A371259 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(26 + x + 10\,n + n^{2}\right)\,P_{n-1}(x) + \left(11 + 11\,x + 2\,n + 2\,n x\right)\,P_{n-1}'(x) + \left(1 + 2\,x + x^{2}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	37	1														38
2	1865	102	1													1968
3	122961	10267	203	1												133432
4	10278281	1168460	35102	348	1											11482192
5	1062714245	154447181	6284362	94370	545	1										1223540704
6	133215991777	23772339810	1230439231	24938492	217695	802	1									158243927808
7	19913088181377	4241107108903	268341129461	6893548643	81018467	450485	1127	1								24429511438464
8	3500271565033393	871016335435864	65528274612492	2050686578056	30467772758	227676328	858732	1528	1							4438897557969152
9	714944028588443461	204425020518981497	17915166495346948	664497423961412	11994858607046	113087610958	572466452	1534452	2013	1						937960821546954240
10	167950356430067342489	54438562275203355230	5469739471988829229	235654330108867816	5028705497842434	57483881795380	366579765522	1317400424	2601765	2590	1					228099399064647802880
11	44970653636552995400721	16339429105258911612115	1858321865779631291439	91562234706698305925	2263960427904702410	30531214161722878	235690194905950	1065780440362	2819761109	4223615	3267	1				63262261570699102369792
12	13617086217015989835215993	5493350304321632174740132	699791893503303997335178	38946678117625257620164	1098804025591194368415	17135280411743399944	155797817133377580	851751159449832	2832889827911	5680880084	6609130	4052	1			19850291188916961072828416
13	4630206565577103694905252965	2057027344143410300429008645	290913526655921536125847662	18103560619503588164813926	575726742556510792616239	10224473401795505709239	107322930187223918996	690150271022126148	2772327947567347	6983567756451	10871587870	10021622	4953	1		6996837056227757206566232064
14	1756940670213816382344684605809	853461702116587221658246111450	132983074166887370270826066571	9174975730243369648845709572	325644705111781868810661641	6506829823178944348330182	77655166596117641807003	575730633535850340792	2717407517228641003	8260626543845702	16143949429465	19904695044	14787227	5978	1	2752892651996092218696895037440

Technical details

Node id: n-f6bd524925313b17 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-a8bb4bb33283a6b7

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 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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:                  1;
  1:                 37,                  1;
  2:               1865,                102,                 1;
  3:             122961,              10267,               203,               1;
  4:           10278281,            1168460,             35102,             348,              1;
  5:         1062714245,          154447181,           6284362,           94370,            545,            1;
  6:       133215991777,        23772339810,        1230439231,        24938492,         217695,          802,         1;
  7:     19913088181377,      4241107108903,      268341129461,      6893548643,       81018467,       450485,      1127,       1;
  8:   3500271565033393,    871016335435864,    65528274612492,   2050686578056,    30467772758,    227676328,    858732,    1528,    1;
  9: 714944028588443461, 204425020518981497, 17915166495346948, 664497423961412, 11994858607046, 113087610958, 572466452, 1534452, 2013, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n^2+10*n+26+x)*P(n-1,x) + (2*n+11+(2*n+11)*x)*d/dx P(n-1,x) + (1+2*x+x^2)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n^2 + 10*n + 26 + x)*P[n - 1, x] + (2*n + 11 + (2*n + 11)*x)*D[P[n - 1, x], x] + (1 + 2*x + x^2)*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n^2 + 10*n + 26 + x)*v[i - 1] + (2*n + 11 + (2*n + 11)*x)*deriv(v[i - 1], x) + (1 + 2*x + x^2)*deriv(deriv(v[i - 1], x), x)); 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows, (2, 3)-Lah numbers.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.