A359363 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(6 + 5\,n + n^{2}\right)\,P_{n}(x) = \left(-6 - 3\,x + 7\,n + 5\,n x + 11\,n^{2} + 10\,n^{2} x\right)\,P_{n-1}(x) + \left(12 + 15\,x + 3\,x^{2} - 12\,n - 27\,n x - 15\,n x^{2}\right)\,P_{n-1}'(x) + \left(12 + 30\,x + 24\,x^{2} + 6\,x^{3}\right)\,P_{n-1}''(x)\)

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

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	1														2
2	2	3	1													6
3	6	12	7	1												26
4	22	55	46	14	1											138
5	92	276	300	140	25	1										834
6	422	1477	1975	1245	365	41	1									5526
7	2074	8296	13167	10465	4305	847	63	1								39218
8	10754	48393	88900	85316	45346	12992	1792	92	1							293586
9	58202	291010	607308	683172	445998	169134	35196	3516	129	1						2293666
10	326240	1794320	4192860	5410470	4192860	1986390	560154	87240	6480	175	1					18557190
11	1882960	11297760	29222600	42550200	38194200	21750960	7791630	1682010	200640	11330	231	1				154584522
12	11140560	72413640	205396400	333130160	340028040	226475040	98741280	27522000	4648710	432795	18942	298	1			1319947866
13	67329992	471309944	1454599016	2600564824	2975285456	2270803392	1169121096	401499384	88950576	11961664	883168	30472	377	1		11512339362
14	414499438	3108745785	10371260991	20263885369	25688834171	22114333241	13146932799	5382483249	1486909281	266234969	28915887	1716897	47411	469	1	102274799958

Technical details

Node id: n-754d50850bfffa7d · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-72cfe65083d14ac3

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,      1;
  2:     2,      3,      1;
  3:     6,     12,      7,      1;
  4:    22,     55,     46,     14,      1;
  5:    92,    276,    300,    140,     25,      1;
  6:   422,   1477,   1975,   1245,    365,     41,     1;
  7:  2074,   8296,  13167,  10465,   4305,    847,    63,    1;
  8: 10754,  48393,  88900,  85316,  45346,  12992,  1792,   92,   1;
  9: 58202, 291010, 607308, 683172, 445998, 169134, 35196, 3516, 129, 1;

%F Formula

The row polynomials satisfy (n^2+5*n+6)*P(n,x) = (11*n^2+7*n-6+(10*n^2+5*n-3)*x)*P(n-1,x) - (12*n-12+(27*n-15)*x+(15*n-3)*x^2)*d/dx P(n-1,x) + (12+30*x+24*x^2+6*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[Together[((11*n^2 + 7*n - 6 + (10*n^2 + 5*n - 3)*x)*P[n - 1, x] - (12*n - 12 + (27*n - 15)*x + (15*n - 3)*x^2)*D[P[n - 1, x], x] + (12 + 30*x + 24*x^2 + 6*x^3)*D[P[n - 1, x], {x, 2}])/(n^2 + 5*n + 6)]];
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] = ((11*n^2 + 7*n - 6 + (10*n^2 + 5*n - 3)*x)*v[i - 1] - (12*n - 12 + (27*n - 15)*x + (15*n - 3)*x^2)*deriv(v[i - 1], x) + (12 + 30*x + 24*x^2 + 6*x^3)*deriv(deriv(v[i - 1], x), x))/(n^2 + 5*n + 6)); 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. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 …?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[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.