A290315 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(-13 - 14\,x + 8\,n + 8\,n x\right)\,P_{n-1}(x) + \left(-2 - 6\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	2														5
3	29	40	12													81
4	471	994	644	120												2229
5	10729	30544	30808	12672	1680											86433
6	314427	1127330	1548872	1002992	297264	30240										4321125
7	11265701	48721656	84835620	75358336	35435376	8121984	665280									264403953
8	477107439	2416417090	5097635340	5766438120	3739152080	1367381088	253638720	17297280								19135067157
9	23316504721	135351795872	335476755120	461392513280	382452665440	193654774272	57552009984	8917770240	518918400							1598633707329
10	1291502224563	8453441839298	24074677795984	39006263624160	39424112588320	25597235800256	10567383142656	2631825980928	348734718720	17643225600						151412820940485
11	79955783163629	582559940345288	1874408048152892	3496892860405504	4173045728699680	3311665543422208	1757533112044416	609639659003904	130162254467328	15017365923840	670442572800					16031550738201489
12	5471210121890631	43915857433393186	157541289603354772	332619865660074328	457875542787444320	429848894551022912	279392364732209792	124849550446258944	37205907905938176	6930207483701760	706219011671040	28158588057600				1876385068325017461
13	410048406225261049	3595091634395158512	14226806423976395016	33536631598360962688	52323521942734337904	56750864357096759808	43708108972595517184	23962229517697523712	9215782432916764416	2400235955207958528	395504504544430080	36014915284992000	1295295050649600			240562135956086710497
14	33404608947510526827	317621040112919744354	1374450982928925490392	3578291562083238825936	6240905819593150535888	7681841132272745928672	6844707071879299515648	4452794627554158892544	2104156723123838391552	708342168465975983616	163446976763856377856	24096666386766090240	1979778391757107200	64764752532480000		33526103923256675890725
15	2939051077157786881781	30122809668125308061848	141500440097266302555732	403236898795697749279616	777536178539328484829968	1071046806085833390549888	1084273414693128190981184	817031123115141474033664	458713068581143291014912	189980792904583640549376	56735833040010714627072	11727949106869760065536	1561601676125151866880	116705324545354137600	3497296636753920000	5046526170001593353355057

Technical details

Node id: n-d1f210a2bc5f4d66 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-a562d7fe2342b9e0

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:             3,             2;
   3:            29,            40,             12;
   4:           471,           994,            644,            120;
   5:         10729,         30544,          30808,          12672,           1680;
   6:        314427,       1127330,        1548872,        1002992,         297264,          30240;
   7:      11265701,      48721656,       84835620,       75358336,       35435376,        8121984,         665280;
   8:     477107439,    2416417090,     5097635340,     5766438120,     3739152080,     1367381088,      253638720,      17297280;
   9:   23316504721,  135351795872,   335476755120,   461392513280,   382452665440,   193654774272,    57552009984,    8917770240,    518918400;
  10: 1291502224563, 8453441839298, 24074677795984, 39006263624160, 39424112588320, 25597235800256, 10567383142656, 2631825980928, 348734718720, 17643225600;

%F Formula

The row polynomials satisfy P(n,x) = (8*n-13+(8*n-14)*x)*P(n-1,x) - (2+6*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[(8*n - 13 + (8*n - 14)*x)*P[n - 1, x] - (2 + 6*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] = (8*n - 13 + (8*n - 14)*x)*v[i - 1] - (2 + 6*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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, -1]
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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

References

[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