A371081 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(1 + 4\,x + 8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x^{2} - 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(x^{3}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	16														17
2	1	52	400													453
3	1	116	2948	14400												17465
4	1	216	12344	203072	705600											921233
5	1	360	38480	1437472	17154432	45158400										63789145
6	1	556	99320	6978592	191088544	1760601600	3657830400									5616599013
7	1	812	224420	26445312	1370470592	29277351936	216690624000	365783040000								613148157073
8	1	1136	459088	83896832	7320665792	297889587968	5164563101184	31569232896000	44259747840000							81298838448001
9	1	1536	869184	232641344	31566850240	2171980030720	72189720393984	1043823600728064	5379460107264000	6373403688960000						12871080897739073
10	1	2020	1546560	580314944	115550375424	12399639508480	699891949272064	19524392021587968	240239770271078400	1060747584712704000	1077105223434240000					2398329378160629861
11	1	2596	2615140	1328849984	371469265728	58619789678080	5176161811833344	246289383585736704	5882789064510001152	62562128774108774400	239745311783792640000	211112623793111040000				519554377953510437129
12	1	3272	4237640	2835170624	1074430907264	238410914290432	31027489059866624	2316754108319074304	94793256538960951296	1968585785675349147648	18320200527501228441600	61585912440444026880000	47500340353449984000000			129472180384695112970705
13	1	4056	6622928	5699815264	2846412547264	857283116874496	157146862719505152	17334058813294520320	1116481818307672719360	39885888401259729666048	728979669156302270742528	5997330883350842041958400	17845829035641773752320000	12160087130483195904000000		36773246580917492621295817
14	1	4956	10034024	10892190816	7001577874720	2781457998824960	692948810766065152	107850651739729487872	10286198930540473968640	580263088462173325836288	18318656454111843053469696	297589198545871750338969600	2182882277925295750899302400	5794208694678417891655680000	3514265180709643616256000000	11807854666147122586977709125

Technical details

Node id: n-21b764a934568fab · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-c0f485d760aa07f8

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: 1,   16;
  2: 1,   52,    400;
  3: 1,  116,   2948,     14400;
  4: 1,  216,  12344,    203072,      705600;
  5: 1,  360,  38480,   1437472,    17154432,      45158400;
  6: 1,  556,  99320,   6978592,   191088544,    1760601600,     3657830400;
  7: 1,  812, 224420,  26445312,  1370470592,   29277351936,   216690624000,     365783040000;
  8: 1, 1136, 459088,  83896832,  7320665792,  297889587968,  5164563101184,   31569232896000,   44259747840000;
  9: 1, 1536, 869184, 232641344, 31566850240, 2171980030720, 72189720393984, 1043823600728064, 5379460107264000, 6373403688960000;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n^2+8*n+4)*x)*P(n-1,x) - (4*n+3)*x^2*d/dx P(n-1,x) + 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[(1 + (4*n^2 + 8*n + 4)*x)*P[n - 1, x] - (4*n + 3)*x^2*D[P[n - 1, x], x] + 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] = (1 + (4*n^2 + 8*n + 4)*x)*v[i - 1] - (4*n + 3)*x^2*deriv(v[i - 1], x) + 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 (−∞, 0]
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

Related ?

Same fingerprint ?

A371081: Triangle read by rows, (2, 2)-Lah numbers.

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.