A266213 → basis A013610 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-96 - 648\,x - 1242\,x^{2} - 324\,x^{3} + 486\,x^{4} + 72\,n x - 108\,n x^{2} - 810\,n x^{3} - 729\,n x^{4} + 216\,n^{2} x^{2} + 162\,n^{2} x^{3} + 243\,n^{2} x^{4} + 162\,n^{3} x^{3}\right)\,P_{n}(x) = \left(96 + 648\,x + 1242\,x^{2} + 324\,x^{3} - 486\,x^{4} - 192\,n - 792\,n x - 648\,n x^{2} + 486\,n x^{3} + 729\,n x^{4} - 432\,n^{2} x - 1512\,n^{2} x^{2} - 1134\,n^{2} x^{3} - 243\,n^{2} x^{4} - 216\,n^{3} x^{2} - 486\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(64 + 672\,x + 2664\,x^{2} + 4860\,x^{3} + 3888\,x^{4} + 972\,x^{5} + 192\,n x + 1584\,n x^{2} + 4752\,n x^{3} + 6318\,n x^{4} + 3645\,n x^{5} + 729\,n x^{6} + 144\,n^{2} x^{2} + 864\,n^{2} x^{3} + 1458\,n^{2} x^{4} + 486\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||
| 1 | 1 | 1 | ||||||
| 2 | 3 | 12 | 15 | |||||
| 3 | 7 | 48 | 54 | 109 | ||||
| 4 | 17 | 168 | 378 | 216 | 779 | |||
| 5 | 41 | 528 | 1782 | 2160 | 810 | 5321 | ||
| 6 | 99 | 1572 | 7074 | 13176 | 10530 | 2916 | 35367 | |
| 7 | 239 | 4512 | 25380 | 63936 | 79380 | 46656 | 10206 | 230309 |
Compute more
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Technical details
- Node id
n-befb1852857a5725· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-2947906bdaa38e4f
Observed on the stored rows ?
- Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
- 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: 3, 12; 3: 7, 48, 54; 4: 17, 168, 378, 216; 5: 41, 528, 1782, 2160, 810; 6: 99, 1572, 7074, 13176, 10530, 2916; 7: 239, 4512, 25380, 63936, 79380, 46656, 10206; 8: 577, 12624, 85212, 270864, 458460, 419904, 193914, 34992; 9: 1393, 34656, 272916, 1048032, 2251800, 2822688, 2030994, 769824, 118098;
%F FormulaThe row polynomials satisfy -(96-(72*n-648)*x-(216*n^2-108*n-1242)*x^2-(162*n^3+162*n^2-810*n-324)*x^3-(243*n^2-729*n+486)*x^4)*P(n,x) = -(192*n-96+(432*n^2+792*n-648)*x+(216*n^3+1512*n^2+648*n-1242)*x^2+(486*n^3+1134*n^2-486*n-324)*x^3+(243*n^2-729*n+486)*x^4)*P(n-1,x) + (64+(192*n+672)*x+(144*n^2+1584*n+2664)*x^2+(864*n^2+4752*n+4860)*x^3+(1458*n^2+6318*n+3888)*x^4+(486*n^2+3645*n+972)*x^5+729*n*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(192*n - 96 + (432*n^2 + 792*n - 648)*x + (216*n^3 + 1512*n^2 + 648*n - 1242)*x^2 + (486*n^3 + 1134*n^2 - 486*n - 324)*x^3 + (243*n^2 - 729*n + 486)*x^4)*P[n - 1, x] + (64 + (192*n + 672)*x + (144*n^2 + 1584*n + 2664)*x^2 + (864*n^2 + 4752*n + 4860)*x^3 + (1458*n^2 + 6318*n + 3888)*x^4 + (486*n^2 + 3645*n + 972)*x^5 + 729*n*x^6)*D[P[n - 1, x], x])/(-96 + (72*n - 648)*x + (216*n^2 - 108*n - 1242)*x^2 + (162*n^3 + 162*n^2 - 810*n - 324)*x^3 + (243*n^2 - 729*n + 486)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(192*n - 96 + (432*n^2 + 792*n - 648)*x + (216*n^3 + 1512*n^2 + 648*n - 1242)*x^2 + (486*n^3 + 1134*n^2 - 486*n - 324)*x^3 + (243*n^2 - 729*n + 486)*x^4)*v[i - 1] + (64 + (192*n + 672)*x + (144*n^2 + 1584*n + 2664)*x^2 + (864*n^2 + 4752*n + 4860)*x^3 + (1458*n^2 + 6318*n + 3888)*x^4 + (486*n^2 + 3645*n + 972)*x^5 + 729*n*x^6)*deriv(v[i - 1], x))/(-96 + (72*n - 648)*x + (216*n^2 - 108*n - 1242)*x^2 + (162*n^3 + 162*n^2 - 810*n - 324)*x^3 + (243*n^2 - 729*n + 486)*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
basis_map_A013610 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercu…?
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Geometric basis map (elementary) polytool ? step ›
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from
Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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from
Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
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from
basis_map_A013610 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercu…?
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.