A113413 → basis A013610 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(96 + 288\,x + 270\,x^{2} + 360\,n x + 540\,n x^{2} - 162\,n x^{3} - 243\,n x^{4} + 432\,n^{2} x^{2} + 324\,n^{2} x^{3} - 243\,n^{2} x^{4} + 162\,n^{3} x^{3}\right)\,P_{n}(x) = \left(480 + 2592\,x + 4158\,x^{2} + 972\,x^{3} - 2430\,x^{4} - 1458\,x^{5} - 192\,n + 360\,n x + 4320\,n x^{2} + 5994\,n x^{3} + 729\,n x^{4} - 1458\,n x^{5} - 432\,n^{2} x - 648\,n^{2} x^{2} + 1944\,n^{2} x^{3} + 2187\,n^{2} x^{4} - 216\,n^{3} x^{2} - 486\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(64 + 480\,x + 1080\,x^{2} + 108\,x^{3} - 2430\,x^{4} - 2673\,x^{5} - 729\,x^{6} + 192\,n x + 1296\,n x^{2} + 2592\,n x^{3} + 810\,n x^{4} - 1701\,n x^{5} - 729\,n x^{6} + 144\,n^{2} x^{2} + 864\,n^{2} x^{3} + 1458\,n^{2} x^{4} + 486\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 3 | 6 | 9 | ||||||
| 2 | 7 | 36 | 27 | 70 | |||||
| 3 | 17 | 138 | 243 | 108 | 506 | ||||
| 4 | 41 | 456 | 1296 | 1296 | 405 | 3494 | |||
| 5 | 99 | 1398 | 5508 | 8856 | 6075 | 1458 | 23394 | ||
| 6 | 239 | 4092 | 20655 | 46224 | 50625 | 26244 | 5103 | 153182 | |
| 7 | 577 | 11610 | 71523 | 205956 | 313875 | 258066 | 107163 | 17496 | 986266 |
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Technical details
- Node id
n-fe0ee07ec2d084d1· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-dbe867ce9571587b
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: 3, 6; 2: 7, 36, 27; 3: 17, 138, 243, 108; 4: 41, 456, 1296, 1296, 405; 5: 99, 1398, 5508, 8856, 6075, 1458; 6: 239, 4092, 20655, 46224, 50625, 26244, 5103; 7: 577, 11610, 71523, 205956, 313875, 258066, 107163, 17496; 8: 1393, 32208, 234360, 826848, 1624050, 1854576, 1214514, 419904, 59049; 9: 3363, 87846, 737424, 3083184, 7434990, 10943748, 9930438, 5388768, 1594323, 196830;
%F FormulaThe row polynomials satisfy (96+(360*n+288)*x+(432*n^2+540*n+270)*x^2+(162*n^3+324*n^2-162*n)*x^3-(243*n^2+243*n)*x^4)*P(n,x) = -(192*n-480+(432*n^2-360*n-2592)*x+(216*n^3+648*n^2-4320*n-4158)*x^2+(486*n^3-1944*n^2-5994*n-972)*x^3-(2187*n^2+729*n-2430)*x^4+(1458*n+1458)*x^5)*P(n-1,x) + (64+(192*n+480)*x+(144*n^2+1296*n+1080)*x^2+(864*n^2+2592*n+108)*x^3+(1458*n^2+810*n-2430)*x^4+(486*n^2-1701*n-2673)*x^5-(729*n+729)*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 - 480 + (432*n^2 - 360*n - 2592)*x + (216*n^3 + 648*n^2 - 4320*n - 4158)*x^2 + (486*n^3 - 1944*n^2 - 5994*n - 972)*x^3 - (2187*n^2 + 729*n - 2430)*x^4 + (1458*n + 1458)*x^5)*P[n - 1, x] + (64 + (192*n + 480)*x + (144*n^2 + 1296*n + 1080)*x^2 + (864*n^2 + 2592*n + 108)*x^3 + (1458*n^2 + 810*n - 2430)*x^4 + (486*n^2 - 1701*n - 2673)*x^5 - (729*n + 729)*x^6)*D[P[n - 1, x], x])/(96 + (360*n + 288)*x + (432*n^2 + 540*n + 270)*x^2 + (162*n^3 + 324*n^2 - 162*n)*x^3 - (243*n^2 + 243*n)*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 - 480 + (432*n^2 - 360*n - 2592)*x + (216*n^3 + 648*n^2 - 4320*n - 4158)*x^2 + (486*n^3 - 1944*n^2 - 5994*n - 972)*x^3 - (2187*n^2 + 729*n - 2430)*x^4 + (1458*n + 1458)*x^5)*v[i - 1] + (64 + (192*n + 480)*x + (144*n^2 + 1296*n + 1080)*x^2 + (864*n^2 + 2592*n + 108)*x^3 + (1458*n^2 + 810*n - 2430)*x^4 + (486*n^2 - 1701*n - 2673)*x^5 - (729*n + 729)*x^6)*deriv(v[i - 1], x))/(96 + (360*n + 288)*x + (432*n^2 + 540*n + 270)*x^2 + (162*n^3 + 324*n^2 - 162*n)*x^3 - (243*n^2 + 243*n)*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 A Riordan array of coordination sequences.?
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Geometric basis map (elementary) polytool ? step ›
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from
A Riordan array of coordination sequences.?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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from
A Riordan array of coordination sequences.?
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from
basis_map_A013610 of A Riordan array of coordination sequences.?
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.