A119308 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(3 + n\right)\,P_{n}(x) = \left(-2\,x - 2\,x^{2} + n x + n x^{2}\right)\,P_{n-3}(x) + \left(-2\,x - 4\,x^{2} + n x + 2\,n x^{2}\right)\,P_{n-3}'(x) + \left(-2\,x^{2} + n x^{2}\right)\,P_{n-3}''(x) + \left(2 + x - 4\,n - 3\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(x - 3\,n x - 2\,n x^{2}\right)\,P_{n-2}'(x) + \left(-n x^{2}\right)\,P_{n-2}''(x) + \left(7 + 3\,x + 5\,n + 2\,n x\right)\,P_{n-1}(x) + \left(3\,x + 2\,n x\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + x\)

\(P_{2}(x) = 9 + 8\,x + x^{2}\)

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	1														4
2	9	8	1													18
3	28	48	15	1												92
4	90	263	149	24	1											527
5	297	1391	1255	355	35	1										3334
6	1001	7256	9731	4240	720	48	1									22997
7	3432	37696	72051	45045	11550	1309	63	1								171147
8	11934	195971	519205	445662	160587	27118	2198	80	1							1362756
9	41990	1021942	3680808	4210753	2038491	477141	57078	3474	99	1						11531777
10	149226	5351847	25839691	38569845	24331000	7543116	1238517	110460	5235	120	1					103139058
11	534888	28160364	180367242	345807946	278069660	110690503	23829729	2895717	199980	7590	143	1				970563763
12	1931540	148895533	1255198011	3054181812	3079362979	1538976252	420999238	66571494	6228255	342925	10659	168	1			9572698867
13	7020405	791033595	8723992069	26689341185	33313778069	20552507547	6983610348	1392356680	168517206	12512214	562133	14573	195	1		98635246220
14	25662825	4221725592	60628320663	231479927952	354121235832	266169879976	110427908591	27130221404	4115196514	393417024	23745722	887068	19474	224	1	1058738148862

Technical details

Node id: n-45d3806cddeb43d7 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-be0e79fafd04dc28

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,       1;
  2:     9,       8,       1;
  3:    28,      48,      15,       1;
  4:    90,     263,     149,      24,       1;
  5:   297,    1391,    1255,     355,      35,      1;
  6:  1001,    7256,    9731,    4240,     720,     48,     1;
  7:  3432,   37696,   72051,   45045,   11550,   1309,    63,    1;
  8: 11934,  195971,  519205,  445662,  160587,  27118,  2198,   80,  1;
  9: 41990, 1021942, 3680808, 4210753, 2038491, 477141, 57078, 3474, 99, 1;

%F Formula

The row polynomials satisfy (n+3)*P(n,x) = ((n-2)*x+(n-2)*x^2)*P(n-3,x) + ((n-2)*x+(2*n-4)*x^2)*d/dx P(n-3,x) + (n-2)*x^2*d^2/dx^2 P(n-3,x) - (4*n-2+(3*n-1)*x+n*x^2)*P(n-2,x) - ((3*n-1)*x+2*n*x^2)*d/dx P(n-2,x) - n*x^2*d^2/dx^2 P(n-2,x) + (5*n+7+(2*n+3)*x)*P(n-1,x) + (2*n+3)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + x, P(2,x) = 9 + 8*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[2, x_] := 9 + 8*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((n - 2)*x + (n - 2)*x^2)*P[n - 3, x] + ((n - 2)*x + (2*n - 4)*x^2)*D[P[n - 3, x], x] + (n - 2)*x^2*D[P[n - 3, x], {x, 2}] - (4*n - 2 + (3*n - 1)*x + n*x^2)*P[n - 2, x] - ((3*n - 1)*x + 2*n*x^2)*D[P[n - 2, x], x] - n*x^2*D[P[n - 2, x], {x, 2}] + (5*n + 7 + (2*n + 3)*x)*P[n - 1, x] + (2*n + 3)*x*D[P[n - 1, x], x])/(n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; v[3] = 9 + 8*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = (((n - 2)*x + (n - 2)*x^2)*v[i - 3] + ((n - 2)*x + (2*n - 4)*x^2)*deriv(v[i - 3], x) + (n - 2)*x^2*deriv(deriv(v[i - 3], x), x) - (4*n - 2 + (3*n - 1)*x + n*x^2)*v[i - 2] - ((3*n - 1)*x + 2*n*x^2)*deriv(v[i - 2], x) - n*x^2*deriv(deriv(v[i - 2], x), x) + (5*n + 7 + (2*n + 3)*x)*v[i - 1] + (2*n + 3)*x*deriv(v[i - 1], x))/(n + 3)); 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 Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from Triangle for first differences of Catalan numbers.?
  - 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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.