717644800	59809610541343703040000	93984597155396557209600	103001238976872421785600	77255008540629663744000	37831844962361357107200	10905212867832918835200	1404104661094367232000	420652053144252247894600

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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.