A191935 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + n + n x + n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(-2\,n - 4\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	13	16	4													33
3	81	148	76	8												313
4	669	1648	1348	384	16											4065
5	6955	21630	24500	11888	2096	32										67101
6	88505	333408	484568	334720	107024	12032	64									1360321
7	1346209	5972936	10646120	9624720	4542544	1007936	70848	128								33211441
8	23998521	122766624	259945144	291980928	184399984	63216896	10042432	421888	256							956772673
9	493956467	2865562458	7039311928	9474847488	7533165328	3535665056	912867136	105749248	2523904	512						31963649525
10	11596542533	75302625264	210809551996	331358112896	318619721904	190914111488	69128982528	13730353152	1165529344	15126528	1024					1222640658657
11	307301505073	2210062964380	6952823174676	12518720515416	14155609976624	10361280982656	4864924153856	1392847046656	215437436160	13275456512	90721280	2048				52992373935337
12	9110471500693	71929647664912	251423875888476	510779906247936	665111322020384	576688127367168	334334765186048	126291255390208	29082767581440	3528218652672	154497385472	544243712	4096			2578435399133217
13	299893197116059	2580070803847318	9926595975615180	22479240048393200	33163888269376864	33294459040702272	23038814010889216	10867448843692032	3364249662539520	630640781071872	60236995490816	1821858590720	3265277952	8192		139707362752611213
14	10888674034993905	101434187500306880	426202279778376944	1065216287489357056	1757636813654064992	2007782936850841088	1617819385325852800	918093077515923456	359162170007112960	92337180174196736	14201012694067200	1068296433467392	21650060873728	19591266304	16384	8371863971110717825

Technical details

Node id: n-0b7ea7d96efe8554 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-af2a7605ed60a01d

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:         3,          2;
  2:        13,         16,          4;
  3:        81,        148,         76,          8;
  4:       669,       1648,       1348,        384,         16;
  5:      6955,      21630,      24500,      11888,       2096,         32;
  6:     88505,     333408,     484568,     334720,     107024,      12032,        64;
  7:   1346209,    5972936,   10646120,    9624720,    4542544,    1007936,     70848,       128;
  8:  23998521,  122766624,  259945144,  291980928,  184399984,   63216896,  10042432,    421888,     256;
  9: 493956467, 2865562458, 7039311928, 9474847488, 7533165328, 3535665056, 912867136, 105749248, 2523904, 512;

%F Formula

The row polynomials satisfy P(n,x) = (n^2+n+1+(n^2+n)*x)*P(n-1,x) - (2*n+4*n*x+2*n*x^2)*d/dx P(n-1,x) + (1+3*x+3*x^2+x^3)*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 + n + 1 + (n^2 + n)*x)*P[n - 1, x] - (2*n + 4*n*x + 2*n*x^2)*D[P[n - 1, x], x] + (1 + 3*x + 3*x^2 + x^3)*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 + n + 1 + (n^2 + n)*x)*v[i - 1] - (2*n + 4*n*x + 2*n*x^2)*deriv(v[i - 1], x) + (1 + 3*x + 3*x^2 + x^3)*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 of Legendre-Stirling numbers of the second kind.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A071951 → x ↦ x/(x+1)

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.