A075502 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + x\right)\,P_{n-1}(x) + \left(7 + 7\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 71 + 23\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	71	23	1															95
1	729	430	45	1														1205
2	8842	7809	1441	74	1													18167
3	125399	146151	41065	3615	110	1												316341
4	2026249	2892574	1125496	148980	7610	153	1											6201063
5	36458010	61171803	30782066	5658996	428295	14231	203	1										134513605
6	719866701	1384983979	857216171	206885210	21575701	1048754	24430	260	1									3191601207
7	15453821461	33495652780	24588329151	7461006889	1020316836	67698183	2282994	39306	324	1								82089147925
8	358100141148	862125221275	731120038771	269526400340	46561849390	4027224453	183773877	4541580	60105	395	1							2271649251335
9	8899677678109	23525659003067	22606820481377	9851117431551	2086705834235	227451003703	13438455723	446009817	8413185	88220	473	1						67211324399461
10	235877034446341	678280049463392	727749175655932	366716989354936	92933312593616	12431514133830	921228270886	39332153652	990011517	14709915	125191	558	1					2114949640919767
11	6634976621814472	20598566235580269	24389818690347552	13964674619570324	4147419237409162	666180007725518	60420203207078	3218601674002	103619615385	2043178852	24517779	172705	650	1				70465379904813749
12	197269776623577659	657070931820384235	850432744434203217	545493121305343484	186588067436693286	35330808104303712	3844901650105516	249683197086752	9978977326318	250401558407	3968863899	39252304	232596	749	1			2476290267958932135
13	6177654735731310917	21959392176008186222	30821946856785068173	21894199150516576261	8499241832676931700	1867298685226184574	239810367266830360	18635666614324188	904113044150745	28035308446145	562995369937	7324847803	60719295	306845	855	1		91479112189559254021
14	203136983117907790890	767075793370845058669	1159820405186127787509	903732493536719867342	393222359109802577551	98865120722609743502	14770964712669587786	1352673694755750796	78265125872079003	2934029619722630	72512960582062	1190707853972	12938945838	91180740	397580	968	1	3542058066323628926839

Technical details

Node id: n-2b086b257119fd8b · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-82f087c518e57ff3

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:            71,             23,              1;
  1:           729,            430,             45,             1;
  2:          8842,           7809,           1441,            74,             1;
  3:        125399,         146151,          41065,          3615,           110,            1;
  4:       2026249,        2892574,        1125496,        148980,          7610,          153,           1;
  5:      36458010,       61171803,       30782066,       5658996,        428295,        14231,         203,         1;
  6:     719866701,     1384983979,      857216171,     206885210,      21575701,      1048754,       24430,       260,       1;
  7:   15453821461,    33495652780,    24588329151,    7461006889,    1020316836,     67698183,     2282994,     39306,     324,     1;
  8:  358100141148,   862125221275,   731120038771,  269526400340,   46561849390,   4027224453,   183773877,   4541580,   60105,   395,   1;
  9: 8899677678109, 23525659003067, 22606820481377, 9851117431551, 2086705834235, 227451003703, 13438455723, 446009817, 8413185, 88220, 473, 1;

%F Formula

The row polynomials satisfy P(n,x) = (8+x)*P(n-1,x) + (7+7*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 71 + 23*x + x^2.

%t Mathematica

P[0, x_] := 71 + 23*x + x^2;
P[n_, x_] := P[n, x] = Expand[(8 + x)*P[n - 1, x] + (7 + 7*x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 71 + 23*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (8 + x)*v[i - 1] + (7 + 7*x)*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 - 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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - 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
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.