A156366 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(2 + n\right)\,P_{n}(x) = \left(-36\,x + 450\,x^{2} - 48\,n x + 837\,n x^{2} - 12\,n^{2} x + 468\,n^{2} x^{2} + 81\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x + 111\,x^{2} - 450\,x^{3} - 2\,n x + 72\,n x^{2} - 189\,n x^{3} + 12\,n^{2} x^{2} + 90\,n^{2} x^{3} + 36\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-3\,x^{2} + 75\,x^{3} - 198\,x^{4} - n x^{2} + 60\,n x^{3} - 171\,n x^{4} + 12\,n^{2} x^{3} - 36\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(2 + 48\,x + n + 48\,n x + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(5\,x - 72\,x^{2} + 2\,n x - 60\,n x^{2} - 12\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 48\,x + 81\,x^{2}\)
\(P_{1}(x) = 1 + 132\,x + 891\,x^{2} + 432\,x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | x15 | x16 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 48 | 81 | 130 | ||||||||||||||
| 1 | 1 | 132 | 891 | 432 | 1456 | |||||||||||||
| 2 | 1 | 312 | 5346 | 11232 | 2025 | 18916 | ||||||||||||
| 3 | 1 | 684 | 24462 | 130464 | 115425 | 8748 | 279784 | |||||||||||
| 4 | 1 | 1440 | 96471 | 1043712 | 2411775 | 1049760 | 35721 | 4638880 | ||||||||||
| 5 | 1 | 2964 | 347733 | 6747408 | 31628475 | 37555164 | 8823087 | 139968 | 85244800 | |||||||||
| 6 | 1 | 6024 | 1183248 | 38117088 | 316284750 | 771871032 | 521812368 | 70263936 | 531441 | 1720069888 | ||||||||
| 7 | 1 | 12156 | 3875040 | 196642944 | 2653466850 | 11462976792 | 16259913432 | 6696069120 | 538349733 | 1968300 | 37813274368 | |||||||
| 8 | 1 | 24432 | 12363597 | 951906816 | 19719680850 | 137555721504 | 347855170194 | 308417808384 | 81117559917 | 4007458800 | 7144929 | 899644839424 | ||||||
| 9 | 1 | 48996 | 38739951 | 4401079920 | 134294909850 | 1421657477928 | 5805101368206 | 9282464168832 | 5414153504085 | 941380809300 | 29172745107 | 25509168 | 23032690361344 | |||||
| 10 | 1 | 98136 | 119857806 | 19670450400 | 857145108375 | 13171176951984 | 81271419154884 | 210738831231744 | 224949162241935 | 89623489635000 | 10572537209454 | 208613975904 | 89813529 | 631412255729152 | ||||
| 11 | 1 | 196428 | 367522434 | 85713459552 | 5207777904375 | 112352863525524 | 999158381182332 | 3915069575245056 | 6825436459175583 | 5061960123052500 | 1417630978804194 | 115743105094752 | 1470157656201 | 312487308 | 18454115815306240 | |||
| 12 | 1 | 393024 | 1119803859 | 366375273984 | 30458489780025 | 899093186622144 | 11123076402839331 | 62641113203920896 | 165484136687140251 | 202295966989982400 | 107468517000249009 | 21634093553481216 | 1241648597093931 | 10234584311616 | 1076168025 | 572828707267059712 | ||
| 13 | 1 | 786228 | 3396552345 | 1543140830160 | 172901058061725 | 6841946554079652 | 114574506606946197 | 893387600000884800 | 3392081043753359475 | 6313289139566423100 | 5588159848044555771 | 2178055376530517232 | 320821582640414895 | 13103450602100460 | 70509452829975 | 3673320192 | 18820559455021662208 | |
| 14 | 1 | 1572648 | 10269262620 | 6426172562400 | 958540434646500 | 50014870174397736 | 1109338978969367748 | 11636550854809868160 | 61057458787560470550 | 163638996395763771000 | 221890435364872924452 | 145843361428543798176 | 42513530431998040740 | 4648373529555180600 | 136437679900885500 | 481403304442368 | 12440502369 | 652525944714771693568 |
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Technical details
- Node id
n-e190c0da2f2fe2af· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-a45daab6f850677d
Observed on the stored rows ?
- Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
- 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, 48, 81; 1: 1, 132, 891, 432; 2: 1, 312, 5346, 11232, 2025; 3: 1, 684, 24462, 130464, 115425, 8748; 4: 1, 1440, 96471, 1043712, 2411775, 1049760, 35721; 5: 1, 2964, 347733, 6747408, 31628475, 37555164, 8823087, 139968; 6: 1, 6024, 1183248, 38117088, 316284750, 771871032, 521812368, 70263936, 531441; 7: 1, 12156, 3875040, 196642944, 2653466850, 11462976792, 16259913432, 6696069120, 538349733, 1968300; 8: 1, 24432, 12363597, 951906816, 19719680850, 137555721504, 347855170194, 308417808384, 81117559917, 4007458800, 7144929; 9: 1, 48996, 38739951, 4401079920, 134294909850, 1421657477928, 5805101368206, 9282464168832, 5414153504085, 941380809300, 29172745107, 25509168;
%F FormulaThe row polynomials satisfy (n+2)*P(n,x) = -((12*n^2+48*n+36)*x-(81*n^3+468*n^2+837*n+450)*x^2)*P(n-2,x) - ((2*n+6)*x-(12*n^2+72*n+111)*x^2-(36*n^3+90*n^2-189*n-450)*x^3)*d/dx P(n-2,x) - ((n+3)*x^2-(12*n^2+60*n+75)*x^3+(36*n^2+171*n+198)*x^4)*d^2/dx^2 P(n-2,x) + (n+2+(12*n^2+48*n+48)*x)*P(n-1,x) + ((2*n+5)*x-(12*n^2+60*n+72)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + 48*x + 81*x^2, P(1,x) = 1 + 132*x + 891*x^2 + 432*x^3.
%t MathematicaP[0, x_] := 1 + 48*x + 81*x^2;
P[1, x_] := 1 + 132*x + 891*x^2 + 432*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-((12*n^2 + 48*n + 36)*x - (81*n^3 + 468*n^2 + 837*n + 450)*x^2)*P[n - 2, x] - ((2*n + 6)*x - (12*n^2 + 72*n + 111)*x^2 - (36*n^3 + 90*n^2 - 189*n - 450)*x^3)*D[P[n - 2, x], x] - ((n + 3)*x^2 - (12*n^2 + 60*n + 75)*x^3 + (36*n^2 + 171*n + 198)*x^4)*D[P[n - 2, x], {x, 2}] + (n + 2 + (12*n^2 + 48*n + 48)*x)*P[n - 1, x] + ((2*n + 5)*x - (12*n^2 + 60*n + 72)*x^2)*D[P[n - 1, x], x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1 + 48*x + 81*x^2; v[2] = 1 + 132*x + 891*x^2 + 432*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-((12*n^2 + 48*n + 36)*x - (81*n^3 + 468*n^2 + 837*n + 450)*x^2)*v[i - 2] - ((2*n + 6)*x - (12*n^2 + 72*n + 111)*x^2 - (36*n^3 + 90*n^2 - 189*n - 450)*x^3)*deriv(v[i - 2], x) - ((n + 3)*x^2 - (12*n^2 + 60*n + 75)*x^3 + (36*n^2 + 171*n + 198)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + 2 + (12*n^2 + 48*n + 48)*x)*v[i - 1] + ((2*n + 5)*x - (12*n^2 + 60*n + 72)*x^2)*deriv(v[i - 1], x))/(n + 2)); 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]
-
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
-
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
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.