A134991 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

\(P_{2}(x) = 2 + 13\,x + 15\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	2	3															5
2	2	13	15														30
3	2	35	120	105													262
4	2	81	595	1365	945												2988
5	2	175	2408	10710	18270	10395											41960
6	2	365	8743	66430	207900	280665	135135										699240
7	2	747	29760	359975	1827210	4369365	4864860	2027025									13478944
8	2	1513	97251	1790195	13758745	51606555	99324225	93918825	34459425								294956736
9	2	3047	309408	8402108	93553460	514383870	1516875360	2435132700	1998646650	654729075							7222035680
10	2	6117	966823	37865828	592311720	4571036470	19315145850	46690223580	64163449350	46485764325	13749310575						195606080640
11	2	12259	2984072	165730565	3560496940	37414086710	217017801000	739087278930	1508932274850	1810325892375	1173274502400	316234143225					5806015203328
12	2	24545	9132019	709935681	20588953385	288062584810	2225888094430	10235161386450	29056876348500	51237569708325	54497684015775	31939648465725	7905853580625				187408052230272
13	2	49119	27780600	2992734706	115588829538	2116516006605	21309158430560	128425816138620	485990697342150	1179237120086025	1827411702316200	1744558356791250	932890722513750	213458046676875			6535516745696000
14	2	98269	84160455	12464109946	634230951068	14993333293575	193356192494465	1494181251422860	7318169626177410	23428537793736075	49532983652777625	68394066382892250	59199031611720000	29101447030280625	6190283353629375		244867697007744000
15	2	196571	254217456	51434587507	3418019375590	103180800241155	1681484916571460	16381360043483435	101604991274696910	416781238037789625	1152875038772457000	2156345281676210625	2682908334730874250	2123267190294875625	965684203166182500	191898783962510625	9809534557384270336

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Technical details

Node id: n-dffea903f52b4d20 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-4ef688dcc1f7919d

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: 2,    3;
   2: 2,   13,     15;
   3: 2,   35,    120,      105;
   4: 2,   81,    595,     1365,       945;
   5: 2,  175,   2408,    10710,     18270,      10395;
   6: 2,  365,   8743,    66430,    207900,     280665,      135135;
   7: 2,  747,  29760,   359975,   1827210,    4369365,     4864860,     2027025;
   8: 2, 1513,  97251,  1790195,  13758745,   51606555,    99324225,    93918825,    34459425;
   9: 2, 3047, 309408,  8402108,  93553460,  514383870,  1516875360,  2435132700,  1998646650,   654729075;
  10: 2, 6117, 966823, 37865828, 592311720, 4571036470, 19315145850, 46690223580, 64163449350, 46485764325, 13749310575;

%F Formula

The row polynomials satisfy P(n,x) = (1+n*x)*P(n-2,x) + (x+x^2)*d/dx P(n-2,x) + (n+2)*x*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + 3*x, P(2,x) = 2 + 13*x + 15*x^2.

%t Mathematica

P[1, x_] := 2 + 3*x;
P[2, x_] := 2 + 13*x + 15*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + n*x)*P[n - 2, x] + (x + x^2)*D[P[n - 2, x], x] + (n + 2)*x*P[n - 1, x] + (x + 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] = 2 + 3*x; v[2] = 2 + 13*x + 15*x^2; for(n = 3, N, my(i = n); v[i] = (1 + n*x)*v[i - 2] + (x + x^2)*deriv(v[i - 2], x) + (n + 2)*x*v[i - 1] + (x + 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-72ea2d9fc8c7fc63

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-72ea2d9fc8c7fc63

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of Ward numbers T(n,k) read by rows.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check ma_wang_interval e-c13d1846cdae7550 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
- premise (sequence): Triangle of Ward numbers T(n,k) read by rows.
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  
  no recorded property covers this requirement

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A269939 → Pₙ + Pₙ₊₁

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.