A089231 → x ↦ x+1 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(\left(2 + 3\,n + 4\,n x + n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(4 + 2\,x + 13\,n + 22\,n x + 11\,n x^{2} + 2\,n x^{3} + 9\,n^{2} + 22\,n^{2} x + 17\,n^{2} x^{2} + 4\,n^{2} x^{3} + 2\,n^{3} + 6\,n^{3} x + 6\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-4 - 10\,x - 8\,x^{2} - 2\,x^{3} - 3\,n - 10\,n x - 12\,n x^{2} - 6\,n x^{3} - n x^{4} - n^{2} - 4\,n^{2} x - 6\,n^{2} x^{2} - 4\,n^{2} x^{3} - n^{2} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 4 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	4	2															6
1	16	20	6														42
2	86	174	114	24													398
3	574	1616	1668	744	120												4722
4	4552	16490	23460	16320	5520	720											67062
5	41684	185172	337530	322200	169200	46080	5040										1106906
6	431986	2277590	5079606	6198360	4456200	1879920	428400	40320									20792382
7	4990906	30500864	80632104	120227184	110344080	63645120	22438080	4394880	362880								437536098
8	63537644	442083186	1353571848	2390379264	2678598720	1970740800	949173120	287400960	49351680	3628800							10188466022
9	883014232	6897209780	24030641070	49125180240	65163752640	58504965120	35924212800	14852678400	3940876800	602380800	39916800						259964828682
10	13294235374	115261174622	450665258010	1047978379800	1608612706800	1709003862720	1280109821760	674514086400	244290816000	57679776000	7943443200	479001600					7209832562286
11	215446563446	2054122962960	8912569380396	23252569917720	40590061993800	49888278665280	44207716291200	28408171841280	13109315788800	4223995776000	898367500800	112565376000	6227020800				215879409078482
12	3737993925616	38887050329114	185511080119404	536987302298304	1051484591174160	1469438944768080	1506117352259520	1144775919006720	644141089912320	264405011270400	76728312460800	14845217587200	1706203699200	87178291200			6938853247102038
13	69108820711324	779295385145396	4055905450240098	12906585475800744	28036073285940720	43943706202556160	51205262336308080	45011670153690240	29953249078072320	14989922134387200	5540837875372800	1462546602316800	259529772902400	27548340019200	1307674368000		238242548587831482
14	1356008021856842	16480066444985670	92961320642136030	322702171788611160	770534609585864760	1339656874421065200	1750642232732593200	1749338331178838400	1346161250767132800	796649068171756800	358977416935737600	120684177694080000	29214099218304000	4787395861248000	472070446848000	20922789888000	8700638016700946462

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Technical details

Node id: n-c94fb4ea603ebbf0 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-437c6ffcd0f1111c

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 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

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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         4,          2;
  1:        16,         20,           6;
  2:        86,        174,         114,          24;
  3:       574,       1616,        1668,         744,         120;
  4:      4552,      16490,       23460,       16320,        5520,         720;
  5:     41684,     185172,      337530,      322200,      169200,       46080,        5040;
  6:    431986,    2277590,     5079606,     6198360,     4456200,     1879920,      428400,       40320;
  7:   4990906,   30500864,    80632104,   120227184,   110344080,    63645120,    22438080,     4394880,     362880;
  8:  63537644,  442083186,  1353571848,  2390379264,  2678598720,  1970740800,   949173120,   287400960,   49351680,   3628800;
  9: 883014232, 6897209780, 24030641070, 49125180240, 65163752640, 58504965120, 35924212800, 14852678400, 3940876800, 602380800, 39916800;

%F Formula

The row polynomials satisfy (n^2+3*n+2+(2*n^2+4*n)*x+(n^2+n)*x^2)*P(n,x) = (2*n^3+9*n^2+13*n+4+(6*n^3+22*n^2+22*n+2)*x+(6*n^3+17*n^2+11*n)*x^2+(2*n^3+4*n^2+2*n)*x^3)*P(n-1,x) - (n^2+3*n+4+(4*n^2+10*n+10)*x+(6*n^2+12*n+8)*x^2+(4*n^2+6*n+2)*x^3+(n^2+n)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 2*x.

%t Mathematica

P[0, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 9*n^2 + 13*n + 4 + (6*n^3 + 22*n^2 + 22*n + 2)*x + (6*n^3 + 17*n^2 + 11*n)*x^2 + (2*n^3 + 4*n^2 + 2*n)*x^3)*P[n - 1, x] - (n^2 + 3*n + 4 + (4*n^2 + 10*n + 10)*x + (6*n^2 + 12*n + 8)*x^2 + (4*n^2 + 6*n + 2)*x^3 + (n^2 + n)*x^4)*D[P[n - 1, x], x])/(n^2 + 3*n + 2 + (2*n^2 + 4*n)*x + (n^2 + n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^3 + 9*n^2 + 13*n + 4 + (6*n^3 + 22*n^2 + 22*n + 2)*x + (6*n^3 + 17*n^2 + 11*n)*x^2 + (2*n^3 + 4*n^2 + 2*n)*x^3)*v[i - 1] - (n^2 + 3*n + 4 + (4*n^2 + 10*n + 10)*x + (6*n^2 + 12*n + 8)*x^2 + (4*n^2 + 6*n + 2)*x^3 + (n^2 + n)*x^4)*deriv(v[i - 1], x))/(n^2 + 3*n + 2 + (2*n^2 + 4*n)*x + (n^2 + n)*x^2)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, 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-b96b8d87f5e7b835

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-b96b8d87f5e7b835

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - preserver_image affine, shift 0 e-cee0a9f908e74de4
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangular array A066667 or A008297 unsigned and transposed.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-db00758dc4aca02d root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  - ↵ preserver_image affine, shift 0 e-cee0a9f908e74de4 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

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