A159854 → Veronese 2 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-400 + 148\,x - 36\,x^{2} + 400\,n + 62\,n x - 24\,n x^{2} - 152\,n^{2} x + 23\,n^{2} x^{2} - 58\,n^{3} x + 27\,n^{3} x^{2} + 9\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(-1200 + 880\,x - 592\,x^{2} + 144\,x^{3} + 400\,n + 1614\,n x - 1122\,n x^{2} + 312\,n x^{3} - 36\,n^{2} x - 611\,n^{2} x^{2} + 241\,n^{2} x^{3} - 58\,n^{3} x - 83\,n^{3} x^{2} + 87\,n^{3} x^{3} + 7\,n^{4} x^{2} + 15\,n^{4} x^{3} + n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(800\,x - 1632\,x^{2} + 1120\,x^{3} - 288\,x^{4} - 652\,n x^{2} + 988\,n x^{3} - 336\,n x^{4} - 116\,n^{2} x^{2} + 262\,n^{2} x^{3} - 146\,n^{2} x^{4} + 28\,n^{3} x^{3} - 28\,n^{3} x^{4} + 2\,n^{4} x^{3} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 4\,x\)

\(P_{1}(x) = 4\,x\)

Rows ?

n	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	4								4
1	4								4
2	4	9							13
3	4	27							31
4	4	54	16						74
5	4	90	80						174
6	4	135	240	25					404
7	4	189	560	175					928
8	4	252	1120	700	36				2112
9	4	324	2016	2100	324				4768
10	4	405	3360	5250	1620	49			10688
11	4	495	5280	11550	5940	539			23808
12	4	594	7920	23100	17820	3234	64		52736
13	4	702	11440	42900	46332	14014	832		116224
14	4	819	16016	75075	108108	49049	5824	81	254976

Technical details

Node id: n-6d6e7a6a8b02d534 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-43f956d2ea970306

Observed on the stored rows ?

Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half 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: 0, 4;
  1: 0, 4;
  2: 0, 4,   9;
  3: 0, 4,  27;
  4: 0, 4,  54,   16;
  5: 0, 4,  90,   80;
  6: 0, 4, 135,  240,   25;
  7: 0, 4, 189,  560,  175;
  8: 0, 4, 252, 1120,  700,  36;
  9: 0, 4, 324, 2016, 2100, 324;

%F Formula

The row polynomials satisfy (400*n-400-(58*n^3+152*n^2-62*n-148)*x+(n^5+9*n^4+27*n^3+23*n^2-24*n-36)*x^2)*P(n,x) = (400*n-1200-(58*n^3+36*n^2-1614*n-880)*x+(n^5+7*n^4-83*n^3-611*n^2-1122*n-592)*x^2+(n^5+15*n^4+87*n^3+241*n^2+312*n+144)*x^3)*P(n-1,x) + (800*x-(116*n^2+652*n+1632)*x^2+(2*n^4+28*n^3+262*n^2+988*n+1120)*x^3-(2*n^4+28*n^3+146*n^2+336*n+288)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 4*x, P(1,x) = 4*x.

%t Mathematica

P[0, x_] := 4*x;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((400*n - 1200 - (58*n^3 + 36*n^2 - 1614*n - 880)*x + (n^5 + 7*n^4 - 83*n^3 - 611*n^2 - 1122*n - 592)*x^2 + (n^5 + 15*n^4 + 87*n^3 + 241*n^2 + 312*n + 144)*x^3)*P[n - 1, x] + (800*x - (116*n^2 + 652*n + 1632)*x^2 + (2*n^4 + 28*n^3 + 262*n^2 + 988*n + 1120)*x^3 - (2*n^4 + 28*n^3 + 146*n^2 + 336*n + 288)*x^4)*D[P[n - 1, x], x])/(400*n - 400 - (58*n^3 + 152*n^2 - 62*n - 148)*x + (n^5 + 9*n^4 + 27*n^3 + 23*n^2 - 24*n - 36)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 4*x; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((400*n - 1200 - (58*n^3 + 36*n^2 - 1614*n - 880)*x + (n^5 + 7*n^4 - 83*n^3 - 611*n^2 - 1122*n - 592)*x^2 + (n^5 + 15*n^4 + 87*n^3 + 241*n^2 + 312*n + 144)*x^3)*v[i - 1] + (800*x - (116*n^2 + 652*n + 1632)*x^2 + (2*n^4 + 28*n^3 + 262*n^2 + 988*n + 1120)*x^3 - (2*n^4 + 28*n^3 + 146*n^2 + 336*n + 288)*x^4)*deriv(v[i - 1], x))/(400*n - 400 - (58*n^3 + 152*n^2 - 62*n - 148)*x + (n^5 + 9*n^4 + 27*n^3 + 23*n^2 - 24*n - 36)*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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from veronese_section_2_0 of Riordan array (1-x,x/(1-x)).?
  - Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    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

[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
[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
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.