A371081 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(10 + x + 6\,n + n^{2}\right)\,P_{n-1}(x) + \left(7 + 7\,x + 2\,n + 2\,n x\right)\,P_{n-1}'(x) + \left(1 + 2\,x + x^{2}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	17	1														18
2	453	54	1													508
3	17465	3183	119	1												20768
4	921233	228412	12998	220	1											1162864
5	63789145	20146261	1555082	39930	365	1										85530784
6	5616599013	2164114530	212625815	7381452	102115	562	1									8000823488
7	613148157073	279463647875	33549691985	1478512315	27579627	229313	819	1								927667819008
8	81298838448001	42821732772088	6103001738364	328020441032	7747076102	86675272	467068	1144	1							130559427619072
9	12871080897739073	7692523626220809	1273743818512804	81197992353684	2333334431166	32981037310	238768724	881508	1545	1						21920912889946624
10	2398329378160629861	1602662564691103182	303138039912181117	22450287619069704	763643821982794	13105215236980	119656053602	592760264	1564785	2030	1					4327357139670584320
11	519554377953510437129	383529329198619596471	81740881022273344475	6920893688824726037	273062584263927114	5535756150243510	61257502555110	382194522490	1352503229	2641155	2607	1				992024141844694499328
12	129472180384695112970705	104523015491601735821748	24818355413995652853474	2371555307484375389572	106850859366740479055	2507986198310076264	32726688571052444	247108937288360	1100138677055	2877727204	4273698	3284	1			261294498418242456612864
13	36773246580917492621295817	32192660883773800453358237	8435013374483355467191950	900040270508517260136398	45733868884858004698067	1223834623246491829415	18458331078282260084	164108284672814612	883158416278879	2903775853979	5772935454	6671678	4069	1		78347937436516701225328640
14	11807854666147122586977709125	11128979265142222048095745530	3191521020522863295069179507	376875993037980419388958836	21378088511546099547808757	644284303098499010400726	11060791177264757298507	113495218505539098264	718298805944560087	2852075059097862	7122055637697	11012986036	10098543	4970	1	26527264492672386732468584448

Technical details

Node id: n-09268dddee3be1b7 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-78aed4e5d7efa0b0

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:                17,                1;
  2:               453,               54,                1;
  3:             17465,             3183,              119,              1;
  4:            921233,           228412,            12998,            220,             1;
  5:          63789145,         20146261,          1555082,          39930,           365,           1;
  6:        5616599013,       2164114530,        212625815,        7381452,        102115,         562,         1;
  7:      613148157073,     279463647875,      33549691985,     1478512315,      27579627,      229313,       819,      1;
  8:    81298838448001,   42821732772088,    6103001738364,   328020441032,    7747076102,    86675272,    467068,   1144,    1;
  9: 12871080897739073, 7692523626220809, 1273743818512804, 81197992353684, 2333334431166, 32981037310, 238768724, 881508, 1545, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n^2+6*n+10+x)*P(n-1,x) + (2*n+7+(2*n+7)*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 + 6*n + 10 + x)*P[n - 1, x] + (2*n + 7 + (2*n + 7)*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 + 6*n + 10 + x)*v[i - 1] + (2*n + 7 + (2*n + 7)*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, 2)-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.