A360282 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(120\,n - 248\,n^{2} + 190\,n^{3} - 64\,n^{4} + 8\,n^{5}\right)\,P_{n}(x) = \left(1620\,x^{2} + 18\,x^{3} - 4968\,n x^{2} + 189\,n x^{3} + 5868\,n^{2} x^{2} - 383\,n^{2} x^{3} - 3336\,n^{3} x^{2} + 262\,n^{3} x^{3} + 912\,n^{4} x^{2} - 76\,n^{4} x^{3} - 96\,n^{5} x^{2} + 8\,n^{5} x^{3}\right)\,P_{n-3}(x) + \left(540\,x^{3} + 54\,x^{4} - 1656\,n x^{3} - 81\,n x^{4} + 1956\,n^{2} x^{3} + 39\,n^{2} x^{4} - 1112\,n^{3} x^{3} - 6\,n^{3} x^{4} + 304\,n^{4} x^{3} - 32\,n^{5} x^{3}\right)\,P_{n-3}'(x) + \left(1800 - 300\,x + 240\,x^{2} - 5640\,n + 1190\,n x - 936\,n x^{2} + 6848\,n^{2} - 1608\,n^{2} x + 1236\,n^{2} x^{2} - 4032\,n^{3} + 992\,n^{3} x - 750\,n^{3} x^{2} + 1152\,n^{4} - 288\,n^{4} x + 216\,n^{4} x^{2} - 128\,n^{5} + 32\,n^{5} x - 24\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(-1800\,x - 1140\,x^{2} - 96\,x^{3} + 5640\,n x + 3226\,n x^{2} + 144\,n x^{3} - 6848\,n^{2} x - 3648\,n^{2} x^{2} - 72\,n^{2} x^{3} + 4032\,n^{3} x + 2056\,n^{3} x^{2} + 12\,n^{3} x^{3} - 1152\,n^{4} x - 576\,n^{4} x^{2} + 128\,n^{5} x + 64\,n^{5} x^{2}\right)\,P_{n-2}'(x) + \left(-540 - 270\,x + 2016\,n + 873\,n x - 2760\,n^{2} - 1101\,n^{2} x + 1776\,n^{3} + 678\,n^{3} x - 544\,n^{4} - 204\,n^{4} x + 64\,n^{5} + 24\,n^{5} x\right)\,P_{n-1}(x) + \left(300\,x + 30\,x^{2} - 1040\,n x - 57\,n x^{2} + 1388\,n^{2} x + 33\,n^{2} x^{2} - 888\,n^{3} x - 6\,n^{3} x^{2} + 272\,n^{4} x - 32\,n^{5} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\,x\)
\(P_{2}(x) = 6\,x + 6\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 0 | 2 | 2 | ||||||
| 2 | 0 | 6 | 6 | 12 | |||||
| 3 | 0 | 20 | 36 | 12 | 68 | ||||
| 4 | 0 | 70 | 180 | 120 | 20 | 390 | |||
| 5 | 0 | 252 | 840 | 840 | 300 | 30 | 2262 | ||
| 6 | 0 | 924 | 3780 | 5040 | 2800 | 630 | 42 | 13216 | |
| 7 | 0 | 3432 | 16632 | 27720 | 21000 | 7560 | 1176 | 56 | 77576 |
Compute more
?
Technical details
- Node id
n-0f7e7e7ef029e461· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-04db0035d045da4f
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: 0, 2; 2: 0, 6, 6; 3: 0, 20, 36, 12; 4: 0, 70, 180, 120, 20; 5: 0, 252, 840, 840, 300, 30; 6: 0, 924, 3780, 5040, 2800, 630, 42; 7: 0, 3432, 16632, 27720, 21000, 7560, 1176, 56; 8: 0, 12870, 72072, 144144, 138600, 69300, 17640, 2016, 72; 9: 0, 48620, 308880, 720720, 840840, 540540, 194040, 36960, 3240, 90;
%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-(8*n^5-76*n^4+262*n^3-383*n^2+189*n+18)*x^3)*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-(32*n^5-288*n^4+992*n^3-1608*n^2+1190*n-300)*x+(24*n^5-216*n^4+750*n^3-1236*n^2+936*n-240)*x^2)*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+(24*n^5-204*n^4+678*n^3-1101*n^2+873*n-270)*x)*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) = 2*x, P(2,x) = 6*x + 6*x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2*x;
P[2, x_] := 6*x + 6*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 - (8*n^5 - 76*n^4 + 262*n^3 - 383*n^2 + 189*n + 18)*x^3)*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 - (32*n^5 - 288*n^4 + 992*n^3 - 1608*n^2 + 1190*n - 300)*x + (24*n^5 - 216*n^4 + 750*n^3 - 1236*n^2 + 936*n - 240)*x^2)*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 + (24*n^5 - 204*n^4 + 678*n^3 - 1101*n^2 + 873*n - 270)*x)*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] = 2*x; v[3] = 6*x + 6*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 - (8*n^5 - 76*n^4 + 262*n^3 - 383*n^2 + 189*n + 18)*x^3)*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 - (32*n^5 - 288*n^4 + 992*n^3 - 1608*n^2 + 1190*n - 300)*x + (24*n^5 - 216*n^4 + 750*n^3 - 1236*n^2 + 936*n - 240)*x^2)*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 + (24*n^5 - 204*n^4 + 678*n^3 - 1101*n^2 + 873*n - 270)*x)*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 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 ›
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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.