A142460 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-4 + 5\,n\right)\,P_{n}(x) = \left(4 - 72\,x - 36\,x^{2} + 20\,n - 320\,n x - 115\,n x^{2} + 200\,n^{2} x + 300\,n^{2} x^{2} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(20\,x - 20\,x^{3} + 100\,n x - 25\,n x^{2} - 75\,n x^{3} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-8 + 32\,x - 15\,n - 80\,n x + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-20\,x + 20\,x^{2} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 1 | 2 | 3 | |||||||||||||
| 2 | 1 | 24 | 3 | 28 | ||||||||||||
| 3 | 1 | 166 | 249 | 4 | 420 | |||||||||||
| 4 | 1 | 1028 | 5478 | 2056 | 5 | 8568 | ||||||||||
| 5 | 1 | 6210 | 84930 | 113240 | 15525 | 6 | 219912 | |||||||||
| 6 | 1 | 37312 | 1129845 | 3623680 | 1883075 | 111936 | 7 | 6785856 | ||||||||
| 7 | 1 | 223934 | 13883463 | 89614540 | 112018175 | 27766926 | 783769 | 8 | 244290816 | |||||||
| 8 | 1 | 1343676 | 163131024 | 1915126024 | 4704763350 | 2872689036 | 380639056 | 5374704 | 9 | 10043066880 | ||||||
| 9 | 1 | 8062138 | 1866999768 | 37384765376 | 161041626130 | 193249951356 | 65423339408 | 4978666048 | 36279621 | 10 | 463989689856 | |||||
| 10 | 1 | 48372920 | 21032818935 | 687772234880 | 4830533807050 | 10048997470512 | 6762747329870 | 1375544469760 | 63098456805 | 241864600 | 11 | 23790016825344 | ||||
| 11 | 1 | 290237622 | 234698739765 | 12154149859860 | 132390960517650 | 440969791855572 | 514464757164834 | 211825536828240 | 27346837184685 | 782329132550 | 1596306921 | 12 | 1340170947827712 | |||
| 12 | 1 | 1741425844 | 2603889315498 | 208861253796680 | 3403110351188475 | 17184504082607352 | 31896814944219708 | 22912672110143136 | 6125598632139255 | 522153134491700 | 9547594156826 | 10448555064 | 13 | 82265878182039552 | ||
| 13 | 1 | 10448555186 | 28789062241374 | 3518844534200744 | 83474839468267075 | 614230135426264122 | 1710550434740319732 | 1954914782560365408 | 921345203139396183 | 166949678936534150 | 9676822469052046 | 115156248965496 | 67915608709 | 14 | 5464804764949770240 | |
| 14 | 1 | 62691331248 | 317635727454633 | 58451095861234496 | 1977297967888906005 | 20577794659731209712 | 82407738288172878861 | 140753864344346309376 | 105952806370507987107 | 34296324432885349520 | 4350055529355593211 | 175353287583703488 | 1376421485636743 | 438839318736 | 15 | 390551380535076913152 |
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Technical details
- Node id
n-3afbc2ff27997910· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-2d3f16c9e7bf1867
Observed on the stored rows ?
- Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
- 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: 1; 1: 1, 2; 2: 1, 24, 3; 3: 1, 166, 249, 4; 4: 1, 1028, 5478, 2056, 5; 5: 1, 6210, 84930, 113240, 15525, 6; 6: 1, 37312, 1129845, 3623680, 1883075, 111936, 7; 7: 1, 223934, 13883463, 89614540, 112018175, 27766926, 783769, 8; 8: 1, 1343676, 163131024, 1915126024, 4704763350, 2872689036, 380639056, 5374704, 9; 9: 1, 8062138, 1866999768, 37384765376, 161041626130, 193249951356, 65423339408, 4978666048, 36279621, 10;
%F FormulaThe row polynomials satisfy (5*n-4)*P(n,x) = (20*n+4+(200*n^2-320*n-72)*x-(125*n^3-300*n^2+115*n+36)*x^2)*P(n-2,x) + ((100*n+20)*x-(125*n^2+25*n)*x^2+(125*n^2-75*n-20)*x^3)*d/dx P(n-2,x) - (15*n+8-(50*n^2-80*n+32)*x)*P(n-1,x) + ((25*n-20)*x-(25*n-20)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((20*n + 4 + (200*n^2 - 320*n - 72)*x - (125*n^3 - 300*n^2 + 115*n + 36)*x^2)*P[n - 2, x] + ((100*n + 20)*x - (125*n^2 + 25*n)*x^2 + (125*n^2 - 75*n - 20)*x^3)*D[P[n - 2, x], x] - (15*n + 8 - (50*n^2 - 80*n + 32)*x)*P[n - 1, x] + ((25*n - 20)*x - (25*n - 20)*x^2)*D[P[n - 1, x], x])/(5*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((20*n + 4 + (200*n^2 - 320*n - 72)*x - (125*n^3 - 300*n^2 + 115*n + 36)*x^2)*v[i - 2] + ((100*n + 20)*x - (125*n^2 + 25*n)*x^2 + (125*n^2 - 75*n - 20)*x^3)*deriv(v[i - 2], x) - (15*n + 8 - (50*n^2 - 80*n + 32)*x)*v[i - 1] + ((25*n - 20)*x - (25*n - 20)*x^2)*deriv(v[i - 1], x))/(5*n - 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
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
References
- [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.