A391533 → Veronese 2 (j=1) → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(48\,x - 12\,x^{2} + 6\,x^{3} - 32\,n + 8\,n x - 32\,n x^{2} + 11\,n x^{3} + 16\,n^{2} x - 28\,n^{2} x^{2} + 6\,n^{2} x^{3} + 16\,n^{3} x - 8\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n}(x) = \left(-32 + 152\,x - 24\,x^{2} + 24\,x^{3} - 32\,n - 40\,n x - 88\,n x^{2} + 10\,n x^{3} + 6\,n x^{4} + 32\,n^{2} x - 12\,n^{2} x^{2} - 19\,n^{2} x^{3} + 5\,n^{2} x^{4} + 16\,n^{3} x + 8\,n^{3} x^{2} - 7\,n^{3} x^{3} + n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-64\,x + 112\,x^{2} - 104\,x^{3} + 68\,x^{4} - 12\,x^{5} + 96\,n x^{2} - 160\,n x^{3} + 74\,n x^{4} - 10\,n x^{5} + 32\,n^{2} x^{2} - 48\,n^{2} x^{3} + 18\,n^{2} x^{4} - 2\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 4\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | Pn(1) |
|---|---|---|---|---|---|---|
| 0 | 4 | 4 | ||||
| 1 | 8 | 2 | 10 | |||
| 2 | 12 | 12 | 24 | |||
| 3 | 16 | 38 | 3 | 57 | ||
| 4 | 20 | 88 | 24 | 132 | ||
| 5 | 24 | 170 | 102 | 4 | 300 | |
| 6 | 28 | 292 | 312 | 40 | 672 | |
| 7 | 32 | 462 | 777 | 212 | 5 | 1488 |
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Technical details
- Node id
n-83c12a3b577fa369· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-6e5b1fb41454792a
Observed on the stored rows ?
- Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 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 = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 4; 1: 8, 2; 2: 12, 12; 3: 16, 38, 3; 4: 20, 88, 24; 5: 24, 170, 102, 4; 6: 28, 292, 312, 40; 7: 32, 462, 777, 212, 5; 8: 36, 688, 1680, 800, 60; 9: 40, 978, 3276, 2424, 380, 6;
%F FormulaThe row polynomials satisfy -(32*n-(16*n^3+16*n^2+8*n+48)*x+(8*n^3+28*n^2+32*n+12)*x^2-(n^3+6*n^2+11*n+6)*x^3)*P(n,x) = -(32*n+32-(16*n^3+32*n^2-40*n+152)*x-(8*n^3-12*n^2-88*n-24)*x^2+(7*n^3+19*n^2-10*n-24)*x^3-(n^3+5*n^2+6*n)*x^4)*P(n-1,x) - (64*x-(32*n^2+96*n+112)*x^2+(48*n^2+160*n+104)*x^3-(18*n^2+74*n+68)*x^4+(2*n^2+10*n+12)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4.
%t MathematicaP[0, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(32*n + 32 - (16*n^3 + 32*n^2 - 40*n + 152)*x - (8*n^3 - 12*n^2 - 88*n - 24)*x^2 + (7*n^3 + 19*n^2 - 10*n - 24)*x^3 - (n^3 + 5*n^2 + 6*n)*x^4)*P[n - 1, x] - (64*x - (32*n^2 + 96*n + 112)*x^2 + (48*n^2 + 160*n + 104)*x^3 - (18*n^2 + 74*n + 68)*x^4 + (2*n^2 + 10*n + 12)*x^5)*D[P[n - 1, x], x])/(-32*n + (16*n^3 + 16*n^2 + 8*n + 48)*x - (8*n^3 + 28*n^2 + 32*n + 12)*x^2 + (n^3 + 6*n^2 + 11*n + 6)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 4; for(n = 1, N, my(i = n + 1); v[i] = (-(32*n + 32 - (16*n^3 + 32*n^2 - 40*n + 152)*x - (8*n^3 - 12*n^2 - 88*n - 24)*x^2 + (7*n^3 + 19*n^2 - 10*n - 24)*x^3 - (n^3 + 5*n^2 + 6*n)*x^4)*v[i - 1] - (64*x - (32*n^2 + 96*n + 112)*x^2 + (48*n^2 + 160*n + 104)*x^3 - (18*n^2 + 74*n + 68)*x^4 + (2*n^2 + 10*n + 12)*x^5)*deriv(v[i - 1], x))/(-32*n + (16*n^3 + 16*n^2 + 8*n + 48)*x - (8*n^3 + 28*n^2 + 32*n + 12)*x^2 + (n^3 + 6*n^2 + 11*n + 6)*x^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]
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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
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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
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
veronese_section_2_1 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…?
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Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
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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.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Iterated real-rootedness preservers (above)
- ↑ Parent-row provers of polytool (above)
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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.?
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from
veronese_section_2_1 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…?
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
- [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.