A360282 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(120\,n - 248\,n^{2} + 190\,n^{3} - 64\,n^{4} + 8\,n^{5}\right)\,P_{n}(x) = \left(1620\,x^{2} + 558\,x^{3} + 54\,x^{4} - 4968\,n x^{2} - 1467\,n x^{3} - 81\,n x^{4} + 5868\,n^{2} x^{2} + 1573\,n^{2} x^{3} + 39\,n^{2} x^{4} - 3336\,n^{3} x^{2} - 850\,n^{3} x^{3} - 6\,n^{3} x^{4} + 912\,n^{4} x^{2} + 228\,n^{4} x^{3} - 96\,n^{5} x^{2} - 24\,n^{5} x^{3}\right)\,P_{n-3}(x) + \left(540\,x^{3} + 54\,x^{4} - 1656\,n x^{3} - 81\,n x^{4} + 1956\,n^{2} x^{3} + 39\,n^{2} x^{4} - 1112\,n^{3} x^{3} - 6\,n^{3} x^{4} + 304\,n^{4} x^{3} - 32\,n^{5} x^{3}\right)\,P_{n-3}'(x) + \left(1800 - 2100\,x - 900\,x^{2} - 96\,x^{3} - 5640\,n + 6830\,n x + 2290\,n x^{2} + 144\,n x^{3} + 6848\,n^{2} - 8456\,n^{2} x - 2412\,n^{2} x^{2} - 72\,n^{2} x^{3} - 4032\,n^{3} + 5024\,n^{3} x + 1306\,n^{3} x^{2} + 12\,n^{3} x^{3} + 1152\,n^{4} - 1440\,n^{4} x - 360\,n^{4} x^{2} - 128\,n^{5} + 160\,n^{5} x + 40\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(-1800\,x - 1140\,x^{2} - 96\,x^{3} + 5640\,n x + 3226\,n x^{2} + 144\,n x^{3} - 6848\,n^{2} x - 3648\,n^{2} x^{2} - 72\,n^{2} x^{3} + 4032\,n^{3} x + 2056\,n^{3} x^{2} + 12\,n^{3} x^{3} - 1152\,n^{4} x - 576\,n^{4} x^{2} + 128\,n^{5} x + 64\,n^{5} x^{2}\right)\,P_{n-2}'(x) + \left(-540 + 30\,x + 30\,x^{2} + 2016\,n - 167\,n x - 57\,n x^{2} - 2760\,n^{2} + 287\,n^{2} x + 33\,n^{2} x^{2} + 1776\,n^{3} - 210\,n^{3} x - 6\,n^{3} x^{2} - 544\,n^{4} + 68\,n^{4} x + 64\,n^{5} - 8\,n^{5} x\right)\,P_{n-1}(x) + \left(300\,x + 30\,x^{2} - 1040\,n x - 57\,n x^{2} + 1388\,n^{2} x + 33\,n^{2} x^{2} - 888\,n^{3} x - 6\,n^{3} x^{2} + 272\,n^{4} x - 32\,n^{5} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)
\(P_{2}(x) = 3 + 7\,x + 2\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 1 | 2 | ||||||
| 2 | 3 | 7 | 2 | 12 | |||||
| 3 | 10 | 34 | 21 | 3 | 68 | ||||
| 4 | 35 | 155 | 150 | 46 | 4 | 390 | |||
| 5 | 126 | 686 | 910 | 450 | 85 | 5 | 2262 | ||
| 6 | 462 | 2982 | 5040 | 3500 | 1085 | 141 | 6 | 13216 | |
| 7 | 1716 | 12804 | 26334 | 23730 | 10500 | 2268 | 217 | 7 | 77576 |
Compute more
?
Technical details
- Node id
n-e638274f54a2a81f· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-31d46e29520200cd
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, 1; 2: 3, 7, 2; 3: 10, 34, 21, 3; 4: 35, 155, 150, 46, 4; 5: 126, 686, 910, 450, 85, 5; 6: 462, 2982, 5040, 3500, 1085, 141, 6; 7: 1716, 12804, 26334, 23730, 10500, 2268, 217, 7; 8: 6435, 54483, 132132, 146916, 85470, 26670, 4284, 316, 8; 9: 24310, 230230, 643500, 852852, 618618, 256410, 60060, 7500, 441, 9;
%F FormulaThe row polynomials satisfy (8*n^5-64*n^4+190*n^3-248*n^2+120*n)*P(n,x) = -((96*n^5-912*n^4+3336*n^3-5868*n^2+4968*n-1620)*x^2+(24*n^5-228*n^4+850*n^3-1573*n^2+1467*n-558)*x^3+(6*n^3-39*n^2+81*n-54)*x^4)*P(n-3,x) - ((32*n^5-304*n^4+1112*n^3-1956*n^2+1656*n-540)*x^3+(6*n^3-39*n^2+81*n-54)*x^4)*d/dx P(n-3,x) - (128*n^5-1152*n^4+4032*n^3-6848*n^2+5640*n-1800-(160*n^5-1440*n^4+5024*n^3-8456*n^2+6830*n-2100)*x-(40*n^5-360*n^4+1306*n^3-2412*n^2+2290*n-900)*x^2-(12*n^3-72*n^2+144*n-96)*x^3)*P(n-2,x) + ((128*n^5-1152*n^4+4032*n^3-6848*n^2+5640*n-1800)*x+(64*n^5-576*n^4+2056*n^3-3648*n^2+3226*n-1140)*x^2+(12*n^3-72*n^2+144*n-96)*x^3)*d/dx P(n-2,x) + (64*n^5-544*n^4+1776*n^3-2760*n^2+2016*n-540-(8*n^5-68*n^4+210*n^3-287*n^2+167*n-30)*x-(6*n^3-33*n^2+57*n-30)*x^2)*P(n-1,x) - ((32*n^5-272*n^4+888*n^3-1388*n^2+1040*n-300)*x+(6*n^3-33*n^2+57*n-30)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 3 + 7*x + 2*x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 3 + 7*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-((96*n^5 - 912*n^4 + 3336*n^3 - 5868*n^2 + 4968*n - 1620)*x^2 + (24*n^5 - 228*n^4 + 850*n^3 - 1573*n^2 + 1467*n - 558)*x^3 + (6*n^3 - 39*n^2 + 81*n - 54)*x^4)*P[n - 3, x] - ((32*n^5 - 304*n^4 + 1112*n^3 - 1956*n^2 + 1656*n - 540)*x^3 + (6*n^3 - 39*n^2 + 81*n - 54)*x^4)*D[P[n - 3, x], x] - (128*n^5 - 1152*n^4 + 4032*n^3 - 6848*n^2 + 5640*n - 1800 - (160*n^5 - 1440*n^4 + 5024*n^3 - 8456*n^2 + 6830*n - 2100)*x - (40*n^5 - 360*n^4 + 1306*n^3 - 2412*n^2 + 2290*n - 900)*x^2 - (12*n^3 - 72*n^2 + 144*n - 96)*x^3)*P[n - 2, x] + ((128*n^5 - 1152*n^4 + 4032*n^3 - 6848*n^2 + 5640*n - 1800)*x + (64*n^5 - 576*n^4 + 2056*n^3 - 3648*n^2 + 3226*n - 1140)*x^2 + (12*n^3 - 72*n^2 + 144*n - 96)*x^3)*D[P[n - 2, x], x] + (64*n^5 - 544*n^4 + 1776*n^3 - 2760*n^2 + 2016*n - 540 - (8*n^5 - 68*n^4 + 210*n^3 - 287*n^2 + 167*n - 30)*x - (6*n^3 - 33*n^2 + 57*n - 30)*x^2)*P[n - 1, x] - ((32*n^5 - 272*n^4 + 888*n^3 - 1388*n^2 + 1040*n - 300)*x + (6*n^3 - 33*n^2 + 57*n - 30)*x^2)*D[P[n - 1, x], x])/(8*n^5 - 64*n^4 + 190*n^3 - 248*n^2 + 120*n)]];
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 + x; v[3] = 3 + 7*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-((96*n^5 - 912*n^4 + 3336*n^3 - 5868*n^2 + 4968*n - 1620)*x^2 + (24*n^5 - 228*n^4 + 850*n^3 - 1573*n^2 + 1467*n - 558)*x^3 + (6*n^3 - 39*n^2 + 81*n - 54)*x^4)*v[i - 3] - ((32*n^5 - 304*n^4 + 1112*n^3 - 1956*n^2 + 1656*n - 540)*x^3 + (6*n^3 - 39*n^2 + 81*n - 54)*x^4)*deriv(v[i - 3], x) - (128*n^5 - 1152*n^4 + 4032*n^3 - 6848*n^2 + 5640*n - 1800 - (160*n^5 - 1440*n^4 + 5024*n^3 - 8456*n^2 + 6830*n - 2100)*x - (40*n^5 - 360*n^4 + 1306*n^3 - 2412*n^2 + 2290*n - 900)*x^2 - (12*n^3 - 72*n^2 + 144*n - 96)*x^3)*v[i - 2] + ((128*n^5 - 1152*n^4 + 4032*n^3 - 6848*n^2 + 5640*n - 1800)*x + (64*n^5 - 576*n^4 + 2056*n^3 - 3648*n^2 + 3226*n - 1140)*x^2 + (12*n^3 - 72*n^2 + 144*n - 96)*x^3)*deriv(v[i - 2], x) + (64*n^5 - 544*n^4 + 1776*n^3 - 2760*n^2 + 2016*n - 540 - (8*n^5 - 68*n^4 + 210*n^3 - 287*n^2 + 167*n - 30)*x - (6*n^3 - 33*n^2 + 57*n - 30)*x^2)*v[i - 1] - ((32*n^5 - 272*n^4 + 888*n^3 - 1388*n^2 + 1040*n - 300)*x + (6*n^3 - 33*n^2 + 57*n - 30)*x^2)*deriv(v[i - 1], x))/(8*n^5 - 64*n^4 + 190*n^3 - 248*n^2 + 120*n)); 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 differential operators
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
- 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 differential operators [SP14] polytool ? step ›
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from
Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, …?
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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
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from
Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, …?
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.