A391533 → Veronese 2 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-1200\,x + 480\,x^{2} - 444\,x^{3} + 108\,x^{4} + 576\,n - 80\,n x + 1528\,n x^{2} - 826\,n x^{3} + 216\,n x^{4} - 672\,n^{2} x + 1000\,n^{2} x^{2} - 652\,n^{2} x^{3} + 171\,n^{2} x^{4} - 352\,n^{3} x + 336\,n^{3} x^{2} - 298\,n^{3} x^{3} + 67\,n^{3} x^{4} + 96\,n^{4} x^{2} - 76\,n^{4} x^{3} + 13\,n^{4} x^{4} + 16\,n^{5} x^{2} - 8\,n^{5} x^{3} + n^{5} x^{4}\right)\,P_{n}(x) = \left(-2352\,x - 896\,x^{2} + 288\,x^{3} - 448\,x^{4} + 144\,x^{5} + 576\,n + 2224\,n x + 1224\,n x^{2} + 1478\,n x^{3} - 1090\,n x^{4} + 312\,n x^{5} - 672\,n^{2} x - 968\,n^{2} x^{2} + 1280\,n^{2} x^{3} - 943\,n^{2} x^{4} + 241\,n^{2} x^{5} - 352\,n^{3} x - 272\,n^{3} x^{2} + 422\,n^{3} x^{3} - 403\,n^{3} x^{4} + 87\,n^{3} x^{5} + 96\,n^{4} x^{2} + 76\,n^{4} x^{3} - 85\,n^{4} x^{4} + 15\,n^{4} x^{5} + 16\,n^{5} x^{2} + 8\,n^{5} x^{3} - 7\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(1152\,x - 2656\,x^{2} + 4064\,x^{3} - 3968\,x^{4} + 1696\,x^{5} - 288\,x^{6} - 2752\,n x^{2} + 5728\,n x^{3} - 4748\,n x^{4} + 2108\,n x^{5} - 336\,n x^{6} - 704\,n^{2} x^{2} + 2080\,n^{2} x^{3} - 2244\,n^{2} x^{4} + 1014\,n^{2} x^{5} - 146\,n^{2} x^{6} + 320\,n^{3} x^{3} - 512\,n^{3} x^{4} + 220\,n^{3} x^{5} - 28\,n^{3} x^{6} + 32\,n^{4} x^{3} - 48\,n^{4} x^{4} + 18\,n^{4} x^{5} - 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 4 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | Pn(1) |
|---|---|---|---|---|---|---|
| 0 | 4 | 4 | 8 | |||
| 1 | 4 | 20 | 24 | |||
| 2 | 4 | 52 | 9 | 65 | ||
| 3 | 4 | 100 | 63 | 167 | ||
| 4 | 4 | 164 | 234 | 16 | 418 | |
| 5 | 4 | 244 | 630 | 144 | 1022 | |
| 6 | 4 | 340 | 1395 | 688 | 25 | 2452 |
| 7 | 4 | 452 | 2709 | 2352 | 275 | 5792 |
Compute more
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Technical details
- Node id
n-85358b629bd9f687· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-3f15f3718acb022d
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 ExampleTriangle begins: 0: 4, 4; 1: 4, 20; 2: 4, 52, 9; 3: 4, 100, 63; 4: 4, 164, 234, 16; 5: 4, 244, 630, 144; 6: 4, 340, 1395, 688, 25; 7: 4, 452, 2709, 2352, 275; 8: 4, 580, 4788, 6496, 1600, 36; 9: 4, 724, 7884, 15456, 6600, 468;
%F FormulaThe row polynomials satisfy (576*n-(352*n^3+672*n^2+80*n+1200)*x+(16*n^5+96*n^4+336*n^3+1000*n^2+1528*n+480)*x^2-(8*n^5+76*n^4+298*n^3+652*n^2+826*n+444)*x^3+(n^5+13*n^4+67*n^3+171*n^2+216*n+108)*x^4)*P(n,x) = (576*n-(352*n^3+672*n^2-2224*n+2352)*x+(16*n^5+96*n^4-272*n^3-968*n^2+1224*n-896)*x^2+(8*n^5+76*n^4+422*n^3+1280*n^2+1478*n+288)*x^3-(7*n^5+85*n^4+403*n^3+943*n^2+1090*n+448)*x^4+(n^5+15*n^4+87*n^3+241*n^2+312*n+144)*x^5)*P(n-1,x) + (1152*x-(704*n^2+2752*n+2656)*x^2+(32*n^4+320*n^3+2080*n^2+5728*n+4064)*x^3-(48*n^4+512*n^3+2244*n^2+4748*n+3968)*x^4+(18*n^4+220*n^3+1014*n^2+2108*n+1696)*x^5-(2*n^4+28*n^3+146*n^2+336*n+288)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 4*x.
%t MathematicaP[0, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((576*n - (352*n^3 + 672*n^2 - 2224*n + 2352)*x + (16*n^5 + 96*n^4 - 272*n^3 - 968*n^2 + 1224*n - 896)*x^2 + (8*n^5 + 76*n^4 + 422*n^3 + 1280*n^2 + 1478*n + 288)*x^3 - (7*n^5 + 85*n^4 + 403*n^3 + 943*n^2 + 1090*n + 448)*x^4 + (n^5 + 15*n^4 + 87*n^3 + 241*n^2 + 312*n + 144)*x^5)*P[n - 1, x] + (1152*x - (704*n^2 + 2752*n + 2656)*x^2 + (32*n^4 + 320*n^3 + 2080*n^2 + 5728*n + 4064)*x^3 - (48*n^4 + 512*n^3 + 2244*n^2 + 4748*n + 3968)*x^4 + (18*n^4 + 220*n^3 + 1014*n^2 + 2108*n + 1696)*x^5 - (2*n^4 + 28*n^3 + 146*n^2 + 336*n + 288)*x^6)*D[P[n - 1, x], x])/(576*n - (352*n^3 + 672*n^2 + 80*n + 1200)*x + (16*n^5 + 96*n^4 + 336*n^3 + 1000*n^2 + 1528*n + 480)*x^2 - (8*n^5 + 76*n^4 + 298*n^3 + 652*n^2 + 826*n + 444)*x^3 + (n^5 + 13*n^4 + 67*n^3 + 171*n^2 + 216*n + 108)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 4 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = ((576*n - (352*n^3 + 672*n^2 - 2224*n + 2352)*x + (16*n^5 + 96*n^4 - 272*n^3 - 968*n^2 + 1224*n - 896)*x^2 + (8*n^5 + 76*n^4 + 422*n^3 + 1280*n^2 + 1478*n + 288)*x^3 - (7*n^5 + 85*n^4 + 403*n^3 + 943*n^2 + 1090*n + 448)*x^4 + (n^5 + 15*n^4 + 87*n^3 + 241*n^2 + 312*n + 144)*x^5)*v[i - 1] + (1152*x - (704*n^2 + 2752*n + 2656)*x^2 + (32*n^4 + 320*n^3 + 2080*n^2 + 5728*n + 4064)*x^3 - (48*n^4 + 512*n^3 + 2244*n^2 + 4748*n + 3968)*x^4 + (18*n^4 + 220*n^3 + 1014*n^2 + 2108*n + 1696)*x^5 - (2*n^4 + 28*n^3 + 146*n^2 + 336*n + 288)*x^6)*deriv(v[i - 1], x))/(576*n - (352*n^3 + 672*n^2 + 80*n + 1200)*x + (16*n^5 + 96*n^4 + 336*n^3 + 1000*n^2 + 1528*n + 480)*x^2 - (8*n^5 + 76*n^4 + 298*n^3 + 652*n^2 + 826*n + 444)*x^3 + (n^5 + 13*n^4 + 67*n^3 + 171*n^2 + 216*n + 108)*x^4)); 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_0 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_0 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.