A122542 → Veronese 2 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 0\)

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

\(P_{4}(x) = 4 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2									0
3	1	1							2
4	4	4							8
5	9	10	1						20
6	20	28	8						56
7	49	83	35	1					168
8	120	232	124	12					488
9	289	628	414	76	1				1408
10	696	1688	1332	356	16				4088
11	1681	4501	4122	1434	133	1			11872
12	4060	11884	12372	5316	788	20			34440
13	9801	31134	36327	18620	3831	206	1		99920
14	23660	81076	104804	62412	16484	1484	24		289944
15	57121	210023	297877	202083	65363	8549	295	1	841312

Technical details

Node id: n-841f1de94bed5ffc · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-421c7df94368dc95

Observed on the stored rows ?

Degrees: 0, -1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   1;
   2:   0;
   3:   1,    1;
   4:   4,    4;
   5:   9,   10,    1;
   6:  20,   28,    8;
   7:  49,   83,   35,   1;
   8: 120,  232,  124,  12;
   9: 289,  628,  414,  76,  1;
  10: 696, 1688, 1332, 356, 16;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 0;
P[3, x_] := 1 + x;
P[4, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 4, x] + (2 + 2*x)*P[n - 3, x] + x*P[n - 2, x] + 2*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 0; v[3] = 1 + x; v[4] = 4 + 4*x; for(n = 5, N, my(i = n); v[i] = (1 + x)*v[i - 4] + (2 + 2*x)*v[i - 3] + x*v[i - 2] + 2*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 veronese_section_2_0 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0…?
  - Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check

References

[ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
[AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
[Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
[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