A075500 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + x\right)\,P_{n-1}(x) + \left(5 + 5\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 41 + 17\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	41	17	1															59
1	331	238	33	1														603
2	3176	3279	781	54	1													7291
3	35451	47055	16585	1935	80	1												101107
4	447981	718906	341440	58820	4040	111	1											1571299
5	6282416	11770347	7064246	1657460	166635	7511	147	1										26948763
6	96546231	206398693	149660183	45203606	6177745	403886	12838	188	1									504403371
7	1611270851	3863533684	3279015711	1222490809	215922126	19083351	872634	20586	234	1								10212209987
8	28985293526	76900298485	74665147195	33269765220	7313549860	833685027	51218685	1726020	31395	285	1							222020715699
9	558413253581	1621540048811	1770589131905	919601214015	244264187255	34694346247	2737968387	123241305	3183015	45980	341	1						5151966620843
10	11458179765541	36101245109552	43744984369516	25967498371320	8137828738820	1401927972522	137574653854	7918182492	271729095	5545045	65131	402	1					126957434503291
11	249255304141006	846021719665773	1125533470591608	751819024943636	272599244893690	55724835502622	6631751898486	473089299826	20667247887	557782940	9214491	89713	468	1				3308079675272147
12	5725640423174901	20810828926380589	30131842623286041	22365717892281764	9232520279745126	2199021702429552	311026029342028	26855103106832	1448882934928	49574902377	1078728651	14715064	120666	539	1			90804952531149059
13	138407987170952351	535963184846221790	838406079283183705	684462323956105605	317386787726393840	86732973935322078	14335887367481680	1470040574119260	95734588720485	4031139388705	110793035353	1983587515	22714055	159005	615	1		2617265143387386043
14	3510263847256823056	14418063813311229091	24217395312718745145	21559850636889278210	11104843153225397395	3436185600745828658	654276339258836658	78436934455190740	6055232927753295	306862349312090	10344646681798	233154717068	3493050470	34046460	205820	696	1	78985688314467096651

Technical details

Node id: n-f20e10cf94d19917 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-a670175e1740e9b4

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:           41,            17,             1;
  1:          331,           238,            33,            1;
  2:         3176,          3279,           781,           54,            1;
  3:        35451,         47055,         16585,         1935,           80,           1;
  4:       447981,        718906,        341440,        58820,         4040,         111,          1;
  5:      6282416,      11770347,       7064246,      1657460,       166635,        7511,        147,         1;
  6:     96546231,     206398693,     149660183,     45203606,      6177745,      403886,      12838,       188,       1;
  7:   1611270851,    3863533684,    3279015711,   1222490809,    215922126,    19083351,     872634,     20586,     234,     1;
  8:  28985293526,   76900298485,   74665147195,  33269765220,   7313549860,   833685027,   51218685,   1726020,   31395,   285,   1;
  9: 558413253581, 1621540048811, 1770589131905, 919601214015, 244264187255, 34694346247, 2737968387, 123241305, 3183015, 45980, 341, 1;

%F Formula

The row polynomials satisfy P(n,x) = (6+x)*P(n-1,x) + (5+5*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 41 + 17*x + x^2.

%t Mathematica

P[0, x_] := 41 + 17*x + x^2;
P[n_, x_] := P[n, x] = Expand[(6 + x)*P[n - 1, x] + (5 + 5*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] = 41 + 17*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + x)*v[i - 1] + (5 + 5*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 Stirling2 triangle with scaled diagonals (powers of 5).?
  - 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.