A391533 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(4 + 5\,x + 2\,x^{2} + n + 2\,n x\right)\,P_{n}(x) = \left(5 + 12\,x + 9\,x^{2} + 2\,x^{3} + n + 3\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 5 + 12\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	5	12	4															21
1	6	21	20	4														51
2	7	32	49	28	4													120
3	8	45	94	89	36	4												276
4	9	60	158	208	141	44	4											624
5	10	77	244	410	390	205	52	4										1392
6	11	96	355	724	885	656	281	60	4									3072
7	12	117	494	1183	1764	1687	1022	369	68	4								6720
8	13	140	664	1824	3206	3752	2940	1504	469	76	4							14592
9	14	165	868	2688	5436	7518	7224	4788	2118	581	84	4						31488
10	15	192	1109	3820	8730	13920	15834	12888	7395	2880	705	92	4					67584
11	16	221	1390	5269	13420	24222	31812	30690	21648	10945	3806	841	100	4				144384
12	17	252	1714	7088	19899	40084	59664	66528	55671	34628	15642	4912	989	108	4			307200
13	18	285	2084	9334	28626	63635	105820	133848	129558	95667	53196	21710	6214	1149	116	4		651264
14	19	320	2503	12068	40131	97552	179179	253396	278421	237952	157157	78988	29393	7728	1321	124	4	1376256

Technical details

Node id: n-786f9b37e5a81d53 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-b6b17fe263ea2de9

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:  5,  12,   4;
  1:  6,  21,  20,    4;
  2:  7,  32,  49,   28,    4;
  3:  8,  45,  94,   89,   36,    4;
  4:  9,  60, 158,  208,  141,   44,    4;
  5: 10,  77, 244,  410,  390,  205,   52,    4;
  6: 11,  96, 355,  724,  885,  656,  281,   60,    4;
  7: 12, 117, 494, 1183, 1764, 1687, 1022,  369,   68,   4;
  8: 13, 140, 664, 1824, 3206, 3752, 2940, 1504,  469,  76,  4;
  9: 14, 165, 868, 2688, 5436, 7518, 7224, 4788, 2118, 581, 84, 4;

%F Formula

The row polynomials satisfy (n+4+(2*n+5)*x+2*x^2)*P(n,x) = (n+5+(3*n+12)*x+(2*n+9)*x^2+2*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 5 + 12*x + 4*x^2.

%t Mathematica

P[0, x_] := 5 + 12*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 5 + (3*n + 12)*x + (2*n + 9)*x^2 + 2*x^3)*P[n - 1, x])/(n + 4 + (2*n + 5)*x + 2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 5 + 12*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 5 + (3*n + 12)*x + (2*n + 9)*x^2 + 2*x^3)*v[i - 1])/(n + 4 + (2*n + 5)*x + 2*x^2)); 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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    no recorded property covers this requirement
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

A159854 → basis A013609 → D+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
[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.